Local and global bifurcations in three-dimensional, continuous, piecewise smooth maps

De, Soma ; Dutta, Partha Sharathi ; Banerjee, Soumitro ; Roy, Akhil Ranjan (2011) Local and global bifurcations in three-dimensional, continuous, piecewise smooth maps International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 21 (6). pp. 1617-1636. ISSN 0218-1274

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Official URL: http://www.worldscinet.com/ijbc/21/2106/S021812741...

Related URL: http://dx.doi.org/10.1142/S0218127411029318

Abstract

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.

Item Type:Article
Source:Copyright of this article belongs to World Scientific Publishing Company.
Keywords:Piecewise-smooth Map; Border-collision Bifurcation; Torus Destruction; Homoclinic Bifurcation
ID Code:81335
Deposited On:06 Feb 2012 04:23
Last Modified:06 Feb 2012 04:23

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