Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

Kuznetsov, Nikolay V.; Леонов, Г. А.; Mokaev, Timur N.; Prasad, Awadhesh; Shrimali, Manish Dev

doi:10.1007/s11071-018-4054-z

Cited by 151 publications

(77 citation statements)

References 117 publications

(175 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We numerically approximate the finite-time Lyapunov exponents and finite-time Lyapunov dimension (see corresponding definitions, e.g. in [14,26]). Integration with t > T 1 ≈ 1.8 · 10 4 leads to the collapse of the "attractor ", i.e.…”

Section: Finite-time Lyapunov Dimension Of a Transient Chaotic Setmentioning

confidence: 99%

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

Kuznetsov

Mokaev

2019

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rössler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rössler system. Using the example of the Vallis system describing the El Ninõ-Southern Oscillation it is demonstrated an analytical approach for localization of self-excited and hidden attractors, which allows to obtain the exact formulas or estimates of their Lyapunov dimensions.

show abstract

Section: Finite-time Lyapunov Dimension Of a Transient Chaotic Setmentioning

confidence: 99%

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

Kuznetsov

Mokaev

2019

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

show abstract

“…e Lyapunov exponents are calculated along the trajectories of system (2) by using the Wolf method [42]. We also can further consider the Lyapunov exponents by using the method proposed in literature [43]. Here, the Lyapunov exponents are used to distinguish the chaotic, periodic, and stable states of system (2).…”

Section: Coexisting Hidden Attractorsmentioning

confidence: 99%

Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

et al. 2020

View full text Add to dashboard Cite

is paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. e circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method.

show abstract

“…To identify whether the chaotic attractors shown in Figure 2 are self-excited or hidden attractors, [59][60][61][62][63][64][65][66] we need to check the stability of the origin with system parameters varying, and all attractors are hidden if all roots of the characteristic equation corresponding to the matrix in (5) have negative real parts, otherwise the attractors are nonhidden. 63 It is easy to see that the system (3) has the two-to-eight-wing self-excited chaotic attractors under the given system parameters.…”

Section: The Eight-wing Chaotic Attractormentioning

confidence: 99%

Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

Zhong

Peng

Shahidehpour

2019

Circuit Theory & Apps

View full text Add to dashboard Cite

This paper constructs a three-dimensional (3D) memristor-based system for creating multiwing chaotic attractors. A second-degree polynomial memristance function and a sixth-order exponent internal state memristor function with one parameter are employed, and the complexity of attractors is increased. A detailed analysis on dynamical behaviors of the proposed system are described, such as the bifurcation diagrams, finite-time local Lyapunov exponents, time series, phase portraits, and Poincaré maps. By adjusting the design parameters, the system displays two-to-eight-wing chaotic attractors, especially the five-wing and seven-wing attractors, which have never been found in the known systems. Further, we provide the calculation formula of the number of wings in the system, discuss the distribution of the involving inner holes on the plane, and design an electronic circuit to realize the proposed system. The experimental results of the circuit implementation agree with the numerical simulations on Matlab well. It indicates the potential engineering applications for various chaos-based information systems. KEYWORDSchaotic system, circuit implementation, Lyapunov exponents, memristor, the number of the wings INTRODUCTIONAfter the first solid state realization of a memristor was successfully fabricated in 2008 by researchers at HP, 1 memristor, the new two-terminal circuit element postulated by Chua 2 in 1971 and classified by Chua and Kang 3 in 1976, has been well known and studied thanks to its potential applications in neural networks, novel storage medium, artificial intelligence, and a wide range of other chaotic circuit. 4-12 Since the mathematical model involving chaos was first suggested by Lorenz 13 in 1963 and the term chaos was first used by Li and Yorke 14 in 1975, the design of new chaotic systems with complex topological structures are attracting more and more interest in the research. [15][16][17][18][19][20][21][22][23][24][25][26] In 1993, Stone and Miranda observed first multiwing chaotic attractor from a proto-Lorenz system. 27 Since then, people has found that a wide variety of memristor-based systems can exhibit the complicated dynamics, particularly in birth of multiwing attractors. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] This problem is interesting, especially in the case of a simple, low-dimensional system with less equilibrium points, more wings, and more coexisting attractors. The multiwing or multiscroll 45-53 chaotic attractors with more complex is of the meaningful applications in many fields, particularly, including chaotic broad band radio, encryption, and secure communication. 54 In the reported multiwing systems, most memristor models are based on 686

show abstract

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

Cited by 151 publications

References 117 publications

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

Contact Info

Product

Resources

About