Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits

2000 ◽  
Vol 47 (3) ◽  
pp. 389-394 ◽  
Author(s):  
S. Banerjee ◽  
M.S. Karthik ◽  
Guohui Yuan ◽  
J.A. Yorke
Keyword(s):  
Piecewise Smooth ◽  
One Dimensional ◽  
Smooth Maps ◽  
Switching Circuits ◽  
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.

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

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).

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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