Dynamics of a piecewise smooth map with singularity (original) (raw)

A Gallery of Bifurcation Scenarios in Piecewise Smooth 1D Maps

Global Analysis of Dynamic Models in Economics and Finance, 2012

We give a brief overview of several bifurcation scenarios occurring in 1D piecewise monotone maps defined on two partitions, continuous or discontinuous. A collection of some basic blocks is proposed, which may be observed in particular bifurcation sequences of a system of interest both in regular and chaotic parameter domains.

Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps

International Journal of Bifurcation and Chaos, 2014

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.

Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps

Discrete and Continuous Dynamical Systems - Series B, 2005

This article deals with a two-parameter family of piecewise smooth unimodal maps with one break point. Using superstable cycles and their symbolic representation we describe the structure of the periodicity regions of the 2D bifurcation diagram. Particular attention is paid to the bistability regions corresponding to two coexisting attractors, and to the border-collision bifurcations.

Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps

Chaos, Solitons & Fractals, 2012

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X 0 = aX if jXj < z, X 0 = bX if jXj > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X 0 = F(X) if jXj < z, X 0 = G(X) if jXj > z is also given, proving the conditions leading to a homeomorphism of the circle.

Period-Doubling Scenario Without Flip Bifurcations in a One-Dimensional Map

International Journal of Bifurcation and Chaos, 2005

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

Degenerate Bifurcations and Border Collisions in Piecewise Smooth 1D and 2D Maps

International Journal of Bifurcation and Chaos, 2010

We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in piecewise smooth maps, for which it is not possible to specify in general the result of the bifurcation, as it strongly depends on the global properties of the map. In fact, the degenerate bifurcations mainly occur in piecewise smooth maps defined in some subspace of the phase space by a linear or linear-fractional function, although not necessarily only by such functions. We also discuss the relation between degenerate bifurcations and border-collision bifurcations.

A Study of the Dynamics of a New Piecewise Smooth Map

International Journal of Bifurcation and Chaos, 2020

In this article, we have studied a 1D map, which is formed by combining the two well-known maps i.e. the tent and the logistic maps in the unit interval i.e. [0, 1]. The proposed map can behave as the piecewise smooth or non-smooth maps (depending on the behaviour of the map just before and after the border) and then the dynamics of the map has been studied using analytical tools and numerical simulations. Characterization has been done by primarily studying the Lyapunov spectra and the corresponding bifurcation diagrams. Some peculiar dynamics of this map have been shown numerically. Finally, a Simulink implementation of the proposed map has been demonstrated.

Non-periodic bifurcations of one-dimensional maps

Ergodic Theory and Dynamical Systems, 2007

We prove that a 'positive probability' subset of the boundary of '{uniformly expanding circle transformations}' consists of Kupka-Smale maps. More precisely, we construct an open class of two-parameter families of circle maps (f a,θ) a,θ such that, for a positive Lebesgue measure subset of values of a, the family (f a,θ) θ crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.

Bifurcations from phase-locked dynamics to chaos in a piecewise-linear map

2011

Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. The purpose of the present paper is to examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of bordercollision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Regular Dynamics

International Journal of Bifurcation and Chaos, 2013

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.