HORSESHOE CHAOS IN A HYBRID PLANAR DYNAMICAL SYSTEM

2012 ◽  
Vol 22 (08) ◽  
pp. 1250202 ◽  
Author(s):  
QING-JU FAN
Keyword(s):  
Dynamical System ◽  
Cross Section ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Buck Converter ◽  
Two Dimensional ◽  
Topological Horseshoe ◽  
Piecewise Linear System ◽  
Shift Map

In this paper, we study the chaotic dynamics of a voltage-mode controlled buck converter, which is typically a switched piecewise linear system. For the two-dimensional hybrid system, we consider a properly chosen cross-section and the corresponding Poincaré map, and show that the dynamics of the system is semi-conjugate to a 2-shift map, which implies the chaotic behavior of this system. The essential tool is a topological horseshoe theory and numerical method.

scholarly journals Chaotic Behavior in a Switched Dynamical System

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Fatima El Guezar ◽  
Hassane Bouzahir
Keyword(s):  
Hybrid Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Piecewise Linear ◽  
Open Loop ◽  
Buck Converter ◽  
Modeling Techniques ◽  
Scientific Laboratory ◽  
Switched Dynamical System

We present a numerical study of an example of piecewise linear systems that constitute a class of hybrid systems. Precisely, we study the chaotic dynamics of the voltage-mode controlled buck converter circuit in an open loop. By considering the voltage input as a bifurcation parameter, we observe that the obtained simulations show that the buck converter is prone to have subharmonic behavior and chaos. We also present the corresponding bifurcation diagram. Our modeling techniques are based on the new French native modeler and simulator for hybrid systems called Scicos (Scilab connected object simulator) which is a Scilab (scientific laboratory) package. The followed approach takes into account the hybrid nature of the circuit.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

NETWORK AS A CHAOTIC DYNAMICAL SYSTEM

2007 ◽  
Vol 17 (10) ◽  
pp. 3529-3533 ◽  
Author(s):  
SYUJI MIYAZAKI ◽  
YASUSHI NAGASHIMA
Keyword(s):  
Dynamical System ◽  
Network Structure ◽  
Transition Matrix ◽  
Chaotic Dynamics ◽  
Large Deviation ◽  
Piecewise Linear ◽  
Directed Network ◽  
Chaotic Dynamical System ◽  
One Dimensional ◽  
Finite Time Lyapunov Exponents

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Chen Kong ◽  
Xue Gao ◽  
Xianbin Liu
Keyword(s):  
Dynamical System ◽  
Averaging Method ◽  
Global Analysis ◽  
Piecewise Linear ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Random Dynamical System ◽  
Piecewise Linear System ◽  
Dynamical Behaviors

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.

scholarly journals No Chaos in Dixon’s System

2021 ◽  
Vol 31 (03) ◽  
pp. 2150044
Author(s):  
Werner M. Seiler ◽  
Matthias Seiß
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Continuous Dynamical System ◽  
Two Parameters

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

Exponential-Algebraic Maps and Chaos in 3D Autonomous Quadratic Systems

2015 ◽  
Vol 25 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Vasiliy Ye. Belozyorov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Quadratic Systems ◽  
Discrete Maps ◽  
Algebraic Maps ◽  
Discrete Map

For some 3D autonomous quadratic dynamical systems an explicit autonomous exponential-algebraic 1D map, generating chaos in mentioned systems, is designed. Examples of the systems, where chaos is generated by such discrete maps, are given. New results about an existence of chaotic dynamics in the quadratic 3D systems are also derived. Besides, for the Lanford system (it is 3D autonomous quadratic dynamical system) the value of some parameter at which the system shows increased chaotic behavior is indicated. This assertion is based on the construction for the Lanford system of 2D exponential-algebraic discrete map which possesses chaotic properties.

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