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Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems
Reviewing real-world processes characterized by instantaneous changes in general spatial variables and parameters, this research monograph explores their asymptotical results for application in stability, instability, and hyperbolicity research and provides realistic models based on unsolved discontinuous problems from the literature and describes how Poincaré-Andronov-Melnikov analysis can be used to solve them
Michal Feckan (Author), Michal Pospíšil (Author)
9780128042946
Hardback, published 17 May 2016
260 pages
23.4 x 19 x 2.2 cm, 1.04 kg
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.
An introductory example I. Piecewise-smooth systems of forced ODEs I.2. Bifurcation from family of periodic orbits in autonomous systems I.3. Bifurcation from single periodic orbit in autonomous systems I.4. Sliding solution of periodically perturbed systems I.5. Weakly coupled oscillators Reference II. Forced hybrid systems II.1. Periodically forced impact systems II.2. Bifurcation from family of periodic orbits in forced billiards Reference III. Continuous approximations of non-smooth systems III.1. Transversal periodic orbits III.2. Sliding periodic orbits III.3. Impact periodic orbits III.4. Approximation and dynamics Reference Appendix
Subject Areas: Real analysis, real variables [PBKB]
