Piecewise Smooth Map of Neuronal Activity: Deterministic and Stochastic Cases

2020 ◽  
pp. 183-193
Author(s):  
A. V. Belyaev ◽  
T. V. Ryazanova
Keyword(s):  
Neuronal Activity ◽  
Piecewise Smooth ◽  
Smooth Map

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Bifurcation analysis of an inductorless chaos generator using 1D piecewise smooth map

2014 ◽  
Vol 95 ◽  
pp. 137-145 ◽  
Author(s):  
Laura Gardini ◽  
Fabio Tramontana ◽  
Soumitro Banerjee
Keyword(s):  
Bifurcation Analysis ◽  
Piecewise Smooth ◽  
Chaos Generator ◽  
Smooth Map

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.

Circle Like Strange Attractor in a Piecewise Smooth Map*

2012 ◽  
Vol 45 (12) ◽  
pp. 81-86
Author(s):  
Biswambhar Rakshit ◽  
Soumitro Banerjee ◽  
Kazuyuki Aihara
Keyword(s):  
Strange Attractor ◽  
Piecewise Smooth ◽  
Smooth Map

MULTIPLE ATTRACTOR BIFURCATIONS IN A PIECEWISE-SMOOTH MAP WITH QUASIPERIODICITY

2006 ◽  
Vol 39 (8) ◽  
pp. 427-432
Author(s):  
Zhanybai T. Zhusubaliyev ◽  
Evgeniy Soukhoterin ◽  
Erik Mosekilde ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Smooth Map

scholarly journals On a piecewise-smooth map arising in ecology

2014 ◽  
Vol 418 (2) ◽  
pp. 753-765
Author(s):  
Chun-Ming Huang ◽  
Jonq Juang
Keyword(s):  
Piecewise Smooth ◽  
Smooth Map

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

scholarly journals Dangerous Border-Collision bifurcations of a piecewise-smooth map

2006 ◽  
Vol 5 (3) ◽  
pp. 493-503 ◽  
Author(s):  
Hun Ki Baek ◽  
◽  
Younghae Do
Keyword(s):  
Piecewise Smooth ◽  
Border Collision Bifurcations ◽  
Smooth Map

scholarly journals On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

2015 ◽  
Vol 22 (1) ◽  
pp. 29-63 ◽  
Author(s):  
Giovanni Bellettini ◽  
Maurizio Paolini ◽  
Lucia Tealdi
Keyword(s):  
Piecewise Smooth ◽  
Smooth Map
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