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.

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).

scholarly journals Robust chaos and the continuity of attractors

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Paul A Glendinning ◽  
David J W Simpson
Keyword(s):  
Normal Form ◽  
Parameter Space ◽  
Hausdorff Metric ◽  
Piecewise Smooth ◽  
Chaotic Attractors ◽  
Unimodal Maps ◽  
Smooth Maps ◽  
Tent Maps ◽  
Piecewise Smooth Maps

Abstract As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for many families of piecewise-smooth maps it provides a way to think about changing structures within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map and the border-collision normal form.

C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

2003 ◽  
Vol 18 (5) ◽  
pp. 953-976 ◽  
Author(s):  
Christopher Halse ◽  
Martin Homer ◽  
Mario di Bernardo
Keyword(s):  
Piecewise Smooth ◽  
One Dimensional ◽  
Smooth Maps ◽  
Piecewise Smooth Maps

CHARACTERIZATION OF CHAOS SCENARIOS WITH PERIODIC INCLUSIONS FOR ONE CLASS OF PIECEWISE-SMOOTH DYNAMICAL MAPS

2011 ◽  
Vol 21 (09) ◽  
pp. 2427-2466 ◽  
Author(s):  
JOHN ALEXANDER TABORDA ◽  
FABIOLA ANGULO ◽  
GERARD OLIVAR
Keyword(s):  
Pulse Width Modulation ◽  
Point Of View ◽  
Piecewise Smooth ◽  
Connected Components ◽  
Open Problems ◽  
Smooth Maps ◽  
Digital Pulse ◽  
Piecewise Smooth Maps ◽  
Strongly Connected ◽  
Nonsmooth Dynamical Systems

In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS3. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

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.

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
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