ON ENTROPY OF CHUA'S CIRCUITS

2005 ◽  
Vol 15 (05) ◽  
pp. 1823-1828 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Computer Simulation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Topological Entropy ◽  
Poincaré Map ◽  
Chua’S Circuit ◽  
Poincare Map ◽  
Chua's Circuit

In this paper we revisit the well-known Chua's circuit and give a discussion on entropy of this circuit. We present a formula for the topological entropy of a Chua's circuit in terms of the Poincaré map derived from the ordinary differential equations of this Chua's circuit by computer simulation arguments.

Positive Topological Entropy of Chua's Circuit: A Computer Assisted Proof

1997 ◽  
Vol 07 (02) ◽  
pp. 331-349 ◽  
Author(s):  
Zbigniew Galias
Keyword(s):  
Existence Theorem ◽  
Topological Entropy ◽  
Poincaré Map ◽  
Periodic Points ◽  
Computer Assisted ◽  
Chua’S Circuit ◽  
Poincare Map ◽  
Chua's Circuit ◽  
Positive Topological Entropy ◽  
Fixed Point Index Theory

In this paper we describe a technique for proving that a particular system is chaotic in the topological sense, i.e. that it has positive topological entropy. This technique combines existence results based on the fixed point index theory and computer-assisted computations, necessary to verify the assumptions of the existence theorem. First we present an existence theorem for periodic points of maps, which could be appropriately homotoped with the deformed horseshoe map. As an example we consider Chua's circuit. We prove the existence of infinitely many periodic points of Poincaré map associated with Chua's Circuit. We also show how to use this result to prove that the topological entropy of the Poincaré map and also of the continuous system is positive.

CHUA’S CIRCUIT: RIGOROUS RESULTS AND FUTURE PROBLEMS

1994 ◽  
Vol 04 (03) ◽  
pp. 489-519 ◽  
Author(s):  
LEONID P. SHIL’NIKOV
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Dynamics ◽  
Multidimensional Systems ◽  
Chua’S Circuit ◽  
Rigorous Results ◽  
Chua's Circuit ◽  
Mathematical Problems ◽  
Homoclinic Tangencies

Mathematical problems arising from the study of complex dynamics in Chua’s circuit are discussed. An explanation of the extreme complexity of the structure of attractors of Chua’s circuit is given. This explanation is based upon recent results on systems with homoclinic tangencies. A number of new dynamical phenomena is predicted for those generalizations of Chua’s circuits which are described by multidimensional systems of ordinary differential equations.

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

A HORSESHOE IN A CELLULAR NEURAL NETWORK OF FOUR-DIMENSIONAL AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 17 (09) ◽  
pp. 3211-3218 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Poincaré Map ◽  
Cellular Neural Network ◽  
Poincare Map ◽  
Invariant Subset ◽  
The Neural Network

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

Computing the Number of Fixed Joints on Poincare Map in Nonlinear Mathieu Equation

1993 ◽  
Author(s):  
Hisato Fujisaka ◽  
Chikara Sato
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Numerical Method ◽  
Topological Degree ◽  
Poincaré Map ◽  
Mathieu Equation ◽  
Time Varying ◽  
Poincare Map ◽  
Newton's Iteration ◽  
Combined Use

Abstract A numerical method is presented to compute the number of fixed points of Poincare maps in ordinary differential equations including time varying equations. The method’s fundamental is to construct a map whose topological degree equals to the number of fixed points of a Poincare map on a given domain of Poincare section. Consequently, the computation procedure is simply computing the topological degree of the map. The combined use of this method and Newton’s iteration gives the locations of all the fixed points in the domain.

scholarly journals Topological Entropy of One Type of Nonoriented Lorenz-Type Maps

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Guo Feng
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Poincaré Map ◽  
Geometric Model ◽  
Dynamical Property ◽  
High Dimensional ◽  
Poincare Map ◽  
One Dimensional

Constructing a Poincaré map is a method that is often used to study high-dimensional dynamical systems. In this paper, a geometric model of nonoriented Lorenz-type attractor is studied using this method, and its dynamical property is described. The topological entropy of one-dimensional nonoriented Lorenz-type maps is also computed in terms of their kneading sequences.

FREQUENCY DOMAIN METHOD FOR THE DICHOTOMY OF MODIFIED CHUA'S EQUATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 2485-2505 ◽  
Author(s):  
ZHISHENG DUAN ◽  
JIN-ZHI WANG ◽  
LIN HUANG
Keyword(s):  
Computer Simulation ◽  
Limit Cycles ◽  
Chaotic Attractors ◽  
Output Coupling ◽  
Frequency Domain Method ◽  
Chua’S Circuit ◽  
Single Variable ◽  
Nonlinear Functions ◽  
Chua's Circuit ◽  
Input And Output

On condition of dichotomy, it is pointed out that in Lorenz and a kind of Rössler-like system chaotic attractors or limit cycles will disappear if nonlinearity of the product of two variables is replaced by some single variable nonlinearity, for example, nonlinearity of Chua's circuit. Furthermore, an extended Chua's circuit with two nonlinear functions is presented. By computer simulation it is shown that oscillating phenomena in the extended Chua's circuit are richer than the single Chua's circuit. The corresponding extension for smooth Chua's equations is also considered. The effects of input and output coupling are analyzed for the extended Chua's circuit.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

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