Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems

By Michal Feckan and 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

• Language: English
• Publisher: Academic Press
• Release date: Jun 7, 2016
• ISBN: 9780128043646

Michal Feckan

Professor Michal Feckan 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. Feckan has written several scientific monographs that have been published at top international publishing houses

Book preview

Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems

Michal Fečkan

Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Department of Mathematical Analysis and Numerical Mathematics, Mlynská dolina, 842 48 Bratislava, Slovak Republic

Michal Pospíšil

Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovak Republic

Table of Contents

Cover image

Title page

Copyright

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

Copyright

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Dedication

To our beloved families

Acknowledgment

Partial support of Grants VEGA-MS 1/0071/14 and VEGA-SAV 2/0153/16, an award from Literárny fond and by the Slovak Research and Development Agency under contract No. APVV-14-0378 are appreciated.

Michal Fečkan; Michal Pospíšil, April 2016

Preface

Discontinuous systems describe many real processes characterized by instantaneous changes, such as electrical switching or impacts of a bouncing ball. This is the reason why many papers and books have appeared on this topic in the last few years. This book is a contribution to this direction; namely, it is devoted to the study of bifurcations of periodic solutions for general n-dimensional discontinuous systems. First, we study these systems under assumptions of transversal intersections with discontinuity/switching boundaries and sufficient conditions are derived for the persistence of single periodic solutions under nonautonomous perturbations from single solutions; or under autonomous perturbations from non-degenerate families of solutions; or from isolated solutions. Furthermore, bifurcations of periodic sliding solutions are studied from sliding periodic solutions of unperturbed discontinuous equations. Then bifurcations of forced periodic solutions are investigated for impact systems from single periodic solutions of unperturbed impact equations. We also study weakly coupled discontinuous systems. In addition, local asymptotic properties of derived perturbed periodic solutions are investigated for all studied problems. The relationship between non-smooth systems and their continuous approximations is investigated as well. Many examples of discontinuous ordinary differential equations and impact systems are given to illustrate the theoretical results. To achieve our results, we mostly use the so-called discontinuous Poincaré mapping, which maps a point to its position after one period of the periodic solution. This approach is rather technical. On the other hand, by this method we can get results for general dimensions of spatial variables and parameters as well as asymptotic results such as stability, instability and hyperbolicity of solutions. Moreover, we explain how this approach can be modified for differential inclusions. These are the aims of this book and make it unique, since no one else in any book has ever before studied bifurcations of periodic solutions in discontinuous systems in such general settings. Therefore, our results in this book are original.

Some parts of this book are related to our previous works. But we are substantially improving these results, give more details in the proofs and present more examples. Needless to say, this book contains brand new parts. So the aim of this book is to collect and improve our previous results, as well as to continue with new results. Numerical computations described by figures are given with the help of the computational software Mathematica.

This book is intended for post-graduate students, mathematicians, physicists and theoretically inclined engineers studying either oscillations of nonlinear discontinuous mechanical systems or electrical circuits by applying the modern theory of bifurcation methods in dynamical systems.

Michal Fečkan; Michal Pospíšil, Bratislava, Slovakia April 2016

About the Authors

Michal Fečkan is Professor of Mathematics at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovak Republic. He obtained his Ph.D. (mathematics) from the Mathematical Institute of Slovak Academy of Sciences in Bratislava, Slovak Republic. He is interested in nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.

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.

An introductory example

Let us consider a reflected pendulum sketched in Figure 0.1.

Figure 0.1 Reflected pendulum

In part A, the ball of radius r . When x , 0 < λ1 ≤ 1. So we consider the case of a heavy ball, which is the reason why the ball does not jump above the surface (see [. At x = l , 0 < λ2 ≤ 1. In part C. Note that in the limit case l = 0, the system is reduced to a single impact at x describe positions of slanted segments so the ball with the center at x = 0 or x = l touches the horizontal and one of the slanted segments simultaneously. Setting λ3 = g sin α cos α and λ4 = g sin β cos β, we get the equation

   (0.1)

where times t1 and t2 are unknown. When λ1 = λ2 = 1, then (0.1) is reduced to

   (0.2)

System (0.2) is piecewise-linear, i.e. the plane (x, y) is divided by the lines x = 0 and x = l into three regions

where on each of these regions the vector field of (only if λ3 = λ4 = 0, i.e. α = β = 0, which we do not consider, since it is trivial. The aim of this book is to study the other kind, the discontinuous/non-smooth systems. Moreover, when either 0 < λ1 < 1 or 0 < λ2 < 1, then (0.1) contains (0.2) with additional impact conditions when the ball is passing through x = 0 and x = l, respectively. We call systems like (0.1) hybrid. Systems of this kind are also studied in Part II.

Now we study in more detail the dynamics of (0.1) and (0.2). It is clear that the function

   (0.3)

for

is a first integral of (0.1) and (0.2) in the region O = A ∪ B ∪ C. First we study (0.2). Any of its solutions with xwhen y(t0) = 0 and x(t0) < 0. So shifting the time, we can consider (0.2) with initial value conditions x(0) = ξ < 0 and y(0) = 0. Then the solution satisfies

   (0.4)

The contour plot of (0.4) consists of periodic curves as can be seen in Figure 0.2.

Figure 0.2 Solutions of ( 0.2) with l = 1, λ3 = 2 and λ4 = 1

Now we switch to the system (0.1) with possibly damping impacts. The general solution is given by

  

(0.5)

where

Note

Taking the section Σ = (−∞, 0), we get the Poincaré mapping P : Σ → Σ of (0.1) given by

   (0.6)

This is a generalization of Poincaré mappings for continuous dynamical systems [2–4] to discontinuous/non-smooth ones. Taking λ1 = λ2 = 1, we get (0.2) with P(ξ) = ξ, ξ ∈ Σ, which corresponds to the above observation on the existence of periodic solutions with periods

  

(0.7)

. Furthermore, the function T(ξ) is decreasing on (−∞, ξ0] from ∞ to T0, and increasing on [ξ0, 0) from T0 to ∞. Thus the equation

, and has two different solutions for

When also l = 0 and there is a string force, then (0.2) is the reflection pendulum mentioned in [5] (see also Section II.1.3) with phase portrait in Figure III.1.6.

Now we consider a periodically forced and weakly damped (0.1) of the form

  

(0.8)

are hitting times, ω > 0, ηi, i = 1, 2, 3, 4 are constants and