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

3 Citations (Scopus)

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

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

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