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.

scholarly journals Calculation of Invariant Manifolds of Piecewise-Smooth Maps

2020 ◽  
Vol 24 (3) ◽  
pp. 166-182
Author(s):  
Z. T. Zhusubaliyev ◽  
V. G. Rubanov ◽  
Yu. A. Gol’tsov
Keyword(s):  
Invariant Manifolds ◽  
Inverse Function ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
First Order ◽  
Piecewise Smooth Maps ◽  
Stable Invariant Manifolds ◽  
Stable Invariant ◽  
Original Approach ◽  
Stable Subspace

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

2021 ◽  
Vol 31 (03) ◽  
pp. 2130009
Author(s):  
Zhanybai T. Zhusubaliyev ◽  
Viktor Avrutin ◽  
Frank Bastian
Keyword(s):  
Chaotic Attractor ◽  
Small Amplitude ◽  
Homoclinic Bifurcation ◽  
Piecewise Smooth ◽  
Invariant Curve ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Invariant Curves ◽  
Piecewise Smooth Systems ◽  
Smooth Map

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map

2020 ◽  
Vol 30 (06) ◽  
pp. 2030014
Author(s):  
Wirot Tikjha ◽  
Laura Gardini
Keyword(s):  
Dynamic Behavior ◽  
Phase Plane ◽  
Piecewise Linear ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Piecewise Linear Map ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps ◽  
Transcritical Bifurcations

Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.

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
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