Paper Info
Title
Existence Of Chaos In The Chen System With Linear Time-Delay Feedback
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.
Year
DOI
Venue
2019
10.1142/S0218127419501141
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Keywords
Field
DocType
Linear time-delay feedback, Shil'nikov-type lemma, homoclinic orbit, single-scroll hyperchaotic attractor
Applied mathematics,Oscillation,Homoclinic orbit,Chen,Mathematical analysis,Chaotic,Time complexity,Lemma (mathematics),Mathematics
Journal
Volume
Issue
ISSN
29
9
0218-1274
Citations 
PageRank 
References 
0
0.34
0
Authors
3
Name
Order
Citations
PageRank
Kun Tian101.01
Hai-Peng Ren2265.41
Celso Grebogi312734.41