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 hyperbolicityPresents a toolbox of critical theoretical techniques for many practical examples and models, including non-smooth dynamical systemsProvides realistic models based on unsolved discontinuous problems from the literature and describes how Poincaré-Andronov-Melnikov analysis can be used to solve themInvestigates the relationship between non-smooth systems and their continuous approximations INDICE: Introduction I Piecewise-smooth systems of forced ODEs Periodically forced discontinuous systems Setting of the problem and main results Geometric interpretation of assumed conditions Nonlinear planar applications Piecewise linear planar applications Non-smooth electronic circuits Bifurcation from family of periodic orbits in autonomous systems Setting of the problem and main results Geometric interpretation of required assumptions On the hyperbolicity of persisting orbits The particular case of the initial value manifold 3-dimensional piecewise-linear application Coupled Van der Pol and harmonic oscillators at 1-1 resonance Bifurcation from single periodic orbit in autonomous systems Setting of the problem and main results The special case for linear switching manifold Planar applications Formulae for the second derivatives Sliding solution of periodically perturbed systems Setting of the problem and main results Piecewise linear applications Mechanical oscillator with dry friction Weakly coupled oscillators Setting of the problem and main results Examples II Forced hybrid systems Periodically forced impact systems Setting of the problem and main results Pendulum hitting moving obstacle Forced reflection pendulum Forced billiard Appendix

• ISBN: 978-0-12-804294-6
• Editorial: Academic Press
• Encuadernacion: Cartoné
• Páginas: 260
• Fecha Publicación: 08/06/2016
• Nº Volúmenes: 1
• Idioma: Inglés