category: literaturenote citekey: munidiscretelorenzattractors3d2025 title: Discrete Lorenz Attractors in 3D Sinusoidal Maps authors: "Muni, Sishu Shankar" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.10788v1" zotero_key: JFSEXMNG zotero_storage: H4IUSQNT collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Muni, Sishu Shankar"

status: converted

Discrete Lorenz Attractors in 3D Sinusoidal Maps

Sishu Shankar Muni^a

^aSchool of Digital Sciences, Digital University Kerala, Technopark phase-IV Campus, Mangalapuram, 695317, Thiruvananthapuram, India

June 13, 2025

Abstract

Discrete Lorenz attractors can be found in three-dimensional discrete maps. Discrete Lorenz attractors have similar topology to that of the continuous Lorenz attractor exhibited by the well studied 3D Lorenz system. However, the routes to the formation of discrete Lorenz attractor in 3D maps is different from that of the routes of formation of continuous Lorenz flow attractor. This paper explores the exotic dynamics of a three-dimensional sinusoidal map, highlighting the existence of discrete Lorenz attractors. Through a detailed bifurcation analysis, we discuss various formation pathways involving supercritical and subcritical Neimark-Sacker bifurcations, and homoclinic butterflies towards the formation of discrete Lorenz attractors. Using a twoparameter Lyapunov chart analysis, we systematically illustrate chaotic regions and periodic regions in the parameter space, illustrating diverse attractor topologies, including hyperchaotic and wavy chaotic structures. Furthermore, we investigate spatiotemporal patterns in a ring-star network of coupled sinusoidal maps, revealing diverse spatiotemporal patterns such as chimera states, traveling waves, and synchronization transitions, controlled by coupling strengths of ring and star networks respectively. A key contribution is the application of hyperchaotic signals to secure video encryption, where we apply YOLOv10 for object detection and CNN-based cryptographic key generation. This selective encryption approach ensures enhanced security, high computational efficiency, and robustness against differential attacks.

Introduction

Discrete maps are remarkable and the simplest examples of systems exhibiting chaotic dynamics. One-dimensional maps can even show chaos [1]. Since the beginning of the field of chaos, chaotic attractors, strange attractors caught a great attention [2]. Researchers started investigating one-dimensional and two-dimensional maps and have till now mostly understood the pathways of formation of chaotic attractors. However, scarce attention has been paid to the dynamics of multidimensional maps such as three-dimensional maps or more. Around 1990's, there were few studies attempting to study the dynamics of three-dimensional maps [3]. Gonchenko et. al. [4] showed various possible routes to chaotic dynamics in three-dimensional maps. Three-dimensional maps are important as they serve as an effective tool to study the four-dimensional flows. The study of multidimensional maps is interesting because they show additional new bifurcations which are absent in lower dimensional maps [5]. They are the doubling bifurcation of ergodic and resonant torus. Three-dimensional maps exhibit the doubling bifurcation of quasiperiodic orbit. Since 1980's, doubling bifurcation of quasiperiodic orbits was reported by Kaneko [6]. Since then, it was reported in various applied dynamical systems like Goodwins oscillator [7], radiophysical generators [8], neuron mappings [5], mechanical systems [9]. More recently, it was shown that doubling bifurcation of resonant torus or mode-locked orbit can also occur [10]. Various examples of 3D maps were shown which can exhibit both types of resonant torus doubling bifurcation such as a) length-doubled resonant torus, and b) disjoint length-doubled resonant torus. Moreover examples of 3D maps were given which shows both doubling bifurcation of quasiperiodic orbit and mode-locked orbit [10].

Another beautiful application of three-dimensional maps is that they showcase discrete Lorenz attractors, very similar in structure to that of the Lorenz attractor in 3D Lorenz system. Lorenz attractor was found by E. Lorenz in 1963 [11]. Since then, it has led to many new research directions. It has led to the theory of singular hyperbolic attractors [12], and also to the theory of pseudohyperbolic attractors [13]. Lorenz attractors are formed after a pitchfork bifurcation of an equilibrium point. Lorenz attractor is singularly hyperbolic that is it does not admit any homoclinic tangencies [14]. Moreover one can completely characterize the dynamics and structure of the chaotic attractor just by using kneading invariant. However, the discrete Lorenz attractors found in the 3D maps have different properties. Discrete Lorenz attractors are formed by the period-doubling bifurcation of a stable fixed point and then via Neimark-Sacker bifurcation the closed invariant curves they merge. Discrete Lorenz attractors admit homoclinic tangencies and hence are not hyperbolic [14]. Moreover, since they possess Newhouse regions, their complete bifurcation structure analysis is not possible. Importantly, discrete Lorenz attractors are robust chaotic attractors implying their robustness of chaotic dynamics after perturbations [15]. Discrete Lorenz attractors were first found in three dimensional generalized H´enon maps [16]. Recently, authors have shown new types of discrete Lorenz attractors after attractor merging crisis of disjoint cyclic Lorenz attractors, coined to be period-two discrete Lorenz attractor [15]. Discrete maps with axial symmetry posesses many new types of discrete Lorenz attractors such as twisted and non-twisted discrete Lorenz attractors [14].

Networks of dynamical systems is an important aspect to understand the behavior of complex systems [17]. Such network of coupled dynamical systems display plethora of spatiotemporal patterns varying both in space and time including spiral waves [18]. Networks of dynamical systems also showcases a novel type of spatiotemporal pattern referred to as a chimera state [19]. Chimera state refers to the coexistence of synchronous and asynchronous state in a network of identically coupled oscillators. Chimeras are not just a theoretical concept but has been observed experimentally in mechanical systems [20], aquatic animals [21] like whale brain which keeps one eye open and other closed. In this present work, we have considered a ring-star network [22] of 3D sinusoidal map and we illustrate variety of spatiotemporal structures exhibited by the network including traveling waves , synchronized state, incoherence, and chimera state. In this work we have illustrated a variety of spatiotemporal patterns. However, a detailed analysis on the spatiotemporal patterns exhibited by a 3D discrete map showcasing discrete Lorenz attractors remains to be analyzed and will be considered in our next works.

In the rapidly evolving realm of digital cybersecurity, video encryption stands out as a vital area of research. The main reason being the transmission of sensitive visual datasets across insecure networks [23]. Well known encryption methods such as AES [24], DES [25] are effective for text based data. However those methods struggle to address the encryption of video data. As an alternative, chaos based encryption methods can be used due to their unpredictable nature and sensitivity to initial conditions. Furthermore, they assure enhanced security, making it relevant for applications in real-time video streaming, military surveillance, IoT security [26]. In the present work, we have considered a 3D sinusoidal discrete map exhibiting hyperchaotic behavior [27] along with YOLOv10 for object detection and particular object segmentation from the video frame along with CNN (Convolutional Neural Network) generated keys. Each video frame is processed to identify objects of interest and bounding boxes are generated around this object which allows selective encryption instead of the entire image. This approach is efficient and enhances computational efficiency and improves security by focusing on key regions. The video encryption method includes normalizing hyperchaotic sequences followed by pixel substitution, pixel shuffling, and key generation via using feature extraction via CNN to derive cryptographic key from each video frame. The cryptographic key is highly secure and reduces the risk of brute-force attacks. Our algorithm shows promising results on security analysis, differential attack analysis, anomaly detection, robustness to noise analysis as security measures.

In this paper, we consider dynamics of a three-dimensional sinusoidal map and showcase that the map posesses discrete Lorenz attractors. Due to the sinusoidal terms in the discrete map, the map exhibits the presence of infinitely many fixed points. They can play an important role in understanding the homoclinic and heteroclinic dynamics of the map. To understand the maximum number of fixed points exhibited by the 3D map, a two-parameter color-coded plot is shown colored according to the number of fixed points. It can be observed that the map exhibits the maximum number of five fixed points in that region. We prove that as a particular parameter is increased, the number of fixed points increases to infinity. We show that infinitely many fixed points can exist for the 3D map. To motivate the existence of discrete Lorenz attractors, we have constructed a two-parameter Lyapunov chart. This revealed diverse topologies of chaotic attractors. Moreover, we discuss the route to formation of a discrete Lorenz attractor. We found that the formation of discrete Lorenz attractor takes the route of the formation of a homoclinic butterfly. First, a stable fixed point under the variation of parameters undergoes a period-doubling bifurcation resulting to period-two points. With further variation of parameters, the period-two point develops two disjoint cyclic closed invariant curves via the supercritical Neimark-Sacker bifurcation. With further variation of parameters, we see the development of homoclinic butterfly and slowly after the discrete Lorenz attractor develops. Next, we showcase the coexistence of nested quasiperiodic closed invariant and mode-locked periodic orbit. Finally, we showcase a variety of topologically distinct chaotic attractors. Wavy like chaotic attractors is also observed whose wavy characteristic vary with changes in parameters.

The paper is organized as follows. In §1, we explore the different formation pathways of the discrete Lorenz attractor. In §2, we introduce the 3D sinusoidal map and discuss the existence of its simplest invariant sets and their bifurcations and also illustrate novel dynamical behaviors including coexistence of ergodic and resonant torus, plethora of topologically distinct chaotic and hyperchaotic attractors. In §3, we identify the regions of existence of chaotic attractors and discrete Lorenz attractors via the use of two-parameter Lyapunov charts and discuss the pseudohyperbolicity property of the discrete Lorenz attraction via the Light Method for Pseudohyperbolicity. In §4, we illustrate various novel spatiotemporal patterns of the ring-star network. In §5, we apply the chaotic signals obtained from the map to perform an efficient video encryption. The paper concludes with conclusion and future directions.

