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Interactions between global and grazing bifurcations in an impacting system.

Joanna F Mason¹, Petri T Piiroinen

¹School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland.

Chaos (Woodbury, N.Y.)

|April 5, 2011

Summary

Boundary crises in impacting systems create periodic windows through a novel mechanism involving grazing bifurcations. Researchers identified a new grazing-crisis vertex and related basin-boundary metamorphosis.

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Area of Science:

Nonlinear Dynamics
Chaos Theory
Bifurcation Theory

Background:

Boundary crises in smooth dynamical systems are known to create gaps leading to periodic windows.
The underlying mechanisms for these phenomena in impacting systems have not been fully explored.

Purpose of the Study:

To investigate the occurrence and mechanisms of boundary crises in impacting systems.
To identify novel bifurcations and topological changes in the phase space of these systems.

Main Methods:

Analysis of dynamical systems exhibiting boundary crises and grazing bifurcations.
Numerical simulations to observe the creation and disappearance of gaps in periodic windows.
Identification of critical points, specifically grazing-crisis vertices.

Main Results:

Boundary crises in impacting systems also exhibit gaps that lead to periodic windows.
The mechanism for gap creation differs from smooth systems, involving intersections with grazing bifurcation branches.
A novel grazing-crisis vertex, a type of double-crisis vertex, was identified.
Several types of basin-boundary metamorphosis intricately linked with grazing bifurcations were illustrated.

Conclusions:

The phenomenon of periodic windows arising from boundary crises is not exclusive to smooth systems.
Grazing bifurcations play a crucial role in the dynamics of boundary crises in impacting systems.
The discovery of the grazing-crisis vertex and associated metamorphoses offers new insights into the complex behavior of impacting systems.