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

2019 ◽  
Vol 29 (09) ◽  
pp. 1950114 ◽  
Author(s):  
Kun Tian ◽  
Hai-Peng Ren ◽  
Celso Grebogi
Keyword(s):  
Time Delay ◽  
Projection Method ◽  
Homoclinic Orbit ◽  
Linear Time ◽  
Autonomous Systems ◽  
Technical Problem ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Real Eigenvalue ◽  
Chen System

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.

CLASSIFICATION OF CHAOS IN 3-D AUTONOMOUS QUADRATIC SYSTEMS-I: BASIC FRAMEWORK AND METHODS

2006 ◽  
Vol 16 (09) ◽  
pp. 2459-2479 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
GUANGRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Classification Problem ◽  
Heteroclinic Orbits ◽  
Orbit Type ◽  
Quadratic Systems ◽  
Hybrid Type

This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

AN EXTENDED ŠIL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS

2009 ◽  
Vol 19 (05) ◽  
pp. 1679-1693 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU ◽  
GUANRONG CHEN
Keyword(s):  
Theoretical Analysis ◽  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
The Other ◽  
Complex Numbers ◽  
New Chaotic System ◽  
Focus Type

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

CHEN'S ATTRACTOR EXISTS

2004 ◽  
Vol 14 (09) ◽  
pp. 3167-3177 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
YUN TANG ◽  
GUANRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Series Expansions ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši'lnikov type that connects two nontrivial singular points. The Ši'lnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen's attractor.

Complementary Sensitivity Design of PID Controllers for Arbitrary-Order Transfer Functions With Time Delay

2008 ◽  
Author(s):  
Tooran Emami ◽  
John M. Watkins
Keyword(s):  
Time Delay ◽  
Transfer Functions ◽  
Linear Time ◽  
Order System ◽  
Pid Controllers ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Single Output ◽  
Plant Transfer ◽  
Time Invariant

A graphical technique for finding all proportional integral derivative (PID) controllers that stabilize a given single-input-single-output (SISO) linear time-invariant (LTI) system of any order system with time delay has been solved. In this paper a method is introduced that finds all PID controllers that also satisfy an H∞ complementary sensitivity constraint. This problem can be solved by finding all PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. A key advantage of this procedure is the fact that it does not require the plant transfer function, only its frequency response.

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