1 Formation pathways of discrete Lorenz attractors

Discrete homoclinic attractors refer to strange attractors in higher-dimensional maps that typically arise from the homoclinic structure of a saddle fixed point. Therefore, it is important to understand the topological behavior of the saddle fixed point which plays a major role in the classification of discrete homoclinic attractors. Suppose O refers to the fixed point of a three-dimensional map f. Let λ1, λ2, λ³ denote the three eigenvalues of the Jacobian J of the 3D map f. The fixed point O can be stable, saddle, or unstable. In the case of a 3D map, if all three eigenvalues are less than one in absolute value, the fixed point is stable. If all eigenvalues are greater than one in absolute value, the fixed point is unstable. The remaining possible cases are:

a) The saddle fixed point can have one eigenvalue with absolute value less than one (stable directions) and two eigenvalues with absolute value greater than one (unstable directions). Such fixed points are of the saddle focus type and they correspond to discrete Shilnikov attractors [28]. It has been shown that such attractors show simple routes to hyperchaos in multidimensional maps [27].
b) The saddle fixed point can have two eigenvalues less than one in absolute value (stable directions) and one eigenvalue greater than one in absolute value (unstable directions). Such fixed points are of the saddle-focus type and correspond to discrete figure-8 spiral attractors.

Discrete Lorenz like attractors are homoclinic attractors that contain only one saddle fixed point with two-dimensional stable manifold and one-dimensional unstable manifold, where the eigenvalues of the saddle fixed point satisfy the following conditions: a) 0 < λ¹ < 1, −1 < λ² < 0, λ³ < −1, |λ1λ2λ3| < 1 b) |λ1| > |λ2| c) saddle value σ = |λ1λ3| > 1, where λ1, λ³ represent the stable and unstable eigenvalues closest to the unit circle. We can observe from the above conditions that saddle fixed point O contains the one-dimensional unstable manifold. Moreover, it is important to note that if condition |λ1| > |λ2| is violated, this might result in the formation of a discrete figureeight attractor [29]. Based on these three conditions, we can analyze the geometric properties of the discrete Lorenz attractor.

Discrete homoclinic attractors occur in one parameter families of three-dimensional maps through simple bifurcation routes. The same holds for a special type of discrete homoclinic attractor: the discrete Lorenz attractor. We explain two scenarios leading to the formation of discrete Lorenz attractor. Suppose we have an asymptotically stable fixed point O. With the variation of a parameter (say µ), the stable fixed point O loses stability through a period-doubling bifurcation leading to the formation of a period-two orbit {p1, p2} periodic points. The previously stable fixed point O now becomes a saddle type fixed point with two eigenvalues less than one in absolute value and one eigenvalue greater than one in absolute value. Further variation of parameters leads to certain bifurcation scenarios: a) The period-two cycle (p1, p2) loses stability via a subcritical Neimark-Sacker bifurcation where a disjoint cyclic closed invariant curve of the saddle type emerges and further merges with the period-two orbit, see bottom of Fig. 1. b) The period-two cycle (p1, p2) loses stability via a super-critical Neimark-Sacker bifurcation, leading to the formation of a stable cyclic closed invariant curve, which then merges with the saddle period-2 cycle and dissappears, see the top part of Fig. 1.

As λ³ < −1, both branches of the one-dimensional unstable manifold emerge from the saddle fixed point O along the unstable eigen direction of λ³ < −1. Since this eigenvalue is negative, the points alternate between the branches. The stable manifold is two-dimensional and consists of both slow and fast directions, meaning it has a strong stable manifold. For the case of −1 < λ² < 0 < λ¹ < 1 with |λ2| < |λ1|, the strong stable manifold emerges from O, corresponding to the eigen

Figure 1: Routes to the formation of discrete Lorenz attractors via a) a subcritical Neimark-Sacker bifurcation and b) a supercritical Neimark-Sacker bifurcation of a period-two point, which arises from the fixed point O via a period-doubling bifurcation.

directions associated with the multiplier λ² < 0. As a result, the point O on the 2D local stable manifold becomes non-orientable. We can observe that the strong stable manifold partitions the 2D stable manifold into distinct regions due to λ² > 0. See Fig. 2 (a), the points cannot jump from one partition to the other partition meaning that iterations are restricted to one side of the partition. From this we can see that point x¹ maps to point x2, and both the points lie on only one side of the strong stable manifold. The points x1, x² result from the intersection of unstable and stable manifold. More importantly, the points x1, x² form homoclinic points to fixed point O. For non-orientable maps, a similar geometric analogy applies. Suppose eigenvalues λ1, λ2, λ³ satisfy 0 < λ² < λ¹ < 1 and λ³ < −1. In this case, we observe that the fixed point O is orientable on the 2D local stable manifold since λ1λ² > 0. Consequently, the homoclinic points x1, x² to O lie on the same invariant curve and don't alternate along the slow manifold, see Fig. 2 (b). An additional observation from Fig. 2 is related to the dimensionality of the saddle fixed points p1, p² and O. The saddle period-two points p¹ and p² have two-dimensional unstable manifold and a one-dimensional stable manifold, whereas the fixed point O has a one-dimensional unstable manifold and a twodimensional stable manifold. Additionally, the period-two points p1, p² reside in the holes of the discrete Lorenz attractor exhibiting dynamics similar to the flow like Lorenz attractor.

2 Three-dimensional sinusoidal map

The three-dimensional sinusoidal map is defined as follows:

$$x_{n+1} = y_n,$$

$y_{n+1} = \sin(zn),$
$z{n+1} = a + bx_n + cy_n - \sin(z_n^2),$ (1)

Figure 2: In (a), a schematic of homoclinic butterfly configuration formed by both branches of the one-dimensional unstable manifold of the saddle fixed point O with one-dimensional unstable manifold and two-dimensional stable manifold with eigenvalues −1 < λ² < 0 < λ¹ < 1 satisfying |λ2| < |λ1|. Observe that the homoclinic points x1, x² lie on same side of the strong stable manifold direction. In (b), a schematic of the homoclinic butterfly configuration by the one-dimensional unstable manifold of the saddle with eigenvalues 0 < λ² < λ¹ < 1, λ³ < −1.

where x, y, z are state variables and a, b, c are parameters.

It can be shown that 3D map in Eq. (1) can be transformed into a third-order difference equation of the form

$$z{n+1} = a + b\sin(z{n-2}) + c\sin(z_{n-1}) - \sin(z_n^2)$$ (2)

For any initial conditions and parameters a, b, c, we have

$$|z_{n+1}| \le 1 + |a| + |b| + |c| \tag{3}$$

implying that all iterates are bounded and lie inside a cuboid of side 1 + |a| + |b| + |c|.

2.1 Fixed points and their bifurcations

The fixed points (x ∗ , y^∗ , z^∗ ) of Eq. (1) are obtained by solving the following three simultaneous equations:

$$x^* = y^*,$$

$y^* = \sin(z^),$
$z^ = a + bx^* + cy^* - \sin(z^{*2}),$ (4)

which leads to finding the roots of the nonlinear equation given by

$$z^* = a + (b+c)\sin(z^*) - \sin(z^{*2})$$ (5)

Under the simultaneous variations of both parameters a and b, we observe that the number of fixed points varies. Moreover, the map exhibits multiple fixed points ranging from one fixed point to

five fixed points, as shown in Fig. 3 (a). We hypothesize that the presence of many fixed points results from the presence of sin terms in the discrete map equations. It is interesting to observe the existence of multiple fixed points, as they can support the occurrence of homoclinic and heteroclinic dynamics. The Jacobian matrix is given by

$$J = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & \cos(z^*) \ b & c & -2z\cos(z^{*2}) \end{bmatrix}$$ (6)

The characteristic polynomial is given by

$$P(\lambda) = \lambda^3 + \lambda^2 (2z^* \cos(z^{2})) + \lambda(c \cos(z^)) + b \cos(z^*) = 0$$

A saddle-node bifurcation occurs when an eigenvalue reaches 1, while a period-doubling bifurcation occurs when an eigenvalue reaches −1. Thus, a saddle-node bifurcation occurs when λ = 1, so P(1) = 1 + 2z ∗ cos(z ∗2 ) + c cos(z ∗ ) + b cos(z ∗ ) = 0. Similarly, a period-doubling bifurcation occurs when λ = −1, so P(−1) = −1 + 2z ∗ cos(z ∗2 ) + c cos(z ∗ ) + b cos(z ∗ ) = 0. However, obtaining fixed points analytically is not always tractable, thus, we can compute the codimension-one bifurcation curves numerically, as shown in Fig. 3 (b).

Figure 3: In (a), two-parameter variation of number of fixed points color coded according to the number of existing fixed points. In (b), a two-parameter stability analysis of fixed points in the a − b plane. The stable fixed point is marked in tile lineated by the codimension-one saddle-node and period-doubling bifurcation curves. The yellow and dark blue region denotes the saddle and unstable fixed point regimes respectively. Parameter c = 0.99 is fixed.

To illustrate the existence of multiple fixed points, we use a nullcline-like plot, as seen in Fig. 4. The orange straight line denotes the left hand side of Eq. (7), while the blue curve denotes the right hand side. The number of intersections between the orange and blue curve represents the existence

Figure 4: The orange straight line denotes the left side of eq. (5) and the blue curve denotes the right of eq. (5). In (a), the curves intersect at a single point and thus a single fixed point exists for a = −0.3, b = 0.416, c = −1.3. In (b), two fixed points exist at a = 0.54, b = 0.725, c = 0. In (c), existence of three fixed points are shown for a = −0.3, b = 0.4, c = 1.5. In (d), existence of four fixed points are shown for a = −0.19, b = 0.4, c = 1.5. In (e), existence of five fixed points are shown for a = −0.3, b = 0.402, c = 2.1.

of number of fixed points. In Fig. 4, we discuss how the number of fixed points changes depending on various values of a, b, c. In order to effectively vary all three parameters simultaneously and observe the variation in the number of fixed points, one can think of assigning three parameters a, b, c on three coordinate axis and then scan the parameter space along each quadrant of the parameter axis. In Fig. 4 (a), we observe one intersection and thus a single fixed point exists for a = −0.3, b = 0.416, c = −1.3. We observe two fixed points when a = 0.54, b = 0.725, c = 0, see Fig. 4 (b). Existence of three fixed points is noted when a = −0.3, b = 0.4, c = 1.5, see Fig. 4 (c). With decrease in the value of a, four fixed points emerge at a = −0.19, b = 0.4, c = 1.5, see Fig. 4 (d). So far, we have observed a maximum of five fixed points for a = −0.3, b = 0.402, c = 2.1, see Fig. 4 (e).

