Poincaré Andronov Melnikov Analysis for Non Smooth Systems 1st Edition by Michal Fečkan, Michal Pospíšil 0128043644 9780128043646

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ISBN 10: 0128043644 

ISBN 13: 9780128043646

Author: Michal Fečkan, Michal Pospíšil

Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems is devoted to the study of bifurcations of periodic solutions for general n-dimensional discontinuous systems. The authors study these systems under assumptions of transversal intersections with discontinuity-switching boundaries. Furthermore, bifurcations of periodic sliding solutions are studied from sliding periodic solutions of unperturbed discontinuous equations, and bifurcations of forced periodic solutions are also investigated for impact systems from single periodic solutions of unperturbed impact equations. In addition, the book presents studies for weakly coupled discontinuous systems, and also the local asymptotic properties of derived perturbed periodic solutions.

The relationship between non-smooth systems and their continuous approximations is investigated as well. Examples of 2-, 3- and 4-dimensional discontinuous ordinary differential equations and impact systems are given to illustrate the theoretical results. The authors use so-called discontinuous Poincaré mapping which maps a point to its position after one period of the periodic solution. This approach is rather technical, but it does produce results for general dimensions of spatial variables and parameters as well as the asymptotical results such as stability, instability, and hyperbolicity.

• Extends Melnikov analysis of the classic Poincaré and Andronov staples, pointing to a general theory for freedom in dimensions of spatial variables and parameters as well as asymptotical results such as stability, instability, and hyperbolicity
• Presents a toolbox of critical theoretical techniques for many practical examples and models, including non-smooth dynamical systems
• Provides realistic models based on unsolved discontinuous problems from the literature and describes how Poincaré-Andronov-Melnikov analysis can be used to solve them
• Investigates the relationship between non-smooth systems and their continuous approximations

Table of contents:

Part I: Piecewise-smooth systems of forced ODEs
Introduction
Chapter I.1: Periodically forced discontinuous systems
I.1.1 Setting of the problem and main results
I.1.2 Geometric interpretation of assumed conditions
I.1.3 Two-position automatic pilot for ship’s controller with periodic forcing
I.1.4 Nonlinear planar applications
I.1.5 Piecewise-linear planar application
I.1.6 Non-smooth electronic circuits
Chapter I.2: Bifurcation from family of periodic orbits in autonomous systems
I.2.1 Setting of the problem and main results
I.2.2 Geometric interpretation of required assumptions
I.2.3 On the hyperbolicity of persisting orbits
I.2.4 The particular case of the initial manifold
I.2.5 3-dimensional piecewise-linear application
I.2.6 Coupled Van der Pol and harmonic oscillators at 1-1 resonance
Chapter I.3: Bifurcation from single periodic orbit in autonomous systems
I.3.1 Setting of the problem and main results
I.3.2 The special case for linear switching manifold
I.3.3 Planar application
I.3.4 Formulae for the second derivatives
Chapter I.4: Sliding solution of periodically perturbed systems
I.4.1 Setting of the problem and main results
I.4.2 Piecewise-linear application
Chapter I.5: Weakly coupled oscillators
I.5.1 Setting of the problem
I.5.2 Bifurcations from single periodic solutions
I.5.3 Bifurcations from families of periodics
I.5.4 Examples
Reference

Part II: Forced hybrid systems
Introduction
Chapter II.1: Periodically forced impact systems
II.1.1 Setting of the problem and main results
Chapter II.2: Bifurcation from family of periodic orbits in forced billiards
II.2.1 Setting of the problem and main results
II.2.2 Application to a billiard in a circle
Reference

Part III: Continuous approximations of non-smooth systems
Introduction
Chapter III.1: Transversal periodic orbits
III.1.1 Setting of the problem and main result
III.1.2 Approximating bifurcation functions
III.1.3 Examples
Chapter III.2: Sliding periodic orbits
III.2.1 Setting of the problem
III.2.2 Planar illustrative examples
III.2.3 Higher dimensional systems
III.2.4 Examples
Chapter III.3: Impact periodic orbits
III.3.1 Setting of the problem
III.3.2 Bifurcation equation
III.3.3 Bifurcation from a single periodic solution
III.3.4 Poincaré-Andronov-Melnikov function and adjoint system
III.3.5 Bifurcation from a manifold of periodic solutions
Chapter III.4: Approximation and dynamics
III.4.1 Asymptotic properties under approximation
III.4.2 Application to pendulum with dry friction
Reference

Appendix A
A.1 Nonlinear functional analysis
A.2 Multivalued mappings
A.3 Singularly perturbed ODEs
A.4 Note on Lyapunov theorem for Hill’s equation
Bibliography

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Tags: Michal Fečkan, Michal Pospíšil, Poincaré, Analysis