Border collision bifurcations in two-dimensional piecewise smooth maps

1999 ◽  
Vol 59 (4) ◽  
pp. 4052-4061 ◽  
Author(s):  
Soumitro Banerjee ◽  
Celso Grebogi
Keyword(s):  
Piecewise Smooth ◽  
Two Dimensional ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps

BORDER COLLISION BIFURCATIONS IN THREE-DIMENSIONAL PIECEWISE SMOOTH SYSTEMS

2008 ◽  
Vol 18 (02) ◽  
pp. 577-586 ◽  
Author(s):  
INDRAVA ROY ◽  
A. R. ROY
Keyword(s):  
Three Dimensional ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
Research Fields ◽  
Border Collision Bifurcations ◽  
Period 2 ◽  
Wide Range ◽  
Different Types ◽  
Piecewise Smooth Maps ◽  
The Stability

Piecewise smooth maps have been a focus of study for scientists in a wide range of research fields. These maps show qualitatively different types of bifurcations than those exhibited by generic smooth maps. We present a theoretical framework for analyzing three-dimensional piecewise smooth maps by deriving a suitable normal form and then finding the stability criteria for periodic orbits. We also show by numerical simulation different types of border collision bifurcations that can occur in such a map. We have also been able to observe a border collision bifurcation from a period-2 to a quasiperiodic orbit.

Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210432
Author(s):  
Viktor Avrutin ◽  
Anastasiia Panchuk ◽  
Iryna Sushko
Keyword(s):  
Geometrical Structure ◽  
Sudden Change ◽  
Piecewise Smooth ◽  
Connected Components ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps ◽  
One Dimensional Maps

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components).

MULTIPLE-ATTRACTOR BIFURCATIONS AND QUASIPERIODICITY IN PIECEWISE-SMOOTH MAPS

2008 ◽  
Vol 18 (06) ◽  
pp. 1775-1789 ◽  
Author(s):  
ZHANYBAI T. ZHUSUBALIYEV ◽  
ERIK MOSEKILDE ◽  
SOUMITRO BANERJEE
Keyword(s):  
Periodic Orbit ◽  
Piecewise Smooth ◽  
Invariant Curve ◽  
Simultaneous Generation ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps ◽  
Stable Periodic ◽  
Attractor Bifurcation ◽  
New Type

It is known that border-collision bifurcations in piecewise-smooth maps can lead to situations where several attractors are created simultaneously in so-called "multiple-attractor" or "multiple-choice" bifurcations. It has been shown that such a situation leads to a fundamental source of uncertainty regarding which attractor the system will follow as a parameter is varied through the bifurcation point. Phenomena of this type have been observed in various physical and engineering systems. We have recently demonstrated that piecewise-smooth systems can exhibit a new type of border-collision bifurcation in which a stable invariant curve, associated with a quasiperiodic or a mode-locked periodic orbit, arises from a fixed point. In this paper we consider a particular variant of the multiple-attractor bifurcation in which a stable periodic orbit arises simultaneously with a closed invariant curve. We also show examples of simultaneously appearing stable periodic orbits and of the simultaneous generation of periodic and chaotic attractors.

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

2005 ◽  
Vol 5 (3) ◽  
pp. 881-897 ◽  
Author(s):  
Iryna Sushko ◽  
◽  
Anna Agliari ◽  
Laura Gardini ◽  
◽  
...  
Keyword(s):  
Piecewise Smooth ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps

DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS

2010 ◽  
Vol 20 (07) ◽  
pp. 2045-2070 ◽  
Author(s):  
IRYNA SUSHKO ◽  
LAURA GARDINI
Keyword(s):  
Phase Space ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
Codimension One ◽  
Border Collision Bifurcations ◽  
Global Properties ◽  
Piecewise Smooth Maps ◽  
Linear Fractional Function ◽  
Fractional Function ◽  
Discrete Time Dynamical Systems

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.

Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

2014 ◽  
Vol 24 (02) ◽  
pp. 1450024 ◽  
Author(s):  
Laura Gardini ◽  
Viktor Avrutin ◽  
Irina Sushko
Keyword(s):  
Discontinuity Point ◽  
Piecewise Smooth ◽  
One Dimensional ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps ◽  
Symbolic Sequences ◽  
Bifurcation Structures ◽  
Local Monotonicity ◽  
Parameter Values

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, we prove that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

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.

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