However, it is interesting enough that the number of fixed points increases as the parameter b increases. We prove that the 3D map has an infinite number of fixed points as parameter b increases.

Theorem 1. The map contains infinite number of fixed points as parameter b increases.

Proof. To understand the existence of an infinite number of fixed points in the 3D map, we analyze the number of real solutions to the equation

$$z^* = a + b\sin(z^*) - \sin(z^{*2}) \tag{7}$$

Our goal is to show that the number of real solutions to the above equation is infinite. We achieve this by demonstrating the existence of at least 2N solutions for arbitrarily large N, implying that the number of solutions can grow indefinitely. We mention the strategy of the proof: the proof takes advantage of the periodic behavior of sin(z). The sin function is periodic and lies in range [−1, +1]. Moreover, on any interval of length 2π, sin(z) completes one oscillation. So if any function f(z) also lies within [−1, +1], then the equation sin(z) = f(z) is guaranteed to have multiple solutions via Intermediate Value Theorem.

The proof is divided into further steps:

Rewrite the transcendental equation: We can rewrite the equation as sin(z) = f(z), where f(z) = ^z−|a|−¹ . The motivation to write in such a way is to split the equation into two parts: a) sin(z), a well understood oscillatory function, b) f(z), a controllable function whose behavior depends on parameters a, b.
Keeping f(z) bounded on [−1, +1]: For the equation sin(z) = f(z) to have solutions, f(z) needs to lie between [−1, +1]. By choosing parameter b large enough, we can control value of f(z) by making it small (make b large) and ensure |f(z)| < 1 over a large domain. To perform this, we consider the inequality ²Nπ−|a|−¹ ^b < 1. This guarantees f(z) is within [−1, +1] for |z| ≤ 2Nπ, where N is an arbitrary positive integer. Note that the numerator in f(z) grows linearly in z and hence the inequality can be preserved by bounding the numerator.
Periodic nature of sin(z): Consider an interval of length 2π for example, [2nπ, 2(n+ 1)π], the sin function completes one full oscillation from −1 to +1. Next, since f(z) is restricted within [−1, +1], there must be at least one solution of sin(z) = f(z) in each such sub-interval.
Counting the number of solutions: We can divide the domain [−2Nπ, 2Nπ] into 2N disjoint intervals each of length 2π as [2nπ, 2(n + 1)π], where −N ≤ n ≤ N. As there exists at least one solution in each interval, there are at least 2N solutions in total.

• Deducing the count when N is arbitrarily large: By allowing N → ∞, we find that the number of solutions grow without bound. Thus, the number of solutions to the equation is infinite.

2.2 Coexistence of ergodic and resonant torus

We illustrate the coexistence of both ergodic and resonant torus in the 3D sinusoidal discrete map. For a = 0.0192, we observe the coexistence of length-doubled quasiperiodic orbit and a mode-locked period-34 orbit for different initial conditions; see Fig. 5 (c). Interestingly, the topology of the modelocked orbit and the quasiperiodic orbit is similar and symmetrical. Moreover, the quasiperiodic orbit and mode-locked orbit are nested to each other. The quasiperiodic orbit orbit resides inside the mode-locked period-34 orbit. In Fig. 5 (a), the symmetrical mode-locked period-34 orbit is depicted, where stable period-34 orbits are shown in blue triangles, and saddle period-34 orbit are shown as black squares. The corresponding one-dimensional unstable manifolds of the saddle periodic orbit are illustrated in red forming a saddle-node connection. A symmetrical length-doubled quasiperiodic orbit with similar topology as that of the mode-locked period-34 orbit is displayed in Fig. 5 (b). Although, the coexistence of mode-locked and quasiperiodic orbits are common [30, 5], to the best of our knowledge, such nested mode-locked periodic orbits and quasiperiodic orbits with similar symmetrical topology are shown for the first time in the context of 3D maps.

Figure 5: For a = 0.0192, we observe coexistence of a length-doubled quasiperiodic closed invariant curve and a mode-locked period-34 orbit. In (a), a mode-locked symmetrical period-34 orbit is displayed. In (b), a length-doubled quasiperiodic closed invariant curve is shown. In (c), both the mode-locked period-34 orbit and the length doubled quasiperiodic closed invariant curve are depicted together. Observe that these structures are nested within each other. The parameters b = 0.8, c = 0.99 are fixed.

2.3 Plethora of chaotic attractors

We also showcase various topologically distinct chaotic attractors exhibited by the 3D sinusoidal map, see Fig. 6. For parameter values a = 0.3, b = 3.9, c = 0.99, the map develops a cube-like hyperchaotic attractor with two positive Lyapunov exponents. For the parameter values a = 0.3, b = 3.9, and c = 0.99, the system generates a hyperchaotic attractor with a distinctive cube-like shape. The presence of two positive Lyapunov exponents confirms the hyperchaotic nature of the

system, as it signifies exponential divergence along two independent directions—one of the key characteristics of hyperchaos.

Additionally, the attractor's Kaplan–Yorke dimension is estimated to be approximately 2, implying that the system's trajectories are primarily confined to a complex, fractal-like structure embedded within a two-dimensional surface in three-dimensional space. The attractor exhibits a dispersed, cloud-like pattern, indicative of significant dynamical complexity and extreme sensitivity to initial conditions.

Unlike conventional Lorenz-type attractors, which typically display a well-defined folded structure, the attractor observed in Fig. 6(a) is more diffuse and cube-shaped, suggesting that the underlying dynamics follow a different organizational pattern. The Lyapunov exponents were esti-

Figure 6: A variety of chaotic attractors. In (a), we observe a hyperchaotic cube like attractor for a = 0.3, b = 3.9, c = 0.99. In (b), a chaotic attractor with different topology for a = −0.3, b = −1.8, c = 0.99. In (c), a wavy chaotic attractor for a = −3, b = −0.5, c = 0.99. In (d), more wavy chaotic attractor for a = −6.5, b = −0.5, c = 0.99. In (e), extreme wavy chaotic attractor for a = −19, b = −0.5, c = 0.99. In (f), more wavy chaotic attractor for a = −15, b = −0.5, c = 0.99.

mated numerically via Wolf's algorithm [31]. The fractal dimension of the chaotic attractor can be determined via the formula of Kaplan and Yorke [32] is given by

$$D{KY} = k + \frac{\sum{i=1}^{k} \lambdai}{|\lambda{k+1}|}, \quad (8)$$

where k is the largest integer for which the sum of decreasingly ordered Lyapunov exponents λⁱ is still positive. For a variety of chaotic attractors illustrated in Fig. 6 from the 3D sinusoidal map, we compute the Kaplan Yorke dimension and confirm that it aligns with the expectations, see Table 1. Observe Fig. 6(a), a cubelike chaotic attractor with two positive Lyapunov exponents is shown and the Kaplan Yorke dimension is 2, as per expectations. Chaotic attractors with exactly one positive Lyapunov exponent have fractal dimension between 1 and 2. It would be interesting to compare the Kaplan-Yorke dimension with the other fractal dimensions as a future work.

For negative values of a, chaotic attractors with diverse topologies are observed. However, we have not observed chaotic attractors for positive values of parameter a. For a = −0.3, we observe a more dense volume spread of chaotic attractor. As parameter a increases more towards negative value, the chaotic attractor starts taking the shape of a wavy shape. It is interesting to note that the fluctuations of the wavy chaotic attractor increases with increase in the negative value of parameter a. For a = −3, we observe three wavy peaks in the chaotic attractor, see Fig. 6 (c). For a = −6.5, we observe a six peak wavy chaotic attractor, see Fig. 6 (d). For a = −19, we observe a heavily fluctuating wavy peak chaotic attractor, see Fig. 6 (e).

Table 1: Lyapunov exponents and Kaplan–Yorke dimension for different chaotic attractors in Fig. 6.

Figure	Lyapunov Exponents	Kaplan–Yorke Dimension
a	(0.880, 0.057, -0.258)	2
b	(0.5006, -0.1095, -0.4654)	1.814
c	(1.0692, -0.9633, -1.4456)	1.066
d	(1.8599, -1.4352, -1.8372)	1.226
e	(2.9420, -2.0256, -2.3335)	1.389
f	(2.7148, -1.8680, -2.1545)	1.390

2.4 Two-parameter Lyapunov charts

Two-parameter Lyapunov charts are more effective than one-parameter diagrams in understanding many novel bifurcation scenarios. Two-parameter Lyapunov charts have been useful in understanding the routes to the presence of hyperchaos [27], routes to the resonant torus doubling bifurcation [10], eyes of chaos [33]. We identify the region of existence of the discrete Lorenz attractors via the use of two-parameter Lyapunov charts in the a − b plane, see Fig. 7. The yellow regions denote regular or periodic behavior with all three positive Lyapunov exponents. The grey regions denote chaotic or hyperchaotic behavior with more than one positive Lyapunov exponents. For each point of the 1000 × 1000 grid in the two-dimensional parameter space, the Lyapunov exponents are computed via Wolf's algorithm [31]. If all Lyapunov exponents are negative, it denotes the regular or periodic behavior and is marked yellow. If at least one Lyapunov exponent is positive, it denotes chaotic or hyperchaotic behavior and is marked grey.

