Labyrinth chaos: Revisiting the elegant, chaotic, and hyperchaotic walks

Basios, Vasileios; Antonopoulos, Chris G.; Latifi, A.

doi:10.1063/5.0022253

Cited by 10 publications

(3 citation statements)

References 43 publications

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“…Ref. [14] designed a feedback controller for a popular chaotic system, introduced by Thomas [38] (see also the recent review [39]), so that the closed-loop system is 2-contracting. Here we show how our results can be used to analyse the closed-loop system with an added perturbation modeled as a decaying exponential.…”

Section: Proof Of Theoremmentioning

confidence: 99%

A sufficient condition for $k$-contraction of the series connection of two systems

Ofir,

Margaliot,

Levron

et al. 2021

Preprint

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The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system.A generalization of contracting systems is k-contracting systems, where k ∈ {1, . . . , n}. The flow of such systems contracts the volume of k-dimensional parallelotopes at an exponential rate, and in particular they reduce to contracting systems when k = 1. It was shown by Muldowney and Li that time-invariant 2-contracting systems have a well-ordered asymptotic behaviour: all bounded trajectories converge to the set of equilibria.Here, we derive a sufficient condition guaranteeing that the system obtained from the series interconnection of two sub-systems is k-contracting. This is based on a new formula for the kth multiplicative and additive compounds of a block-diagonal matrix, which may be of independent interest. As an application, we find conditions guaranteeing that 2-contracting systems with an exponentially decaying input retain the well-ordered behaviour of time-invariant 2-contracting systems.

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Section: Proof Of Theoremmentioning

confidence: 99%

A sufficient condition for $k$-contraction of the series connection of two systems

Ofir,

Margaliot,

Levron

et al. 2021

Preprint

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“…Example 10. A popular example for a chaotic system, introduced by Thomas [47] (see also the recent review [4]), is f = trace(J f (x)) = −3b. This implies that the system is 3 contracting (that is, dissipative), w.r.t.…”

Section: Matrix Measures Of the α Additive Compoundmentioning

confidence: 99%

Generalization of the multiplicative and additive compounds of square matrices and contraction in the Hausdorff dimension

Wu,

Pines,

Margaliot

et al. 2020

Preprint

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The k multiplicative and k additive compounds of a matrix play an important role in geometry, multi-linear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for integer values of k. Here, we introduce generalizations called the α multiplicative and α additive compounds of a square matrix, with α real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterlé Theorem. This leads to a generalization of contracting systems to α contracting systems, with α real. Roughly speaking, the dynamics of such systems contracts any set with Hausdorff dimension larger than α. For α = 1 they reduce to standard contracting systems.

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“…He used the cubic and sine nonlinearities and showed the pathway to chaos using phase portraits with multistability. Later on, many others were introduced [21][22][23][24][25]. In most of these systems, studies were focused on the stability of equilibria, the diferent routes to chaos, and multistability.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Analysis and Offset Boosting in a 4-Dimensional Quintic Chaotic Oscillator with Circulant Connection of Space Variables

Sandrine,

Jacques

2023

Complexity

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In recent years, much energy has been devoted to the study of chaotic models with specific features particularly those with cyclic connection of the variables. Previous ones provide multistability, amplitude control, and so on. Concerning the first phenomenon, models with ring connection of variables presented a coexistence of up to twelve disconnected attractors. In order to emphasize the complexity of circulant chaotic oscillators and their use in the engineering domain, a quintic chaotic model with cyclic connection of variables is considered and studied, which has complex equilibria located on the line x = y = z = w . Therefore, it experiences, amongst other, the phenomenon of offset boosting obtained by introducing four constants into the equations of the model, which has not be done in the past. Multistability is also revealed and the coexistence of eight and sixteen attractors is demonstrated using phase portraits. The system’s dynamics has been investigated considering its two parameters. Nonlinear dynamical tools such as bifurcation diagrams, phase portraits, time evolutions, two-parameter diagram, and Lyapunov exponents help to highlight the important phenomena encountered. The numerical results are confirmed using PSpice and particularly show the double-band chaotic attractor. Moreover, total amplitude control (TAC) is shown, proving that our oscillator can be used as an attenuator or amplifier in the engineering domain. The method of adaptive synchronization has been applied to the considered oscillator to emphasize the possible implication into the secure of communication systems.

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Labyrinth chaos: Revisiting the elegant, chaotic, and hyperchaotic walks

Cited by 10 publications

References 43 publications

A sufficient condition for $k$-contraction of the series connection of two systems

A sufficient condition for $k$-contraction of the series connection of two systems

Generalization of the multiplicative and additive compounds of square matrices and contraction in the Hausdorff dimension

Dynamical Analysis and Offset Boosting in a 4-Dimensional Quintic Chaotic Oscillator with Circulant Connection of Space Variables

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