Smale Horseshoe in a Piecewise Smooth Map

2019 ◽  
Vol 29 (04) ◽  
pp. 1950051
Author(s):  
Denghui Li ◽  
Hebai Chen ◽  
Jianhua Xie
Keyword(s):  
Chaotic Dynamics ◽  
Piecewise Smooth ◽  
Two Dimensional ◽  
Bounded Region ◽  
Smale Horseshoe ◽  
Time Representation ◽  
Impact Oscillators ◽  
Shift Map ◽  
Smooth Map ◽  
Nonwandering Set

We investigate the chaotic dynamics of a two-dimensional piecewise smooth map. The map represents the normal form of a discrete time representation of impact oscillators near grazing states. It is proved that, in certain region of the parameter space, the nonwandering set of the map is contained in a bounded region and that, restricted to the nonwandering set, the map is topologically conjugate to the two-sided shift map on two symbols.

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 Multi-Baker Map as a Model of Digital PD Control

2016 ◽  
Vol 26 (02) ◽  
pp. 1650023 ◽  
Author(s):  
Gábor Csernák ◽  
Gergely Gyebrószki ◽  
Gábor Stépán
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Linear Systems ◽  
Small Amplitude ◽  
Chaotic Dynamics ◽  
Largest Lyapunov Exponent ◽  
Two Dimensional ◽  
Smale Horseshoe ◽  
Analytical Proof ◽  
Baker Map

Digital stabilization of unstable equilibria of linear systems may lead to small amplitude stochastic-like oscillations. We show that these vibrations can be related to a deterministic chaotic dynamics induced by sampling and quantization. A detailed analytical proof of chaos is presented for the case of a PD controlled oscillator: it is shown that there exists a finite attracting domain in the phase-space, the largest Lyapunov exponent is positive and the existence of a Smale horseshoe is also pointed out. The corresponding two-dimensional micro-chaos map is a multi-baker map, i.e. it consists of a finite series of baker’s maps.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

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