3 Discrete Lorenz attractors

We demonstrate the prevalence of discrete Lorenz attractors in the 3D map under consideration. Importantly, we illustrate the routes leading to the formation of a discrete Lorenz attractor by the computation of the one-dimensional unstable manifolds of the saddle fixed point absorbed in the discrete Lorenz attractor. For a = 0.3, b = 0.8, c = 0.99, the stable fixed point O undergoes a perioddoubling bifurcation leading to the formation of a stable period-two orbit {p1, p2}, see Fig. 8 (a). The saddle fixed point O has one eigenvalue greater than one in absolute value and two eigenvalues less than one in absolute value. Fixed point O develops one-dimensional unstable manifold which converges towards the period-two points spirally shown in a zoomed version. The period-two point persists as parameter a dereases till 0.08, after which the stable period-two point undergoes a super-critical Neimark-Sacker bifurcation resulting in the formation of the disjoint cyclic two closed

Figure 7: A two-parameter Lyapunov dynamical chart in the a−b plane. The yellow regions denote regular or periodic behavior, while the grey regions denote chaotic or hyperchaotic behavior. Three different phase portraits are shown in the chaotic regions. We can observe that the topology of the chaotic attractors varies significantly. One of these chaotic attractors represent the discrete Lorenz attractor. The parameters b = 0.8, c = 0.99 are fixed.

invariant curves. Considering both the branches of the one-dimensional unstable manifold of O, we can see that formation of homoclinic butterfly has taken place. The one-dimensional unstable manifold has grown from the saddle fixed point O covering the closed invariant curve until it returns to the fixed point O where the branches of the unstable manifold appear non-twisted. Further as parameter a decreases to 0.069, discrete Lorenz attractor develops, see Fig. 8 (c). A

Figure 8: In (a), the fixed point O (shown in red) undergoes a supercritical period-doubling bifurcation resulting in a period-two orbit denoted by {p1, p2} at a = 0.3. Both branches of the one-dimensional unstable manifold are also shown as Wu(O) for the saddle fixed point O. In (b), the formation of disjoint cyclic closed invariant curves is shown along with the one-dimensional unstable manifolds of the saddle fixed point at a = 0.08. Observe that the one-dimensional unstable manifolds form a homoclinic butterfly structure before the formation of the discrete Lorenz attractor. In (c), the formation of the discrete Lorenz attractor is shown, which has absorbed the saddle fixed point O for a = 0.069. The parameters b = 0.8, c = 0.99 are fixed.

detailed structure of the discrete Lorenz attractor is shown in Fig. 9. In Fig. 9 (a), a detailed structure of discrete Lorenz attractor is shown along with the saddle fixed point O in red, with one-dimensional unstable manifold and two-dimensional stable manifold. In Fig.9 (b), shows only the one-dimensional unstable manifold of the saddle fixed point O forming the homoclinic butterfly. In Fig.9 (c), a zoomed version of the unstable manifold near by the saddle fixed point O is shown demonstrating the twistedness of the unstable manifold.

Figure 9: A detailed structure of the discrete Lorenz attractor is shown. In (a), the discrete Lorenz attractor is displayed along with the saddle fixed point O (in red). In (b), both branches of the one-dimensional unstable manifold of the saddle fixed point O are shown. In (c), a zoomed in view of the one-dimensional unstable manifold is presented. We observe the excursions taken by the one-dimensional unstable manifold, where the branches cross each other but do not intersect.

We show detailed routes leading to the formation of discrete Lorenz attractor with a different topology in Fig. 10. In Fig. 10 (a), we observe a symmetric closed invariant curve at a = 0.063. Further variation of parameter a leads to a saddle-node bifurcation of the closed invariant curve resulting in a mode-locked period-19 orbit at a = 0.062, see Fig. 10 (b). The blue triangles denote the stable period-19 orbit while the black squares denote the saddle period-19 orbit. The red curve represents both branches of the one-dimensional unstable manifold of the saddles, forming a saddle-node connection of the mode-locked period-19 orbit. At a = 0.0192, formation of asymmetric length-doubled quasiperiodic closed invariant curve takes place, see Fig. 10(c). The closed invariant curve is destroyed by the saddle-node bifurcation of invariant curves, leading to the formation of a mode-locked period-34 orbit at a = 0.019, see Fig. 10(d). A further increase in parameter a leads to the formation of chaotic like attractor pieces for a = 0.01527, see Fig. 10(e). The chaotic pieces merge at a = 0.015231 taking the form of a chaotic Lorenz like attractor but it has not formed fully yet, see Fig. 10(f). With further variation in parameter a leads to the formation of discrete Lorenz attractor, see Fig. 10(g) at a = 0.01524. The discrete Lorenz attractor has fully formed with absorption of the saddle fixed point O at a = 0.015, see Fig. 10(h). A zoomed version of the attractor is shown, highlighting the twisted nature of the iterates near the saddle fixed point.

In Fig. 11(a), the twisted Lorenz attractor is shown along with the saddle fixed point O. In Fig 11(b), both the branches of the one-dimensional unstable manifold of saddle fixed point O is shown in Fig. 11(b). We can observe the twisted nature of the one-dimensional unstable manifold. A zoomed version near the saddle fixed point O is shown in Fig. 11(c).

So far we have shown the presence of discrete Lorenz attractors in the case of positive values of parameter a. Next, we show that the discrete Lorenz attractor also exists for negative values of parameter a, see Fig. 12. At a = −0.08, three layered closed invariant curve exists. Further increase in parameter a to −0.076 leads to the formation of a triple length closed invariant curve, see Fig. 12 (a), (b). When parameter a is increased to −0.075, the closed invariant curve is destroyed and

Figure 10: In (a), a quasiperiodic closed symmetric invariant curve is shown for a = 0.063. In (b), a symmetric mode-locked period-19 orbit is shown after a saddle-node bifurcation of the invariant curve for a = 0.062. In (c), a length-doubled quasiperiodic closed invariant curve is shown for a = 0.0192. In (d), a symmetric length-doubled mode-locked orbit of period 34 is shown for a = 0.019. In (e), presence of chaotic pieces is shown at a = 0.01527. In (f), merger of various chaotic pieces is shown at a = 0.015231. In (g), formation of discrete Lorenz attractor is shown at a = 0.01524. In (h), fully developed discrete Lorenz attractor is shown at a = 0.015. The parameters b = 0.8, c = 0.99 are fixed.

Figure 11: In (a), a twisted discrete Lorenz attractor is shown along with the saddle fixed point O. In (b), both branches of the one-dimensional unstable manifold (in red) are shown forming the homoclinic butterfly structure. In (c), a zoomed version of the twisted nature of the discrete Lorenz attractor is shown. The one-dimensional unstable manifold appears to be twisted.

Figure 12: Formation of a discrete Lorenz attractor for a < 0. The detailed formation of the discrete Lorenz attractor is shown. It results from the closed invariant curve.

the formation of symmetric chaotic attractor takes place, see Fig. 12 (c). At a = −0.06, we observe the merger of symmetric heads of the chaotic attractor leading to the formation of a discrete Lorenz attractor, see Fig. 12 (d) and it persists for a longer parameter range of a. For a = −0.003, we observe the merger of wings of the discrete Lorenz attractor leading to the formation of a two eye discrete Lorenz attractor, see Fig. 12 (e). At a = −0.001, those wings get detached, see Fig. 12 (f) and the discrete Lorenz attractor has similar topology as that of Fig. 12 (d).

3.1 On verification of pseudohyperbolicity

Pseudohyperbolicity is a generalization of the hyperbolicity property which ensures that the maximal Lyapunov exponent is positive for every orbit of the attractor. It also guarantees that every orbit in the attractor is unstable. Pseudohyperbolicity is an important aspect to understand the aspect of robustness of chaotic dynamics [34]. A compact set A is Pseudohyperbolic if it satisfies the following three conditions:

a) For each point x ∈ A, there exists two continuously dependent transversal linear subspaces N1(x) and N2(x) with dimensions dim(N1) = k and dim(N2) = n−k for 1 ≤ k ≤ n−1. These subspaces are invariant under the differential of the map f that is Df(N1(x)) = N1(f(x)), Df(N2(x)) = N2(f(x)). In the context of dynamics and hyperbolicity, transversal subspaces usually arise while analyzing the decomposition of the tangent space at points in the attractor. These are associated with splitting of the space into contracting and expanding subspaces. For pseudohyperbolic attractors, such transversality ensures robust instability and further prevents tangencies that could compromise chaotic dynamics.
b) The transversal linear subspaces N1(x) and N2(x) depend continuously on each point of the attractor A. This continuous dependence ensures that such splitting is well behaved throughout the attractor A without sudden changes. To verify the continuity of N1(x) and N2(x), we can measure the angle and distance between corresponding points of the attractor. The latter can shed light on whether the variations are smooth and bounded away from zero helping to determine the transversality across attractor A.
c) The diferential of the map Df is exponentially contracting in N¹ and Df is expanding exponentially in remaining (n − k) dimensions. This means that for v ∈ N1, the differential Df satisfies contraction:

$$||Df(v)|| \le Ce^{-\lambda}||v||, \quad \lambda > 0, C > 0.$$

For w ∈ N2, the differential Df satisfies the expansion:

$$||Df(w)|| \ge Ce^{\sigma}||w||, \quad \sigma > 0, C > 0.$$

For a pseudohyperbolic attractor, the fixed point contained within it must be pseudohyperbolic. That is, the eigenvalues λ1, λ2, λ³ should satisfy |λ1| > 1, 0 < λ² < 1, 0 < |λ3| < |λ2| and |λ1λ2| > 1.

For a 3D map, if the attractors are pseudohyperbolic, then dim(N1) = 1 and dim(N2) = 2. Three Lyapunov exponents on the attractor Λ1,Λ2,Λ³ are determined. They satisfy Λ¹ > 0,Λ¹ + Λ² > 0,Λ¹ + Λ² + λ³ < 0 with Λ¹ > Λ² > Λ3.

