Nonlinear Dynamics of the Chua’s Circuit System Revisited

2008 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bing Xue
Keyword(s):  
Local Stability ◽  
Normal Vector ◽  
Chua’S Circuit ◽  
Separation Boundary ◽  
Normal Vector Field ◽  
Chua's Circuit ◽  
System Parameters ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic and chaotic behaviors in the Chua’s circuit system are discussed. The solutions of the system in different regions with different parameters are obtained. The switching boundaries are introduced for systems switching because of different system parameters. In the vicinity of the switching boundary, the normal vector-field product is introduced to measure the flow switching on the separation boundary, and the grazing and passable conditions to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from the mapping structures. The local stability and bifurcation analysis are carried out.

AN ANALYTICAL PREDICTION OF PERIODIC FLOWS IN THE CHUA CIRCUIT SYSTEM

2009 ◽  
Vol 19 (07) ◽  
pp. 2165-2180 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
BING XUE
Keyword(s):  
Local Stability ◽  
Normal Vector ◽  
Analytical Prediction ◽  
Separation Boundary ◽  
Normal Vector Field ◽  
Periodic Flows ◽  
System Parameters ◽  
Chua Circuit ◽  
Discontinuous Boundary ◽  
Mapping Structures

In this paper, periodic and chaotic behaviors in the Chua circuit system are investigated, and the analytical prediction of periodic flows in such a system is carried out. The solutions of the system in different regions with different parameters are first obtained. The switching boundaries are introduced for systems switching because of different system parameters in different domains. In the vicinity of the switching boundaries, the normal vector-field product is introduced to measure flow switching on the separation boundary, and the conditions for grazing and passable flows to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from mapping structures. The local stability and bifurcation analysis are carried out.

Dynamics of a Simplified van der Pol Oscillator Revisited

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Bifurcation Analysis ◽  
Local Stability ◽  
Van Der Pol Oscillator ◽  
Nonsmooth Dynamics ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

Switching Dynamics of a Periodically Forced Discontinuous System With an Inclined Boundary

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon M. Rapp
Keyword(s):  
Dynamical System ◽  
Vector Fields ◽  
Normal Vector ◽  
Periodic Motions ◽  
Separation Boundary ◽  
Discontinuous System ◽  
Normal Vector Field ◽  
Periodically Driven ◽  
Switching Dynamics ◽  
Straight Line

This paper presents the switching dynamics of flow from one domain into another one in the periodically driven, discontinuous dynamical system. The simple inclined straight line boundary in phase space is considered as a control law for the dynamical system to switch. The normal vector-field product for flow switching on the separation boundary is introduced, and the passability condition of flow to the discontinuous boundary is presented. The sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the local stability and bifurcation analysis are carried out. Numerical illustrations of periodic motions with grazing to the boundary and/or sliding on the boundary are given, and the normal vector fields are illustrated to show the analytical criteria.

scholarly journals PERIODIC MOTIONS AND CHAOS WITH IMPACTING CHATTER AND STICK IN A GEAR TRANSMISSION SYSTEM

2009 ◽  
Vol 19 (06) ◽  
pp. 1975-1994 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
DENNIS O'CONNOR
Keyword(s):  
Stability Analysis ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Local Stability Analysis ◽  
Gear Transmission System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in [Luo & O'Connor, 2007] are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.

Estimating system parameters of Chua's circuit from synchronizing signal

2004 ◽  
Vol 324 (1) ◽  
pp. 36-41 ◽  
Author(s):  
Ling Liu ◽  
Xiaogang Wu ◽  
Hanping Hu
Keyword(s):  
Chua’S Circuit ◽  
Chua's Circuit ◽  
System Parameters

Nonlinear Dynamics of a Gear Transmission System: Part II — Periodic Impacting Chatter and Stick

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Motion Mechanism ◽  
Local Stability Analysis ◽  
Gear Transmission System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
System Part

In Part I, the motion mechanism of impacting chatter and stick motion in the gear transmission dynamical system was discussed. This paper focuses on periodic motions relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in Part I are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.

STOCHASTIC RESONANCE IN CHUA’S CIRCUIT

1992 ◽  
Vol 02 (02) ◽  
pp. 397-401 ◽  
Author(s):  
V.S. ANISHCHENKO ◽  
M.A. SAFONOVA ◽  
L.O. CHUA
Keyword(s):  
Stochastic Resonance ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Physical Mechanism ◽  
Sinusoidal Signal ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
System Parameters ◽  
Coherent Interaction ◽  
Chaotic Electronic Circuit

In this paper, we report numerical observations of the stochastic resonance (SR) phenomenon in a bistable chaotic electronic circuit (namely, Chua’s circuit) driven simultaneously by noise and a sinusoidal signal. It is shown that the noise-induced “chaos-chaos” type intermittency is a physical mechanism of the SR-phenomenon in chaotic systems. The resulting amplification of the sinusoidal signal intensity is due to a coherent interaction of three characteristic frequencies of the system. The SR-phenomenon can be controlled by a variation of either the noise intensity or the system parameters in the absence of noise.

Impacting chatter and stick in a transmission system with two oscillators

2009 ◽  
Vol 223 (3) ◽  
pp. 159-188 ◽  
Author(s):  
A C J Luo ◽  
D O'Connor
Keyword(s):  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Transmission Systems ◽  
Motion Mechanism ◽  
System Parameters ◽  
Mapping Structures ◽  
Non Linear ◽  
The Relationship

In this article, an investigation on non-linear dynamical behaviours of a transmission system with a gear pair is conducted. The transmission system is described through an impact model with a possible stick between the two gears. From the theory of discontinuous dynamical systems, the motion mechanism of impacting chatter with stick is investigated. The onset and vanishing conditions for stick motions are developed, and the condition for maintaining stick motion is obtained as well. The corresponding physical interpretation is given for a better understanding of non-linear behaviours of gear transmission systems. A parameter map is presented to provide a global picture of the relationship between system parameters and corresponding motion. Grazing and stick conditions are utilized to determine the existence of periodic motions. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analyses are carried out. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. A better understanding of the motion mechanism between two gears may be helpful for improving the efficiency of gear transmission systems.

Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities

2008 ◽  
Vol 4 (1) ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Local Stability ◽  
Singularity Theory ◽  
Periodic Motions ◽  
Sliding Motion ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Local Singularity ◽  
Stability And Bifurcations ◽  
Bifurcation And Stability

The local stability and existence of periodic motions in a periodically forced oscillator with multiple discontinuities are investigated. The complexity of periodic motions and chaos in such a discontinuous system is often caused by the passability, sliding, and grazing of flows to discontinuous boundaries. Therefore, the corresponding analytical conditions for such singular phenomena to discontinuous boundary are presented from the local singularity theory of discontinuous systems. To develop the mapping structures of periodic motions, basic mappings are introduced, and the sliding motion on the discontinuous boundary is described by a sliding mapping. A generalized mapping structure is presented for all possible periodic motions, and the local stability and bifurcations of periodic motions are discussed. From mapping structures, the switching points of periodic motions on the boundaries are predicted analytically. Two periodic motions are presented for illustrations of the passability, sliding, and grazing of periodic motions on the boundary.

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