Minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits

Huaraca, Walter; Mendoza, Valentín

doi:10.1016/j.physd.2015.10.009

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…Four scenarios are presented in this research study in which the systems are assumed to be subject to small non-autonomous perturbations, yielding four new bifurcation diagrams. Then, in [ 63 ], minimal topological chaos is discovered with respect to finite sets of homoclinic and periodic orbits.…”

Section: Related Workmentioning

confidence: 99%

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

Azar

Serrano

Zhu

et al. 2021

Entropy

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In this paper, the robust stabilization and synchronization of a novel chaotic system are presented. First, a novel chaotic system is presented in which this system is realized by implementing a sigmoidal function to generate the chaotic behavior of this analyzed system. A bifurcation analysis is provided in which by varying three parameters of this chaotic system, the respective bifurcations plots are generated and evinced to analyze and verify when this system is in the stability region or in a chaotic regimen. Then, a robust controller is designed to drive the system variables from the chaotic regimen to stability so that these variables reach the equilibrium point in finite time. The robust controller is obtained by selecting an appropriate robust control Lyapunov function to obtain the resulting control law. For synchronization purposes, the novel chaotic system designed in this study is used as a drive and response system, considering that the error variable is implemented in a robust control Lyapunov function to drive this error variable to zero in finite time. In the control law design for stabilization and synchronization purposes, an extra state is provided to ensure that the saturated input sector condition must be mathematically tractable. A numerical experiment and simulation results are evinced, along with the respective discussion and conclusion.

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Section: Related Workmentioning

confidence: 99%

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

Azar

Serrano

Zhu

et al. 2021

Entropy

View full text Add to dashboard Cite

show abstract

“…Chen et al improved the perturbation method based on nonlinear time transformation to achieve explicit homoclinic solutions, but it can only deal with the powerlaw nonlinear oscillator [Chen et al, 2017]. By finding the set of pruning domains, Huaraca & Mendoza identified a finite set of homoclinic orbits, if there exists a pronged singularity without rotation [Huaraca & Mendoza, 2016]. Particularly, references [Chen et al, 2017] and [Huaraca & Mendoza, 2016] gave an invariant formal expression, but failed to analytically describe the homoclinic motion for a general system.…”

Section: Introductionmentioning

confidence: 99%

“…By finding the set of pruning domains, Huaraca & Mendoza identified a finite set of homoclinic orbits, if there exists a pronged singularity without rotation [Huaraca & Mendoza, 2016]. Particularly, references [Chen et al, 2017] and [Huaraca & Mendoza, 2016] gave an invariant formal expression, but failed to analytically describe the homoclinic motion for a general system. Lin et al used Lyapunov-Schmidt reduction and exponential dichotomies to derive general conditions under which the perturbed system have transverse homoclinic solutions, but it did not apply to infinite dimensional system [Lin et al, 2015].…”

Section: Introductionmentioning

confidence: 99%

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

Tian

Ren

Grebogi

2019

Int. J. Bifurcation Chaos

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It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

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Markov shifts and topological entropy of families of homoclinic tangles

Garcia

Mendoza

2019

Journal of Differential Equations

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Minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits

Cited by 4 publications

References 40 publications

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

Markov shifts and topological entropy of families of homoclinic tangles

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