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Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling

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Invariant polygons in systems with grazing-sliding.

R Szalai¹, H M Osinga

¹Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, University Walk, Bristol, BS8 1TR, United Kingdom. r.szalai@bristol.ac.uk

Chaos (Woodbury, N.Y.)

|July 8, 2008

Summary

This study analyzes unstable periodic orbits in 3D nonsmooth systems near grazing-sliding. The research reveals that attractors, formed by sliding orbits, are structured along lines intersecting at polygon vertices.

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Area of Science:

Dynamical Systems
Nonlinear Dynamics
Chaos Theory

Background:

Investigates generic three-dimensional nonsmooth systems.
Focuses on systems with an unstable periodic orbit near grazing-sliding, exhibiting two dominant frequencies.
Addresses dimension loss and trajectory instability leading to sliding behavior.

Purpose of the Study:

Analyze the attractor formed by forward sliding orbits in such systems.
Characterize the structure and dynamics of this attractor.
Classify the attractor's formation based on system parameters.

Main Methods:

Utilizes a Poincaré section for analysis.
Employs a three-parameter generalized map as a normal form.
Investigates the one-dimensional dynamics on the attractor for fixed parameters.

Main Results:

The attractor is shown to be contained within a finite number of lines intersecting at polygon vertices.
The attractor is typically larger than the associated polygon.
Classification of the number of lines forming the attractor is provided as a function of parameters.

Conclusions:

The study provides a detailed characterization of attractors in specific nonsmooth dynamical systems.
Understanding the structure of these attractors is crucial for predicting system behavior.
The findings contribute to the broader theory of nonlinear and nonsmooth dynamical systems.