It is not trivial how to implement the above conditions to establish the pseudohyperbolic nature of attractors analytically. A numerical approach to verifying the pseudohyperbolic nature of attractor was established using the Light Method of pseudohyperbolicity (LMP method) and was proposed [35] to study the pseudohyperbolic nature of aattractor in multidimensional systems. Since it is known that for 2d maps for instance, confirming pseudohyperbolicity is same as confirming uniform hyperbolicity property of the attractor. Then it remains challenging to verify pseudohyperbolicity in multidimensional systems where LMP method can be quite useful. LMP method deals with the most important condition for verifying pseudohyperbolicity: the continuity of subspaces N¹ and N2. Using the LMP method, a field of vectors related to the strongest contracting directions is computed. The method involves in constructing an LMP graph which shows the dependencies of the angles dϕ between the vectors N1(x1) and N2(x2) at orbit points x¹ and x² of the attractor A. Let us discuss how this method is implemented numerically: the map is computed for a large number of iterations and transients are discarded. Suppose pⁱ set of points are found. We then compute the corresponding Lyapunov vectors for the map f −1 . For points pⁱ and p^j , we compute the distance between them (denoted by ρ(pⁱ , p^j )) and angles between them ϕ(pⁱ , p^j ) and we plot them on a (ρ, ϕ) axis. Such dependencies plot will result in a cloud of points in the (ρ, ϕ) plane and is referred to as (ρ, ϕ)− cloud. If the (ρ, ϕ)-cloud has no points other than (0, 0) and (0, π) on the ρ = 0 axis, then we can conclude that A is pseudohyperbolic. Note that the presence of (ρ, ϕ) at (0, π) does not affect pseudohyperbolicity. This situation typically arises in maps whose fixed points, or periodic orbits have negative eigenvalues and also when the attractor itself is non-orientable. In Fig. 13, examples of discrete Lorenz attractors is shown. The continuity diagrams were computed for iterations of the map (discarding the transients). As we can observe, the points touch many points on the ϕ axis (y-axis) implying the discontinuity of the field of subspaces Ess(x) (subspace spanned by strongly stable direction) and Ncu(x) (subspace spanned by strongly contracting direction). This implies the discontinuity of the subspaces and therefore the non-pseudohyperbolicity of the discrete Lorenz attractor.

4 Ring-star network behaviors

Researchers have studied phenomena of chaotic synchronization in generalized continuous Lorenz systems [36]. Chaotic synchronization based on Reservoir Computing was performed for the purpose of secure communication[37]. A novel control scheme for the chaos synchronization in stochastic nonlinear systems was explored [38]. Higher dimensional chaotic systems were stabilized via optimized linear feedback control and sliding mode control [39]. A combination of ring and star network result in ring-star network. Moreover, by simply tuning the coupling strengths of the network, we can switch between a ring network, star network, and ring-star network. Ring-star network was discussed in the context of Chua circuits where the prevalence of single and double-well chimeras was studied [22]. Subsequently, ring-star network has been applied in various systems such as neuron systems [40, 41], heterogeneous neuron networks [42] among others. In this network configuration, the network takes the configuration of a ring-star illustrated in Fig. 14. The ring-star network consists of a central node which connects to other distant nodes of the network. This is beneficial in understanding the information flow throughout the network. In the ring-star sinusoidal map based ring-star network, each node i = 1, . . . , N follows chaotic dynamics. The coupling between nodes is bidirectional as indicated by the double arrows. The coupling strengths of the star and ring networks are denoted by µ and σ, respectively. The ring-star network model of the 3D sinusoidal map is described as follows:

Figure 13: Test of pseudohyperbolicity via the Light Method of Pseudohyperbolicity for three discrete Lorenz attractors. The parameters are fixed as follows: In (a), a = 0.069, (b) a = 0.015, (c) a = −0.001. The parameters b = 0.8, c = 0.99 is fixed.

$$x{m}(n+1) = y{m}(n)$$

$$+ \mu(x{m}(n) - x{1}(n)) + \frac{\sigma}{2R} \sum{i=m-R}^{m+R} \left( x{i}(n) - x_{m}(n) \right),$$

$$y{m}(n+1) = \sin(z{m}(n)),$$

$$z{m}(n+1) = a + bx{m}(n) + cy{m}(n) - \sin(z{m}(n)^{2}),$$

$$(9)$$

where the central node is defined as

$$x_1(n+1) = y1(n) + \mu \sum{i=1}^{N} \left( x_i(n) - x_1(n) \right),$$

$$y_1(n+1) = \sin(z_1(n))$$

$$z_1(n+1) = a + bx_1(n) + cy_1(n) - \sin(z_1(n)^2),$$ (10)

The boundary conditions are given by

$$x_{m+N}(n) = xm(n), \quad y{m+N}(n) = ym(n), \quad z{m+N}(n) = z_m(n),$$ (11)

where N denotes the total number of nodes in the network, and R represents the number of neighbors considered for each node.

Figure 14: A sketch of ring-star network constituting N nodes with ring coupling strength σ and star coupling strength denoted by µ. The double sided arrows indicate bidirectional coupling between the oscillators in the network.

Next, we examine various spatiotemporal patterns exhibited by three different ring-star networks just by tuning the coupling strengths µ and σ. For a ring network we can tune σ ̸= 0, µ = 0. For star network, we can tune σ = 0, µ ̸= 0. For ring-star network we can tune σ ̸= 0, µ ̸= 0.

4.1 Ring-star network

We consider the ring-star network of the 3D sinusoidal map in the regime of discrete Lorenz attractor, exhibited by uncoupled discrete map at a = 0.069, b = 0.8, c = 0.99. For σ = 0.01, µ = 0.07, we observe complete synchronization among all the nodes in the ring-star network, see Fig. 15 (a). The left plot represents the evolution of x-state of the nodes with respect to time, where a uniform pattern can be observed. Corroborating this, the middle plot shows that all oscillators are aligned at the same level or value showing complete synchronization. The rightmost plot represents the recurrence plot where the distance between the ith node and jth node is color coded according to the distance. Observe that it is completely blue indicating that the distance between nodes is zero and thus denotes complete synchronization. For σ = 0.03, µ = 0.05, we observe a two-cluster chimera state, where nodes upto 40 are in a two-cluster state while the remaining nodes are oscillating asynchronously, see Fig. 15 (b). The left most plot shows the evolution of the x-state over time, while the righmost recurrence plot displays small square like structures of different colours which denotes the two cluster state. For σ = 0.04, µ = 0.12, we observe an unsynchronized state in which all oscillators oscillate out of phase. The evolution of the x-state with respect to time shows no discernible pattern and the recurrence plot reveals a range of colours, further confirming the unsynchronized state.

To understand the sensitivity to spatiotemporal patterns with the variations of ring coupling strength σ and star coupling strength µ, we perform a two parameter regime map with simultaneous variation of σ and µ. Fig. 16 illustrates the simultaneous impact of ring coupling strength σ and star coupling strength µ on the emergent spatiotemporal patterns in a ring-star network. The analysis reveals distinct dynamical regimes, where different coupling strengths give rise to synchronized, unsynchronized, chimera, traveling wave, and diverging states. At lower values of σ and µ, the system exhibits synchronization and chimera states, indicating partial coherence. As σ and µ increase, unsynchronized and traveling wave patterns emerge, reflecting more complex dynamical behavior. For sufficiently high coupling strengths, the system enters a diverging state, suggesting instability and loss of bounded dynamics. This study highlights the detailed dependence of organization of various spatiotemporal patterns on coupling parameters, offering deeper explanations into the control and predictability of complex networked systems.

4.2 Ring network

We consider a ring network of the 3D sinusoidal map by setting σ ̸= 0, µ = 0. For σ = 0.02, µ = 0, we observe a traveling wave, see Fig. 17 (a). On the leftmost plot, we can see wavy like patterns in the evolution of the x-state of the nodes of the network over time discarding transients. This coincides with the middle plot which shows wavy like pattern. In the rightmost recurrence plot, we observe a unique pattern which differs from earlier observed box like structures. This unique structures of the recurrence plots resembles that of the traveling wave state. For σ = 0.01, µ = 0, we observe another traveling wave. Notice the similarity between the evolution of the x-state over time in the leftmost plot and also similarity of the recurrence plot on the right. When the ring coupling strength σ is increased and tuned to σ = 0.05, we observe that the nodes oscillate asynchronously as all oscillators oscillate out of phase with their neighboring oscillators.

Figure 15: Spatiotemporal patterns exhibited by the ring-star network (σ ̸= 0, µ ̸= 0). In (a), for σ = 0.01, µ = 0.07, the network develops a synchronized state. In (b), for σ = 0.03, µ = 0.05, the network exhibits a two cluster chimera state. In (c), for σ = 0.04, µ = 0.12, the network displays an unsynchronized state. The dynamical parameters are set as a = 0.069, b = 0.8, c = 0.99.

Figure 16: Regime map in the ring and star coupling strength plane. Sensitivity Analysis of Coupling Parameters on Spatiotemporal Patterns.

4.3 Star network

For a star network, we set σ = 0, µ ̸= 0. For µ = 0.02, all oscillators were in sync, see Fig. 18 (a). Observe the regularity in the evolution of the x-state over time as well as in the rightmost plot which shows a recurrence plot with a complete blue colour indicating synchronization. With a further increase in µ to 0.04, all oscillators were out of sync leading to desynchronization, see Fig. 18 (b). This agrees with the evolution of the x−state variable over time in the leftmost plot and with the recurrence plot in the rightmost plot.

5 Application to video encryption

We showcase an application of hyperchaotic signals generated by the 3D sinusoidal map with two positive Lyapunov exponents for the purpose of general multimedia encryption specifically video encryption. We take advantage of YOLOv10 and CNN to segment the necessary subparts of the video frames for encryption and to generate the secret key respectively. Below each subsection outlines the important steps for video encryption.

5.1 Implementation of YOLOv10

To accurately identify and extract specific elements from video frames, we employ YOLOv10 for object detection. For our application, YOLOv10 is ideal because it can detect and locate objects with high precision and real-time performance. Bounding boxes are created around objects of inter-

Figure 17: Spatiotemporal patterns exhibited by the ring network. In (a), for σ = 0.02, the network develops a traveling wave. In (b), for σ = 0.01, the network develops another traveling wave of different configuration. In (c), for σ = 0.05, the network develops an unsynchronization behavior. The dynamical parameters are set as a = 0.069, b = 0.8, c = 0.99.

Figure 18: Spatiotemporal patterns exhibited by the star network (σ = 0, µ ̸= 0). In (a), for µ = 0.02, the network develops synchronized state. In (b), for σ = 0.04, the network develops unsynchronized state. The dynamical parameters are set as a = 0.069, b = 0.8, c = 0.99.

est that are identified by processing each video frame. By separating target areas, these bounding boxes enable sensitive items to be encrypted selectively rather than the full image. This method focuses on important areas to improve security, decrease computational load, and increase encryption efficiency. High-accuracy real-time processing is made possible by YOLOv10's integration.

