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Saddle-point solutions 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 3, 2012
Summary
This study examines how grazing bifurcations impact boundary saddle-point solutions in impacting systems. These solutions are crucial for understanding system dynamics and their existence is affected by these bifurcations.
Area of Science:
- Dynamical Systems
- Nonlinear Dynamics
- Mechanical Vibrations
Background:
- Impacting systems exhibit complex dynamics arising from interactions with boundaries.
- Boundary saddle-point solutions are critical for characterizing global system behavior.
- Nonsmooth phenomena, such as folds, can generate unique solution types.
Purpose of the Study:
- To analyze the formation and characteristics of boundary saddle-point solutions originating from nonsmooth folds.
- To investigate the influence of grazing bifurcations on the existence of these solutions.
- To elucidate the role of accessible boundary saddle-point solutions in the overall dynamics of impacting systems.
Main Methods:
- Analysis of a specific impacting system model.
- Identification and characterization of boundary saddle-point solutions.
- Investigation of bifurcation phenomena, specifically grazing bifurcations.
Main Results:
- A boundary saddle-point solution born in a nonsmooth fold was identified and analyzed.
- Grazing bifurcations were shown to significantly affect the existence of these boundary saddle-point solutions.
- The accessibility of these solutions was linked to their role in global dynamics.
Conclusions:
- Boundary saddle-point solutions are sensitive to grazing bifurcations in impacting systems.
- Understanding these bifurcations is essential for predicting the global dynamics of systems with boundary interactions.
- The study highlights the interplay between smooth and nonsmooth dynamics in complex systems.
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