scholarly journals Horseshoe Chaos in a Simple Memristive Circuit

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lei Wang ◽  
XiaoSong Yang ◽  
WenJie Hu ◽  
Quan Yuan
Keyword(s):  
Stability Analysis ◽  
Poincaré Map ◽  
Circuit Model ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Topological Horseshoe ◽  
Memristive Circuit ◽  
The Stability

A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION

2008 ◽  
Vol 18 (08) ◽  
pp. 2391-2396 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Poincaré Map ◽  
Chaotic Behavior ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Numerical Studies ◽  
Topological Horseshoes

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Analysis of a Mass-Spring-Relay System With Periodic Forcing

2021 ◽  
Author(s):  
János Lelkes ◽  
Tamás KALMÁR-NAGY
Keyword(s):  
Poincaré Map ◽  
Pitchfork Bifurcation ◽  
Global Dynamics ◽  
Poincare Map ◽  
Periodic Excitation ◽  
Periodic Forcing ◽  
Relay System ◽  
Saddle Type ◽  
The Stability ◽  
Mass Spring

Abstract The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. A Poincare map is constructed to simplify the mathematical analysis. The stability of the xed points of the Poincare map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle type xed point. The global dynamics of the system is investigated, showing discontinuity induced bifurcations of the xed points.

scholarly journals Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hu Wang ◽  
Yongguang Yu ◽  
Guoguang Wen
Keyword(s):  
Atmospheric Circulation ◽  
Chaotic Behavior ◽  
Circulation Model ◽  
Dynamical Analysis ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Normal Form Theory ◽  
Computer Assisted Proof ◽  
Parameter Interval ◽  
The Stability

The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.

HYPERCHAOS IN A SPACECRAFT POWER SYSTEM

2011 ◽  
Vol 21 (06) ◽  
pp. 1719-1726 ◽  
Author(s):  
QINGDU LI ◽  
XIAO-SONG YANG ◽  
SHU CHEN
Keyword(s):  
Power System ◽  
Numerical Computation ◽  
Lyapunov Exponents ◽  
Cross Section ◽  
Strange Attractor ◽  
Poincaré Map ◽  
Poincare Map ◽  
Topological Horseshoe

This paper presents some rigorous arguments on a chaotic strange attractor in a spacecraft power system, which is a 3D system with hysteresis switching. By carefully picking a suitable cross-section with respect to the attractor, we find a topological horseshoe of the corresponding third-returned Poincaré map, thus giving a rigorous verification of the existence of chaos in this system. Numerical computation shows that the two Lyapunov exponents of the Poincaré map are both positive and that there exists a two-directional expansion in this horseshoe, suggesting that this attractor is hyperchaotic in nature.

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