5.2 Encryption using 3D Hyperchaotic Map

Our motivation for using a 3D discrete map was to highlight its ability to exhibit discrete Lorenz attractors and demonstrate an application in video encryption using hyperchaotic signals derived from the system. While simpler chaotic maps, such as adaptive symmetric chaotic maps [43] and minimal digital chaotic maps [44], may offer similar encryption performance, they often lack the ability to generate discrete Lorenz attractors. This highlights a fundamental trade-off: simpler maps provide lower computational complexity, but they may not capture the intricate dynamics observed in our proposed system. This paper introduces a novel video encryption method that generates hyperchaotic sequences for pixel transformation and shuffling utilizing signals from a 3D sinusoidal hyperchaotic map.

The encryption process occurs in two stages:

1. Pixel intensities are combined with the hyperchaotic sequences to create a transformed frame.
2. Pixel shuffling is then carried out, using the hyperchaotic map to rearrange pixel locations.

By altering the values and positions of the pixels, this transformation scrambles the image. Without understanding the characteristics of the chaotic system, the combination of these strategies makes it challenging to decipher the encrypted image. Encryption enhances security and resilience by ensuring that even minor modifications to the key or initial conditions produce significantly different outcome.

5.3 Three Dimensional Hyperchaotic Map

The encryption algorithm generates chaotic sequences xn, yn, and zⁿ for every pixel using a 3D hyperchaotic system (1). The system is governed by the following equation (1), where x0, y0, z⁰ are the initial conditions and the parameters a, b, and c govern the dynamics of the system. These sequences are used for pixel transformation and shuffling to ensure high security and robustness.

Normalization of Chaotic Sequences

The chaotic sequences are normalized in the following manner to match the image dimensions:

$$x{\text{indices}} = x \cdot h$$ , $y{\text{indices}} = y \cdot w$ , $z_{\text{indices}} = z \cdot 256$ ,

where the height and width of the frame are denoted by h and w, respectively. Here xindices and yindices specify the row and column indices, and zindices modifies the pixel intensity.

Pixel Substitution (Transformation)

Each pixel is transformed by adding the corresponding value from zindices, which enhances randomness. The transformation is given by:

Transformed Pixel $$[i, j] = (\text{Original Pixel}[i, j] + z_{\text{indices}}) \mod 256,$$

where Original Pixel[i, j] represents the intensity at position (i, j).

Pixel Shuffling

Pixels are rearranged based on xindices and yindices. The new position is determined as:

Shuffled $$Pixel[i, j] = Transformed $Pixel[x{indices}[i], y{indices}[j]].$$$

Final Encrypted Video Frame Construction

The encrypted frame is reconstructed by reshaping the shuffled pixels into a 2D format:

Encrypted Image $$[i, j]$$ = Shuffled Pixel $[i, j]$ ,

where (i, j) represents the position of the pixels in the frame.

Key Generation

The key generation via feature extraction method uses CNN to get a deterministic cryptographic key for each video frame. The CNN produces a feature vector, which is mapped to a fixed-length hexadecimal key to achieve uniformity across frames. The approach focuses on frame-level features, ensuring unpredictability and strength for the encryption. Since keys of CNNs are dynamically computed based on the input data's feature, each example is encrypted distinctively, and therefore the encryption process is safer against brute-force and cryptanalysis attacks through pixel shuffling and linearity of the key. Besides, CNNs are most competent in processing multimedia data, and therefore are most suitable for encrypting images and videos. With the deployment of deep learning, this method enhances security without sacrificing performance, providing a resistant and responsive solution to the challenges of modern-day encryption. Feature extraction highlights critical frame characteristics, strengthening encryption and safeguarding against unauthorized access. Feature extraction is covered in detail in the next section.

Feature Extraction and Key Generation

The following outlines the procedure for feature extraction and key generation:

Feature Extraction

The CNN's convolutional layers process the input image I in order to produce feature maps:

$$f_1 = \text{ReLU}(W_1 * I + b_1), \quad f_2 = \text{ReLU}(W_2 * f_1 + b_2),$$

where W1, W² are convolutional weights, b1, b² are biases, and f¹ and f² are feature maps.

Feature Vector Creation

A feature vector k is created by flattening the feature maps and passing them through fully connected layers:

$$f_3 = \text{ReLU}(W_3 \cdot \text{flatten}(f_2) + b_3), \quad k = W_4 \cdot f_3 + b_4,$$

where W3, W⁴ are weights, b3, b⁴ are biases, and f³ is an intermediate feature map.

5.4 Algorithm : Encryption of Image

Input : Image I of size (h, w) with pixel values in [0, 255], Key K (string of at least 3
          characters)

Output: Encrypted Image E

STEP 1: Initialize chaotic system:
Extract the first three characters from key K: K0, K1, K2.
Compute normalized initial conditions:
x⁰ = ASCII(K0)/255, y⁰ = ASCII(K1)/255, z⁰ = ASCII(K2)/255
STEP 2: Generate chaotic sequences:
Compute the total number of iterations T = h · w.
Initialize arrays for chaotic sequences x, y, and z of length T.

9 Set $$x[0] = x_0, y[0] = y_0, z[0] = z_0.$$

for n = 1 to T − 1 do

11 $$x[n] = a_1x[n-1] + a_2y[n-1] + a_3y[n-1]^2$$

12 $y[n] = b_1 - b_2z[n-1]$
13 $z[n] = c \cdot x[n-1]$

# Ensure a1, a2, a3, b1, b2, c are constants suitable for the map.
STEP 3: Normalize sequences for indices:
Normalize sequences to valid ranges for indexing:
xindices = (⌊x · h⌋) mod h
yindices = (⌊y · w⌋) mod w
zvalues = (⌊z · 256⌋) mod 256
STEP 4: Apply pixel-wise transformation:
Reshape zvalues into a 2D array of size (h, w).
foreach i ∈ [0, h − 1], j ∈ [0, w − 1] do
I ′ [i, j] = (I[i, j] + zvalues[i, j]) mod 256
STEP 5: Perform pixel shuffling:
Initialize E as a blank image of size (h, w).
foreach i ∈ [0, h − 1], j ∈ [0, w − 1] do
E[xindices[i], yindices[j]] = I ′ [i, j]
STEP 6: Output encrypted image E.

Key Normalization and Conversion

The feature vector k is normalized and converted into a string representation of the key:

$$k_{\text{normalized}} = \frac{\text{flatten}(k)}{\max(\text{flatten}(k))},$$

$$k{\text{str}} = \sum{i=1}^{n} \text{int}(k_i \cdot 255),$$

where kⁱ represents each component of the normalized feature vector.

Hexadecimal Key Generation

The final cryptographic key is generated by applying SHA-256 hashing to the key string:

$$k{\text{final}} = \text{SHA-256}(k{\text{str}})[0:32],$$

resulting in a 256-bit hexadecimal key.

Chaotic Sequence Generation for Weights and Biases

Chaotic sequences from the logistic map initialize the network's weights and biases:

$$U_{n+1} = \mu U_n (1 - Un), \quad V{n+1} = \mu V_n (1 - V_n),$$

where µ = 3.9999. After sufficient iterations, weights W1, W2, W3, W⁴ and biases b1, b2, b3, b⁴ are sequentially selected from these sequences.

Decryption using 3D Hyperchaotic Map

The decryption process accurately reconstructs the original video frame from its encrypted version using the chaotic key stream and neural network-based techniques. Decryption reverses the encryption steps, ensuring precise recovery. The key steps are:

The same chaotic key stream is regenerated using the decryption key, and hyperchaotic signals are subtracted from the encrypted pixel values to restore the original frame.
Reconstruction is only possible with the correct key, making it essential for decryption.

This approach takes advantage of the sensitivity and unpredictability of hyperchaotic systems, ensuring strong security and making it highly challenging for unauthorized parties to decrypt the video without the key.

Decryption Process

To decrypt the encrypted video, the encryption method based on the chaotic sequence from the 3D hyperchaotic map is reversed. The decryption steps are as follows:

Generate Hyperchaotic Sequences: Using the same initial conditions derived from the key, hyperchaotic sequences xn, yn, and zⁿ are regenerated by applying the 3D hyperchaotic map.
Normalization of Hyperchaotic Sequences: The chaotic sequence zⁿ is normalized to pixel indices as:

$$z_{\text{indices}} = (z_n \cdot 256)$$

This sequence is reshaped into a matrix of size h × w, where h and w represent the height and width of the encrypted image, respectively.

• Pixel Transformation (Reversing Encryption): The chaotic sequence is subtracted from the pixel values of the encrypted image:

$$I{\text{decrypted}}[i, j] = (I{\text{encrypted}}[i, j] - z_{\text{indices}}[i, j]) \mod 256$$

where the encrypted image's pixel value is Iencrypted[i, j], and the decrypted image's pixel value is Idecrypted[i, j].

• Reconstruction of the Original Image: For RGB images, this procedure is applied to every pixel in the image and for every channel. The result is a reconstructed image that accurately matches the original video frame before encryption.

¹ 5.5 Algorithm : Decryption of Image

Input : encrypted image, key

Output: decrypted image

2 STEP 1: Generate initial conditions from key (ASCII values):
³ x⁰ = ASCII(k0), y⁰ = ASCII(k1), z⁰ = ASCII(k2)
⁴ STEP 2: Generate chaotic sequences xn+1, yn+1, zn+1 using the chaotic map:
⁵ xn+1 = a1xⁿ + a2yⁿ + a3y 2 n
⁶ yn+1 = b¹ − b2zⁿ
⁷ zn+1 = cxⁿ
8 STEP 3: Normalize sequences to pixel indices:
⁹ zindices = ⌊zⁿ · 256⌋
10 STEP 4: Reverse pixel transformation:
¹¹ I ′ [i, j] = (E[i, j] − zindices[i, j]) mod 256
¹² STEP 5: Reverse pixel shuffling:
¹³ I[i, j] = I ′ [x −1 indices[i], y⁻¹ indices[j]]
¹⁴ STEP 6: Output the decrypted image I

Figure 19: Pathway depicting the process of extracting key information from video frames for secure encryption and decryption

Security Analysis

The security analysis of video encryption evaluates the effectiveness of encryption techniques in protecting video data from manipulation and unauthorized access. This assessment ensures the confidentiality, integrity, authenticity, and overall security of video content.

