Global Bifurcations in a Complementarity Game

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.

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Some Aspects of Global Bifurcations and Chaos

Introduction to Applied Nonlinear Dynamical Systems and Chaos - Texts in Applied Mathematics ◽

10.1007/978-1-4757-4067-7_5 ◽

1990 ◽

pp. 420-650 ◽

Cited By ~ 3

Author(s):

Stephen Wiggins

Keyword(s):

Global Bifurcations

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Global Dynamics in the Leslie–Gower Model with the Allee Effect

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501511 ◽

2018 ◽

Vol 28 (12) ◽

pp. 1850151 ◽

Cited By ~ 2

Author(s):

Valery A. Gaiko ◽

Cornelis Vuik

Keyword(s):

Singular Point ◽

Bifurcation Analysis ◽

Limit Cycles ◽

Allee Effect ◽

Global Bifurcation ◽

Global Dynamics ◽

Global Bifurcations ◽

Biomedical System

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

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Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497001631 ◽

1997 ◽

Vol 07 (11) ◽

pp. 2437-2457 ◽

Cited By ~ 5

Author(s):

W. Szemplińska-Stupnicka ◽

E. Tyrkiel

Keyword(s):

Nonlinear Oscillator ◽

Duffing Oscillator ◽

Nonlinear Resonance ◽

Basins Of Attraction ◽

Global Bifurcations ◽

System Behavior ◽

Coexisting Attractors ◽

System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

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