Transitions from phase-locked dynamics to chaos in a piecewise-linear map

Zhusubaliyev, Zhanybai T.; Mosekilde, Erik; De, Suparna; Banerjee, Soumitro

doi:10.1103/physreve.77.026206

Cited by 27 publications

(14 citation statements)

References 25 publications

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“…As τ further increases, the 14-band chaotic attractor merge first into a 7-band chaotic attractor and subsequently into a single chaotic band. The above scenario is similar in its appearance to the transition observed in [Zhusubaliyev et al, 2008b] for the twodimensional normal form map and in [Maistrenko et al, 1995] for the skew tent map. However, in our case the transition involves the destruction of a torus for the three-dimensional map.…”

Section: Torus Destruction Via Period-doubling Route To Chaossupporting

confidence: 68%

Local and Global Bifurcations in Three-Dimensional, Continuous, Piecewise Smooth Maps

Dutta

Banerjee

et al. 2011

Int. J. Bifurcation Chaos

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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 modelocked invariant circles. We demonstrate the transition to chaos through the breakdown of modelocked 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.

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Section: Torus Destruction Via Period-doubling Route To Chaossupporting

confidence: 68%

Local and Global Bifurcations in Three-Dimensional, Continuous, Piecewise Smooth Maps

Dutta

Banerjee

et al. 2011

Int. J. Bifurcation Chaos

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“…Finally, at a shrinking point, the map has an invariant polygon that may continue as an invariant circle. This circle can be destroyed by various mechanisms [26,79,80]; however, analogies of a number of the rigorous results for the smooth case remain to be obtained for PWSC maps, perhaps due, in part, to a lack of normal hyperbolicity.…”

Section: Discussionmentioning

confidence: 96%

Aspects of bifurcation theory for piecewise-smooth, continuous systems

Simpson

Meiss

2012

Physica D: Nonlinear Phenomena

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Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.

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“…There is at least numerical evidence that the two-dimensional discontinuous normal form (Dutta et al 2008) demonstrates the bandcount increment scenario as presented in this work. Furthermore, the continuous normal form (Banerjee & Grebogi 1999;Bernardo et al 1999;Zhusubaliyev et al 2008) shows similar, although not completely identical, bifurcation structures. The investigation of these bifurcation structures represents a challenging task, as there is a significant difference between one-and multidimensional maps regarding the complexity of the analytical calculation of the crisis curves.…”

Section: Discussionmentioning

confidence: 90%

The bandcount increment scenario. III. Deformed structures

2008

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Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

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Transitions from phase-locked dynamics to chaos in a piecewise-linear map

Cited by 27 publications

References 25 publications

Local and Global Bifurcations in Three-Dimensional, Continuous, Piecewise Smooth Maps

Local and Global Bifurcations in Three-Dimensional, Continuous, Piecewise Smooth Maps

Aspects of bifurcation theory for piecewise-smooth, continuous systems

The bandcount increment scenario. III. Deformed structures

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