Confidentiality: Evaluates how well the encryption algorithm prevents unauthorized decryption attempts, particularly in scenarios where the key remains unknown.
Integrity: Ensures that the encrypted video remains unaltered during transmission or storage, safeguarding it from tampering.
Authenticity: Verifies that the encrypted video originates from a legitimate source and has not been compromised or subjected to fraudulent actions.
Efficiency: Focuses on computational efficiency, ensuring that the encryption and decryption processes do not impose excessive overhead, especially for large video files or real-time streaming applications.

Differential Attack Analysis

Differential attack analysis assesses the strength of an encryption algorithm by examining how small changes in the input affect the output. In this test, the original frame is slightly altered, and the resulting encrypted frame is analyzed to see if it shows any patterns. A strong encryption algorithm should produce entirely different outputs for even minor changes in the input. The frames were encrypted and then decrypted to observe any variations. Metrics like NPCR (Number of Pixels Change Rate) and UACI (Unified Average Changing Intensity) were used to quantify these differences. The results, summarized in the Table2, show high NPCR values ranging from 99.589 to 99.641 and UACI values from 33.893 to 35.011. These high NPCR values indicate that even slight

Figure 20: Encryption of frame 10, 50, 100, 250 in the leftmost verticals. In the second column, the image segmentation via YOLOv10 is performed. In the third column, encryption of the segmented part is performed. In the rightmost column, decryption of the encrypted frame is performed and can be compared with the leftmost vertical as they are identical which shows the efficiency of the video encryption-decryption process.

changes in the input lead to significant variations in the encrypted image, proving the encryption's resistance to differential attacks. The UACI values reflect the intensity changes between the original and decrypted images, further confirming the robustness of the encryption.

Table 2: NPCR (Number of Pixels Change Rate) and UACI (Unified Average Changing Intensity) values for encrypted images shown in Figure 6.

Figure	NPCR	UACI
Figure 6(a)	99.589	33.893
Figure 6(b)	99.633	34.978
Figure 6(c)	99.637	33.938
Figure 6(d)	99.641	35.011
Figure 6(e)	99.634	33.955
Figure 6(f)	99.603	33.893

Table 3: Comparison of various video encryption approaches along with their correlation analysis, NPCR, and UACI values with the proposed method.

Methods	Horizontal	Vertical	Diagonal	NPCR (%)	UACI (%)
Proposed Method	0.0019	-0.0010	-0.0008	99.634	33.955
[45]	0.0025	0.0037	0.0049	98.6327	34.0338
[46]	-0.0012	-0.0002	-0.0006	99.65	50.02
[47]	-0.2533	-0.2557	-0.2126	NA	NA
[48]	0.0073	0.0052	0.0064	99.634	40.955

The Table demonstrates the encryption algorithm's performance through NPCR and UACI metrics. High NPCR values (above 99.5%) and UACI values (near 33%) indicate strong sensitivity to pixel changes and robust encryption capability.The proposed method in Table3 demonstrates strong encryption performance with a low correlation across horizontal, vertical, and diagonal directions, ensuring minimal statistical dependency in encrypted data. It achieves a high NPCR of 99.634% and a UACI of 33.955%, indicating strong resistance to differential attacks. Compared to existing methods, it maintains better unpredictability while offering competitive encryption strength.

Anomaly Detection

Anomaly detection evaluates the effectiveness of encryption and decryption by identifying differences between the original and processed frames. Key metrics such as Mean Squared Error (MSE) and Structural Similarity Index Measure (SSIM) are used. Low MSE values indicate minimal pixel differences, while SSIM values close to 1 demonstrate preserved structural similarity, ensuring encryption robustness and data integrity, see Table 4 for MSE and SSIM values of various frames.

Table 4: Performance of anomaly detection for given frames. The Mean Squared Error (MSE) and Structural Similarity Index (SSIM) data demonstrate the accuracy of the anomaly detection methods when applied to different frames of the given video.

Frame	MSE	SSIM
Frame 50	0.54	0.995620
Frame 80	0.28	0.996712
Frame 100	0.45	0.997120
Frame 120	0.51	0.997548

Information Entropy

Information Entropy measures the uncertainty or randomness of a system. It quantifies how much information is required to represent a system's state. The formula

$$H(X) = -\sum P(x_i)\log_2 P(x_i)$$

calculates entropy, with higher values indicating more unpredictability. In Table 5, the original values represent initial data, while the encrypted values show the data after encryption, introducing uncertainty. The decrypted values indicate the recovered data. The difference between the original and encrypted values highlights the encryption's impact on entropy, while the decrypted values reflect the effectiveness of the decryption process.

Table 5: Information entropy values for frames at different intervals (50, 80, 100, and 120) in their original, encrypted, and decrypted states. The entropy values of the encrypted frames are consistently close to the ideal value of 8, indicating a high level of randomness introduced by the encryption process.

Frame	Original	Encrypted	Decrypted
50	6.89111	7.95549	6.98696
80	6.93351	7.95641	7.02830
100	6.91632	7.96185	7.01676
120	6.98219	7.95392	7.09219

Robustness to noise analysis

Robustness to noise is a key factor in evaluating the strength of an encryption method. This analysis measures how well the encrypted image retains its quality when subjected to noise, such as compression artifacts or transmission errors, is introduced. Two key metrics—Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM)—are used to assess the impact. Higher PSNR values indicate less noise interference, while higher SSIM values suggest better preservation of the image's structure. In the experiments, the encrypted image was subjected to varying noise levels, and the results, as shown in Table 6, indicate that the encryption technique's performance is impacted by noise, with PSNR and SSIM values remaining low across different clipping ranges. This suggests that the encryption is vulnerable to noise disturbances.

Table 6: Robustness of the encryption method under varying clip ranges. The table evaluates the impact on PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index). Each row corresponds to a specific clip range, with consistent PSNR values, indicating minimal degradation in image quality.

Clip Range	PSNR (dB)	SSIM
10–245	27.93	0.0081
20–235	27.93	0.0077
30–225	27.93	0.0071

Performance Comparison of Encryption Methods

Table7 shows a comparison of encryption times between various approaches, with "Proposed Method" showing better results. The encryption time, varying from 2.2 to 3.8 over various approaches from related work. The proposed method takes less time that is 2.275, well surpassing other methods, varying from 2.8 to 3.7. This signifies that the proposed method is computationally efficient in encryption processes in comparison to other alternatives. The information highlights the performance of our method, making it a better solution for applications that prioritize optimal encryption performance.

Table 7: Comparison of encryption time for different methods. The proposed method demonstrates improved efficiency.

Encryption Methods	Encryption Time (sec)
Proposed Method	2.275
[49]	2.881
[50]	3.120
[51]	3.702

6 Conclusion

The paper discussed the prevalence of discrete Lorenz attractors in a three-dimensional discrete sinusoidal map. We examined the existence of infinitely many fixed points of the map, followed by various codimension-one bifurcation scenarios exhibited by the fixed points. We demonstrated the existence of discrete Lorenz attractors of different topologies. Various routes to the formation of discrete Lorenz attractors were illustrated via phase portraits and two-parameter Lyapunov charts. We also discussed the formation of the butterfly homoclinic structure via the computation of onedimensional unstable manifold. Furthermore, we explored the spatiotemporal patterns exhibited by the ring-star network of 3D sinusoidal map. We demonstrated the existence of various types of traveling waves, chimera states, and cluster states. Finally, we considered an application of the hyperchaotic signals generated from the 3D sinusoidal map to a general multimedia encryption specifically video encryption. We show that the proposed algorithm efficiently encrypts and decrypts video frames. In general, this work bridges the theoretical understanding of chaotic dynamics with practical applications in cybersecurity, setting the stage for further research on 3D discrete maps, their bifurcations, and innovative real-world implementations. Further classification of discrete Lorenz attractors as orientable or non-orientable remains to be discussed. A detailed unfolding of the spatiotemporal patterns exhibited by the ring network of 3D maps exhibiting discrete Lorenz attractors still remains to be explored which will be considered in our future work. It would be interesting to explore the dynamics of implicit and semi-implicit maps. Indeed it is of importance to conduct physical experiments for the verification of theoretical observations obtained in this study. Electronic circuits, microcontroller implementation can be performed to account for experimental setup. Subsequent research can investigate the application of dynamic key management[52] to improve the security and performance of the encryption-decryption process. Cryptographic key storage and transmission securely are important to prevent unauthorized access and reduce potential security threats. With the inclusion of strong key distribution mechanisms and encryption protocols, the overall security of the system can be enhanced significantly. In addition, embracing innovative methods like blockchain-based key management[53] or quantum key distribution[54] can also increase the framework's resistance to emerging cyber threats.

Acknowledgements

S.S.M would like to thank Alexey Kazakov for discussions regarding the pseudohyperbolicity of discrete Lorenz attractors. S.S.M also thanks his students Vismaya V S and Anandik N Anand for discussing preliminary results on networks and encryption techniques.

Conflict of interest

The authors declare that they have no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available within the article.

