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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.
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.
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