Existence of chaos in the Chen system with linear time-delay feedback

Kun Tian, Hai Peng Ren (Corresponding Author), Celso Grebogi

Research output: Contribution to journalArticle

Abstract

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.
Original languageEnglish
Article number1950114
JournalInternational Journal of Bifurcation and Chaos
Volume29
Issue number9
Early online date27 Aug 2019
DOIs
Publication statusPublished - Aug 2019

Fingerprint

Chen System
Homoclinic Orbit
Chaos theory
Linear Time
Time Delay
Time delay
Chaos
Orbits
Feedback
Trajectories
Trajectory
Projection Method
Eigenvalue
Attractor
Lemma
Heteroclinic Orbit
Local Time
Autonomous Systems
Lyapunov Exponent
Experiment

Keywords

  • linear time-delay feedback
  • Shil’nikov-type lemma
  • homoclinic orbit
  • single-scroll hyperchaotic attractor
  • SILNIKOV-TYPE ORBITS
  • LORENZ
  • HIDDEN ATTRACTOR
  • OSCILLATORS
  • HETEROCLINIC ORBITS
  • SHILNIKOV TYPE
  • Shil'nikov-type lemma
  • HOMOCLINIC ORBITS
  • 2ND-ORDER HAMILTONIAN-SYSTEMS
  • Linear time-delay feedback
  • CIRCUIT

Cite this

Existence of chaos in the Chen system with linear time-delay feedback. / Tian, Kun ; Ren, Hai Peng (Corresponding Author); Grebogi, Celso.

In: International Journal of Bifurcation and Chaos, Vol. 29, No. 9, 1950114, 08.2019.

Research output: Contribution to journalArticle

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