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.
1. Introduction - p. 7 2. Chapter I.1: Periodically forced discontinuous systems - p. 9 3. I.1.1 Setting of the problem and main results - p. 9 4. I.1.2 Geometric interpretation of assumed conditions - p. 21 5. I.1.3 Two-position automatic pilot for ship’s controller with periodic forcing - p. 22 6. I.1.4 Nonlinear planar applications - p. 25 7. I.1.5 Piecewise-linear planar application - p. 33 8. I.1.6 Non-smooth electronic circuits - p. 36 9. Chapter I.2: Bifurcation from family of periodic orbits in autonomous systems - p. 39 10. I.2.1 Setting of the problem and main results - p. 39 11. I.2.2 Geometric interpretation of required assumptions - p. 51 12. I.2.3 On the hyperbolicity of persisting orbits - p. 53 13. I.2.4 The particular case of the initial manifold - p. 57 14. I.2.5 3-dimensional piecewise-linear application - p. 58 15. I.2.6 Coupled Van der Pol and harmonic oscillators at 1-1 resonance - p. 62 16. Chapter I.3: Bifurcation from single periodic orbit in autonomous systems - p. 71 17. Chapter I.4: Sliding solution of periodically perturbed systems - p. 87 18. Chapter I.5: Weakly coupled oscillators - p. 105 19. Reference - p. 121 20. Part II: Forced hybrid systems - p. 122 21. Introduction - p. 123 22. Chapter II.1: Periodically forced impact systems - p. 125 23. Chapter II.2: Bifurcation from family of periodic orbits in forced billiards - p. 143 24. Reference - p. 153 25. Part III: Continuous approximations of non-smooth systems - p. 153 26. Introduction - p. 155 27. Chapter III.1: Transversal periodic orbits - p. 157 28. Chapter III.2: Sliding periodic orbits - p. 171 29. Chapter III.3: Impact periodic orbits - p. 189 30. III.3.1 Setting of the problem - p. 189 31. III.3.2 Bifurcation equation - p. 190 32. III.3.3 Bifurcation from a single periodic solution - p. 194 33. III.3.4 Poincaré-Andronov-Melnikov function and adjoint system - p. 198 34. III.3.5 Bifurcation from a manifold of periodic solutions - p. 205 35. Chapter III.4: Approximation and dynamics - p. 221 36. Reference - p. 227 37. Appendix A - p. 229 38. Bibliography - p. 239 39. Index - p. 243
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.
1. Introduction - p. 7 2. Chapter I.1: Periodically forced discontinuous systems - p. 9 3. I.1.1 Setting of the problem and main results - p. 9 4. I.1.2 Geometric interpretation of assumed conditions - p. 21 5. I.1.3 Two-position automatic pilot for ship’s controller with periodic forcing - p. 22 6. I.1.4 Nonlinear planar applications - p. 25 7. I.1.5 Piecewise-linear planar application - p. 33 8. I.1.6 Non-smooth electronic circuits - p. 36 9. Chapter I.2: Bifurcation from family of periodic orbits in autonomous systems - p. 39 10. I.2.1 Setting of the problem and main results - p. 39 11. I.2.2 Geometric interpretation of required assumptions - p. 51 12. I.2.3 On the hyperbolicity of persisting orbits - p. 53 13. I.2.4 The particular case of the initial manifold - p. 57 14. I.2.5 3-dimensional piecewise-linear application - p. 58 15. I.2.6 Coupled Van der Pol and harmonic oscillators at 1-1 resonance - p. 62 16. Chapter I.3: Bifurcation from single periodic orbit in autonomous systems - p. 71 17. Chapter I.4: Sliding solution of periodically perturbed systems - p. 87 18. Chapter I.5: Weakly coupled oscillators - p. 105 19. Reference - p. 121 20. Part II: Forced hybrid systems - p. 122 21. Introduction - p. 123 22. Chapter II.1: Periodically forced impact systems - p. 125 23. Chapter II.2: Bifurcation from family of periodic orbits in forced billiards - p. 143 24. Reference - p. 153 25. Part III: Continuous approximations of non-smooth systems - p. 153 26. Introduction - p. 155 27. Chapter III.1: Transversal periodic orbits - p. 157 28. Chapter III.2: Sliding periodic orbits - p. 171 29. Chapter III.3: Impact periodic orbits - p. 189 30. III.3.1 Setting of the problem - p. 189 31. III.3.2 Bifurcation equation - p. 190 32. III.3.3 Bifurcation from a single periodic solution - p. 194 33. III.3.4 Poincaré-Andronov-Melnikov function and adjoint system - p. 198 34. III.3.5 Bifurcation from a manifold of periodic solutions - p. 205 35. Chapter III.4: Approximation and dynamics - p. 221 36. Reference - p. 227 37. Appendix A - p. 229 38. Bibliography - p. 239 39. Index - p. 243