References

[1] Robert M. May. Simple mathematical models with very complicated dynamics. Nature, 261(5560):459–467, June 1976.
[2] Christian Oestreicher. A history of chaos theory. Dialogues in Clinical Neuroscience, 9(3):279–289, September 2007.
[3] James C Robinson. All possible chaotic dynamics can be approximated in three dimensions. Nonlinearity, 11(3):529–545, May 1998.
[4] Alexander Gonchenko, Sergey Gonchenko, Alexey Kazakov, and Dmitry Turaev. Simple scenarios of onset of chaos in three-dimensional maps. International Journal of Bifurcation and Chaos, 24(08):1440005, August 2014.
[5] Sishu Shankar Muni. Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model. Physica Scripta, 98(8):085205, July 2023.
[6] K. Kaneko. Doubling of Torus. Progress of Theoretical Physics, 69(6):1806–1810, 1983.
[7] Z. T. Zhusubaliyev, V. Avrutin, and A. Medvedev. Doubling of a closed invariant curve in an impulsive Goodwin's oscillator with delay. Chaos, Solitons & Fractals, 153:111571, 2021.
[8] A. P. Kuznetsov and Yu. V. Sedova. Anishchenko–astakhov quasiperiodic generator excited by external harmonic force. Technical Physics Letters, 50(2):146–148, February 2024.
[9] Mingwu Li and George Haller. Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, part ii: Bifurcation and quasi-periodic response. Nonlinear Dynamics, 110(2):1045–1080, May 2022.
[10] Sishu Shankar Muni. Ergodic and resonant torus doubling bifurcation in a three-dimensional quadratic map. Nonlinear Dynamics, 112(6):4651–4661, January 2024.
[11] Edward N. Lorenz. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2):130–141, March 1963.
[12] Vitor Ara´ujo and Edvan Trindade. Robust exponential mixing and convergence to equilibrium for singular-hyperbolic attracting sets. Journal of Dynamics and Differential Equations, 35(3):2487–2536, November 2021.
[13] M. Kainov and A. Kazakov. On examples of pseudohyperbolic attractors in flows and maps. Lobachevskii Journal of Mathematics, 42(14):3451–3467, December 2021.
[14] S.V. Gonchenko and A.S. Gonchenko. Discrete Lorenz attractors of new types in threedimensional maps with axial symmetry. Partial Differential Equations in Applied Mathematics, 11:100904, September 2024.
[15] Sergey Gonchenko, Alexander Gonchenko, Alexey Kazakov, and Evgeniya Samylina. On discrete Lorenz-like attractors. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(2):023117, February 2021.
[16] S. V. Gonchenko, I. I. Ovsyannikov, C. Sim´o, and D. Turaev. Three-dimensional H´enonlike maps and wild Lorenz-like attractors. International Journal of Bifurcation and Chaos, 15(11):3493–3508, November 2005.
[17] Leo Torres, Ann S. Blevins, Danielle Bassett, and Tina Eliassi-Rad. The why, how, and when of representations for complex systems. SIAM Review, 63(3):435–485, January 2021.
[18] J.S. Ram, Sishu Shankar. Muni, and I.A. Shepelev. Spatiotemporal patterns in a 2d lattice of Hindmarsh–Rose neurons induced by high-amplitude pulses. Chaos, Solitons & Fractals, 189:115613, December 2024.
[19] O E Omel'chenko. The mathematics behind chimera states. Nonlinearity, 31(5):R121–R164, April 2018.
[20] Sindre W Haugland. The changing notion of chimera states, a critical review. Journal of Physics: Complexity, 2(3):032001, June 2021.
[21] Tera A. Glaze and Sonya Bahar. Neural synchronization, chimera states and sleep asymmetry. Frontiers in Network Physiology, 1, October 2021.
[22] Sishu Shankar Muni and Astero Provata. Chimera states in ring–star network of Chua circuits. Nonlinear Dynamics, 101(4):2509–2521, September 2020.
[23] Deepti Dhingra and Mohit Dua. A chaos-based novel approach to video encryption using dynamic s-box. Multimedia Tools and Applications, 83(1):1693–1723, May 2023.
[24] SistlaVasundhara Devi and Harika Devi Kotha. AES encryption and decryption standards. Journal of Physics: Conference Series, 1228(1):012006, May 2019.
[25] Ratnadewi, Roy Pramono Adhie, Yonatan Hutama, A. Saleh Ahmar, and M I Setiawan. Implementation cryptography data encryption standard (des) and triple data encryption standard (3des) method in communication system based near field communication (nfc). Journal of Physics: Conference Series, 954:012009, January 2018.
[26] E. Geo Francis and S. Sheeja. Chaotic Resilience: Enhancing IoT Security Through Dynamic Data Encryption, page 331–344. Springer Nature Singapore, October 2024.
[27] Sishu Shankar Muni. Pathways to hyperchaos in a three-dimensional quadratic map. The European Physical Journal Plus, 139(7), July 2024.
[28] Efrosiniia Karatetskaia, Aikan Shykhmamedov, and Alexey Kazakov. Shilnikov attractors in three-dimensional orientation-reversing maps. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(1), January 2021.
[29] A. S. Gonchenko, S. V. Gonchenko, and L. P. Shilnikov. Towards scenarios of chaos appearance in three-dimensional maps. Nelineinaya Dinamika, 8:3–28, 2012.
[30] Sishu Shankar Muni, Hammed Olawale Fatoyinbo, and Indranil Ghosh. Dynamical effects of electromagnetic flux on Chialvo neuron map: Nodal and network behaviors. International Journal of Bifurcation and Chaos, 32(09), July 2022.
[31] Alan Wolf, Jack B. Swift, Harry L. Swinney, and John A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317, July 1985.
[32] James L. Kaplan and James A. Yorke. Chaotic behavior of multidimensional difference equations, page 204–227. Springer Berlin Heidelberg, 1979.
[33] Gonzalo Marcelo Ram´ırez-Avila, Sishu Shankar Muni, and Tomasz Kapitaniak. Unfolding the ´ distribution of periodicity regions and diversity of chaotic attractors in the Chialvo neuron map. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(8), August 2024.
[34] Elhadj Zeraoulia and Julien Clinton Sprott. Robust Chaos and Its Applications. World Scientific, October 2011.
[35] A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, and A. D. Kozlov. Elements of contemporary theory of dynamical chaos: A tutorial. part i. pseudohyperbolic attractors. International Journal of Bifurcation and Chaos, 28(11):1830036, October 2018.
[36] Sungju Moon, Jong-Jin Baik, and Jaemyeong Mango Seo. Chaos synchronization in generalized lorenz systems and an application to image encryption. Communications in Nonlinear Science and Numerical Simulation, 96:105708, May 2021.
[37] Jiayue Liu, Jianguo Zhang, and Yuncai Wang. Secure communication via chaotic synchronization based on reservoir computing. IEEE Transactions on Neural Networks and Learning Systems, 35(1):285–299, 2024.
[38] Zilong Tan, Yang liu, Jiayue Sun, Huaguang Zhang, and Xiangpeng Xie. Chaos synchronization control for stochastic nonlinear systems of interior pmsms based on fixed-time stability theorem. Applied Mathematics and Computation, 430:127115, October 2022.
[39] Guessas Laarem. A new 4-d hyper chaotic system generated from the 3-d r¨osslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it's fractional order model and chaos synchronization using optimized fractional order sliding mode control. Chaos, Solitons & Fractals, 152:111437, November 2021.
[40] Sishu Shankar Muni, Karthikeyan Rajagopal, Anitha Karthikeyan, and Sundaram Arun. Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling. Chaos, Solitons & Fractals, 155:111759, February 2022.
[41] Sishu Shankar Muni, Zeric Njitacke, Cyrille Feudjio, Th´eophile Fozin, and Jan Awrejcewicz. Route to chaos and chimera states in a network of memristive Hindmarsh-Rose neurons model with external excitation. Chaos Theory and Applications, 4(3):119–127, November 2022.
[42] Zeric Tabekoueng Njitacke, Sishu Shankar Muni, Soumyajit Seth, Jan Awrejcewicz, and Jacques Kengne. Complex dynamics of a heterogeneous network of Hindmarsh-Rose neurons. Physica Scripta, 98(4):045210, March 2023.
[43] Aleksandra V. Tutueva, Erivelton G. Nepomuceno, Artur I. Karimov, Valery S. Andreev, and Denis N. Butusov. Adaptive chaotic maps and their application to pseudo-random numbers generation. Chaos, Solitons & Fractals, 133:109615, April 2020.
[44] Erivelton G. Nepomuceno, Arthur M. Lima, Janier Arias-Garc´ıa, Matjaˇz Perc, and Robert Repnik. Minimal digital chaotic system. Chaos, Solitons & Fractals, 120:62–66, March 2019.
[45] Hassan Elkamchouchi, Wessam M Salama, and Yasmine Abouelseoud. New video encryption schemes based on chaotic maps. IET Image Processing, 14(2):397–406, 2020.
[46] Deepti Dhingra and Mohit Dua. A chaos-based novel approach to video encryption using dynamic s-box. Multimedia Tools and Applications, 83(1):1693–1723, 2024.
[47] Ahmed I Sallam, El-Sayed M El-Rabaie, and Osama S Faragallah. Efficient hevc selective stream encryption using chaotic logistic map. Multimedia Systems, 24:419–437, 2018.
[48] Omar El Ogri, Hicham Karmouni, Mhamed Sayyouri, and Hassan Qjidaa. A new image/video encryption scheme based on fractional discrete tchebichef transform and singular value decomposition. Multimedia Tools and Applications, 82(22):33465–33497, 2023.
[49] Qin Liang and Congxu Zhu. A new one-dimensional chaotic map for image encryption scheme based on random dna coding. Optics & Laser Technology, 160:109033, 2023.
[50] Jan Sher Khan and Jawad Ahmad. Chaos based efficient selective image encryption. Multidimensional Systems and Signal Processing, 30(2):943–961, May 2018.
[51] Hongyan Zang, Mengdan Tai, and Xinyuan Wei. Image encryption schemes based on a class of uniformly distributed chaotic systems. Mathematics, 10(7):1027, 2022.
[52] Uma Meena and Promila Sharma. Secret dynamic key authentication and decision trust secure routing framework for internet of things based wsn. Wireless Personal Communications, 125(2):1753–1781, 2022.
[53] Jiaxing Li, Jigang Wu, Long Chen, Jin Li, and Siew-Kei Lam. Blockchain-based secure key management for mobile edge computing. IEEE Transactions on Mobile Computing, 22(1):100– 114, 2021.
[54] Yuan Cao, Yongli Zhao, Qin Wang, Jie Zhang, Soon Xin Ng, and Lajos Hanzo. The evolution of quantum key distribution networks: On the road to the qinternet. IEEE Communications Surveys & Tutorials, 24(2):839–894, 2022.

munidiscretelorenzattractors3d2025.md 92 KB Cronologia Originale