Effect of stochastically moving border on basins of attraction in a class of piecewise smooth maps

2017 ◽  
Vol 381 (27) ◽  
pp. 2161-2166 ◽  
Author(s):  
Dhrubajyoti Mandal ◽  
Soumitro Banerjee
Keyword(s):  
Basins Of Attraction ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
Piecewise Smooth Maps

scholarly journals Nested Closed Invariant Curves in Piecewise Smooth Maps

2019 ◽  
Vol 29 (07) ◽  
pp. 1930017
Author(s):  
Viktor Avrutin ◽  
Zhanybai T. Zhusubaliyev
Keyword(s):  
Piecewise Linear ◽  
Phase Locking ◽  
Basins Of Attraction ◽  
Piecewise Smooth ◽  
Phase Portraits ◽  
Invariant Curves ◽  
Smooth Maps ◽  
Piecewise Smooth Maps ◽  
Smooth Map ◽  
Stable Invariant

The paper describes how several coexisting stable closed invariant curves embedded into each other can arise in a two-dimensional piecewise-linear normal form map. Phenomena of this type have been recently reported for a piecewise smooth map, modeling the behavior of a power electronic DC–DC converter. In the present work, we demonstrate that this type of multistability exists in a more general class of models and show how it may result from the well-known period adding bifurcation structure due to its deformation so that the phase-locking regions start to overlap. We explain how this overlapping structure is related to the appearance of coexisting stable closed invariant curves nested into each other. By means of detailed, numerically calculated phase portraits we hereafter present an example of this type of multistability. We also demonstrate that the basins of attraction of the nested stable invariant curves may be separated from each other not only by repelling closed invariant curves, as previously reported, but also by a chaotic saddle. It is suggested that the considered kind of multistability is a generic phenomenon in piecewise smooth dynamical systems.

Bifurcations in general piecewise-smooth maps

2007 ◽  
pp. 171-217 ◽  
Author(s):  
Mario di Bernardo ◽  
Alan R. Champneys ◽  
Christopher J. Budd ◽  
Piotr Kowalczyk
Keyword(s):  
Piecewise Smooth ◽  
Smooth Maps ◽  
Piecewise Smooth Maps

ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

2009 ◽  
Vol 19 (05) ◽  
pp. 1709-1732 ◽  
Author(s):  
B. M. BAKER ◽  
M. E. KIDWELL ◽  
R. P. KLINE ◽  
I. POPOVICI
Keyword(s):  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Stimulus Period ◽  
Smooth Maps ◽  
Periodic Stimulation ◽  
Piecewise Smooth Maps ◽  
Dimensional Version ◽  
Bifurcation Law ◽  
The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

Influence of a square-root singularity on the behaviour of piecewise smooth maps

Nonlinearity ◽  
2010 ◽  
Vol 23 (2) ◽  
pp. 445-463 ◽  
Author(s):  
Viktor Avrutin ◽  
Partha Sharathi Dutta ◽  
Michael Schanz ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Square Root ◽  
Smooth Maps ◽  
Piecewise Smooth Maps ◽  
Square Root Singularity

scholarly journals Shilnikov-type dynamics in three-dimensional piecewise smooth maps

2020 ◽  
Vol 133 ◽  
pp. 109655
Author(s):  
Indrava Roy ◽  
Mahashweta Patra ◽  
Soumitro Banerjee
Keyword(s):  
Three Dimensional ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
Piecewise Smooth Maps

Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps

2014 ◽  
Vol 24 (08) ◽  
pp. 1440012 ◽  
Author(s):  
Viktor Avrutin ◽  
Laura Gardini ◽  
Michael Schanz ◽  
Iryna Sushko
Keyword(s):  
Chaotic Attractor ◽  
Point Of View ◽  
Piecewise Smooth ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Smooth Maps ◽  
Homoclinic Bifurcations ◽  
Piecewise Smooth Maps

In this work, we classify the bifurcations of chaotic attractors in 1D piecewise smooth maps from the point of view of underlying homoclinic bifurcations of repelling cycles which are located before the bifurcation at the boundary of the immediate basin of the chaotic attractor.

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