Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems

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