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Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems
1st Edition - May 17, 2016
Authors: Michal Feckan, Michal Pospíšil
Hardback ISBN:9780128042946
9 7 8 - 0 - 1 2 - 8 0 4 2 9 4 - 6
eBook ISBN:9780128043646
9 7 8 - 0 - 1 2 - 8 0 4 3 6 4 - 6
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… Read more
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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
Postgraduate students, mathematicians, physicists and theoretically inclined engineers either studying oscillations of nonlinear discontinuous mechanical systems or electrical circuits by applying the modern theory of bifurcation methods in dynamical systems
Dedication
Acknowledgment
Preface
About the Authors
An introductory example
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
Index
No. of pages: 260
Language: English
Published: May 17, 2016
Imprint: Academic Press
Hardback ISBN: 9780128042946
eBook ISBN: 9780128043646
MF
Michal Feckan
Professor Michal Fečkan works at the Department of Mathematical Analysis and Numerical Mathematics at the Faculty of Mathematics, Physics, and Informatics at Comenius University. He specializes in nonlinear functional analysis, and dynamic systems and their applications. There is much interest in his contribution to the analysis of solutions of equations with fractional derivatives. Fečkan has written several scientific monographs that have been published at top international publishing houses
Affiliations and expertise
Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, Department of Mathematical Analysis and Numerical Mathematics, Bratislava, Slovak Republic
MP
Michal Pospíšil
Michal Pospíšil is senior researcher at the Mathematical Institute of Slovak Academy of Sciences in Bratislava, Slovak Republic. He obtained his Ph.D. (applied mathematics) from the Mathematical Institute of Slovak Academy of Sciences in Bratislava, Slovak Republic. He is interested in discontinuous dynamical systems and delayed differential equations.
Affiliations and expertise
Slovak Academy of Sciences, Mathematical Institute, Bratislava, Slovakia