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Numerical Visualization of Attractors: Self-Exciting and Hidden Attractors
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Nikolay Kuznetsov, Gennady Leonov
For a hidden attractor, its basin of attraction is not connected with any equilibria. For example, the hidden attractors are attractors in systems with no equilibria or with only one equilibrium, which is stable (a special case of multistability: coexistence of attractors in multistable systems). Remark that multistability may cause inconvenience in various practical applications (see, e.g., discussion on the problems related to the synchronization of coupled multistable systems in References 23, 24, 33, and 57). Coexisting self-excited attractors can be found by the standard computational procedure while there is no effective regular way to predict the existence or coexistence of hidden attractors in a system.
Hidden and self-excited attractors in Chua circuit: synchronization and SPICE simulation
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
M. A. Kiseleva, E. V. Kudryashova, N. V. Kuznetsov, O. A. Kuznetsova, G. A. Leonov, M. V. Yuldashev, R. V. Yuldashev
Until recently there had been found Chua attractors, which are excited from unstable equilibria only and, thus, can be easily computed (see, e.g. a gallery of Chua attractors in [9]). Note that L. Chua [8], analyzing various cases of attractors in Chua’s circuit, did not admit the existence of attractors of another type — so called hidden attractors, being discovered later in his circuits. An attractor is called a self-excited attractor if its basin of attraction intersects an arbitrarily small open neighborhood of equilibrium, otherwise it is called a hidden attractor [10–13]. Hidden attractor has basin of attraction which does not overlap with an arbitrarily small vicinity of equilibria.
Dynamics and circuit of a chaotic system with a curve of equilibrium points
Published in International Journal of Electronics, 2018
Viet–Thanh Pham, Christos Volos, Tomasz Kapitaniak, Sajad Jafari, Xiong Wang
Researchers have shown an increasing interest in proposing new chaotic systems with new features. Some examples are chaotic systems with no equilibrium points (Jafari, Sprott, & Golpayegani, 2013; Wei, 2011), only stable equilibria (Molaie, Jafari, Sprott, & Golpayegani, 2013; Wei & Pehlivan, 2012), surfaces of equilibria (Jafari, Sprott, & Molaie, 2016), non-hyperbolic equilibria (Wei, Sprott, & Chen, 2015), multi-scroll attractors (Ma, Wu, Chu, & Zhang, 2014; Tlelo-Cuautle, Rangel-Magdaleno, Pano-Azucena, Obeso-Rodelo, & Nunez-Perez, 2012), and with extreme multistability (Bao, Bao, Wang, Chen, & Xu, 2017; Bao et al., 2017, 2016; Bao, Xu, Bao, & Chen, 2016). In the last four years, finding new chaotic systems with an infinite number of equilibrium points is an attractive topic (Chen & Yang, 2015; Jafari & Sprott, 2013; Li, Sprott, & Thio, 2014; C. Li & Sprott, 2014; C. Li, Sprott, Yuan, & Li, 2015; Q. Li, Hu, Tang, & Zeng, 2014; P.; Zhou & Yang, 2014). As illustrated in Figure 1, different chaotic systems with line equilibrium, open curve equilibrium, and closed curve equilibrium have been discovered. Some typical systems are listed as simple chaotic flows with a line equilibrium (Jafari & Sprott, 2013; C. Li & Sprott, 2014; C. Li et al., 2015), chaotic systems with circular equilibrium (Gotthans & Petržela, 2015), or chaotic flow with square equilibrium (Gotthans, Sprott, & Petrzela, 2016). It is widely recognised that most of such systems with infinite equilibria are classified as systems with hidden attractor from a computational point of view (Dudkowski et al., 2016; Leonov & Kuznetsov, 2013; Leonov, Kuznetsov, & Vagaitsev, 2011, 2012). Despite the fact that there are intersections of the basin of attraction and equilibrium points in such systems, the rest of uncountably equilibrium points is outside the basin of the chaotic attractor (Dudkowski et al., 2016; Jafari & Sprott, 2013).
A novel Sine–Tangent–Sine chaotic map and dynamic S-box-based video encryption scheme
Published in The Imaging Science Journal, 2023
In recent years, many works have been done to address the weaknesses of existing chaotic maps using trigonometric functions. Zhou et al. [39] designed a three-dimensional (3D) hyperchaotic map by using a sinusoidal function with three positive Lyapunov coefficients. The proposed 3D chaotic map decreased the correlation between the pixels and increased the entropy when applied to image encryption. Li et al. [40] applied sinusoidal functions to a discrete map for hyperchaos generation and attractor self-reproduction. The constructed map exhibits compound lattice dynamics with controllable and conditional symmetry. The authors generated a pseudorandom number generator using their hyperchaotic map. Islam et al. [41] found a hidden attractor in a 3D chaotic system with an amplitude and frequency controller. To validate the efficiency of the chaotic system the author designed a circuit simulation and also applied the map in image encryption. Kumar and Dua [42] used Sine–Cosine (SC) Map, Logistic Sine Cosine (LSC) map and double DNA operation to encrypt audio. The author claimed that the proposed scheme is secure and resistant to various threats. Hua et al. [43] proposed two-dimensional Sine Logistic modulation map (2D-SLMM) chaotic map. The map was derived from two 1D maps, Logistic and Sine maps. The results presented in the work showed that the map has a wider chaotic range and better chaotic properties than existing maps. Pankaj and Dua [9] proposed a novel chaotic map named Tangent over Cosine Cosine (ToCC). The results showed that the ToCC map is less predictable and exhibits complex chaotic behaviour. Hua et al. [44] designed a 2D modular chaotification system (2D-MCS) to increase the chaotic complexity of any 2D chaotic map. In this work, the chaotic behaviour of three maps: Henon map, the Zeraoulia-Sprott map, and Duffing map was improved using 2D-MCS. In [8], the authors proposed a sine chaotification model (SCM) in which a sine function was applied as a nonlinear chaotification transform to an existing 1-D chaotic map to enhance its chaotic properties. The results presented in this paper verified the effectiveness and efficiency of SCM. Belazi et al. [20] proposed a one-dimensional improved Sine–Tangent chaotic map (IST) map with a combination of a sine map, a tangent function and a Chebyshev polynomial. In the study, the IST map was assessed by various measures such as bifurcation diagram, Lyapunov exponent and trajectory analysis, and it outperformed various existing maps. Zhu et al. [45] derived a 2D (two-dimensional) Logistic-modulated-Sine-coupling-Logistic chaotic map (LSMCL) from 1D chaotic maps, Sine map and Logistic map. In the work, LSMCL was compared with the 2D Logistic map, 2D-SLMM and Hénon map, and it showed complex chaotic behaviour and a wider chaotic range than the other three chaotic maps. In [46], the authors proposed a general framework cosine-transform-based chaotic system (CTBCS) that can generate new chaotic maps using any two existing chaotic maps as seed maps. The work generated three chaotic maps that exhibit complex behaviour.