KNOTTED PERIODIC ORBITS IN CHUA’S CIRCUIT

1994 ◽  
Vol 04 (03) ◽  
pp. 609-621
Author(s):  
Lj. KOCAREV ◽  
Z. TASEV ◽  
D. DIMOVSKI ◽  
L.O. CHUA
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Topological Properties ◽  
Chua's Circuit ◽  
Spiral Type ◽  
Lower Period

Induced templates for two members of Chua’s attractors: spiral-type and double-scroll chaotic attractors are computed using the orbits of lower period. The template describes the topological properties of periodic orbits embedded in the attractor. It is identified by a set of integers which characterize the attractor. The templates are confirmed by investigating orbits of higher period.

CHARACTERISATION OF CHAOS IN CHUA'S OSCILLATOR IN TERMS OF UNSTABLE PERIODIC ORBITS

1993 ◽  
Vol 03 (02) ◽  
pp. 411-429 ◽  
Author(s):  
MACIEJ J. OGORZAŁEK ◽  
ZBIGNIEW GALIAS
Keyword(s):  
Periodic Orbits ◽  
Picture Book ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Chua’S Oscillator ◽  
Canonical Chua’S Circuit

We present a picture book of unstable periodic orbits embedded in typical chaotic attractors found in the canonical Chua's circuit. These include spiral Chua's, double-scroll Chua's and double hook attractors. The "skeleton" of unstable periodic orbits is specific for the considered attractor and provides an invariant characterisation of its geometry.

EXPERIMENTAL VERIFICATION OF PYRAGAS’S CHAOS CONTROL METHOD APPLIED TO CHUA’S CIRCUIT

1994 ◽  
Vol 04 (06) ◽  
pp. 1703-1706 ◽  
Author(s):  
P. CELKA
Keyword(s):  
Periodic Orbits ◽  
Experimental Verification ◽  
Chaos Control ◽  
Control Method ◽  
Experimental Results ◽  
Experimental Setup ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit

We have built an experimental setup to apply Pyragas’s [1992, 1993] control method in order to stabilize unstable periodic orbits (UPO) in Chua’s circuit. We have been able to control low period UPO embedded in the double scroll attractor. However, experimental results show that the control method is useful under some restrictions we will discuss.

CONTROL OF n-SCROLL CHUA'S CIRCUIT

2009 ◽  
Vol 19 (11) ◽  
pp. 3813-3822 ◽  
Author(s):  
ABDELKRIM BOUKABOU ◽  
BILEL SAYOUD ◽  
HAMZA BOUMAIZA ◽  
NOURA MANSOURI
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Predictive Control ◽  
Control Method ◽  
Sufficient Conditions ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Necessary And Sufficient

This paper addresses the control of unstable fixed points and unstable periodic orbits of the n-scroll Chua's circuit. In a first step, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed predictive control method. In addition, we show how a chaotic system with multiple unstable periodic orbits can be stabilized by taking the system dynamics from one UPO to another. Control performances of these approaches are demonstrated by numerical simulations.

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 SYSTEMATIC APPROACH TO GENERATING N-SCROLL ATTRACTORS

2002 ◽  
Vol 12 (12) ◽  
pp. 2907-2915 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM-FUNG MAN ◽  
GUANRONG CHEN
Keyword(s):  
Systematic Approach ◽  
Building Blocks ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Nonlinear Resistor ◽  
Construction Scheme ◽  
Break Points ◽  
Set Up

A new circuitry design based on Chua's circuit for generating n-scroll attractors (n = 1, 2, 3, …) is proposed. In this design, the nonlinear resistor in Chua's circuit is constructed via a systematical procedure using basic building blocks. With the proposed construction scheme, the slopes and break points of the v–i characteristic of the circuit can be tuned independently, and chaotic attractors with an even or an odd number of scrolls can be easily generated. Distinct attractors with n-scrolls (n = 5, 6, 7, 8, 9, 10) obtained with this simple experimental set-up are demonstrated.

scholarly journals Localization of Hidden Attractors in Chua’s System With Absolute Nonlinearity and Its FPGA Implementation

2021 ◽  
Vol 9 ◽  
Author(s):  
Xianming Wu ◽  
Huihai Wang ◽  
Shaobo He
Keyword(s):  
Basin Of Attraction ◽  
Digital Circuit ◽  
Fpga Implementation ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Chua's Circuit ◽  
The Mathematical Model ◽  
The Basin Of Attraction

Investigation of the classical self-excited and hidden attractors in the modified Chua’s circuit is a hot and interesting topic. In this article, a novel Chua’s circuit system with an absolute item is investigated. According to the mathematical model, dynamic characteristics are analyzed, including symmetry, equilibrium stability analysis, Hopf bifurcation analysis, Lyapunov exponents, bifurcation diagram, and the basin of attraction. The hidden attractors are located theoretically. Then, the coexistence of the hidden limit cycle and self-excited chaotic attractors are observed numerically and experimentally. The numerical simulation results are consistent with the FPGA implementation results. It shows that the hidden attractor can be localized in the digital circuit.

EXPERIMENTAL CONTROL OF CHAOS IN CHUA’S CIRCUIT VIA TUNNELS

1994 ◽  
Vol 04 (03) ◽  
pp. 741-750 ◽  
Author(s):  
MAKOTO ITOH ◽  
HIROYUKI MURAKAMI ◽  
LEON O. CHUA
Keyword(s):  
Periodic Orbit ◽  
Experimental Method ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Chua’S Circuit ◽  
Control Of Chaos ◽  
Experimental Control ◽  
Chua's Circuit ◽  
Tunnel Mechanism ◽  
Very High

In this letter, we propose a new experimental method for converting a chaotic attractor in Chua’s circuit to a periodic orbit. A tunnel mechanism is used to achieve this conversion. Using this method, we were able to demonstrate experimentally that periodic orbits of very high periods (e.g., greater than 30) can be robustly stabilized.

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