One-parameter bifurcations in planar Filippov systems

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 13, Issue 8, 2003, Pages 2157-2188

  Kuznetsov, Yu A ^a   Rinaldi, S ^b   Gragnani, A ^c  

^a INSTITUTE OF MATHEMATICAL PROBLEMS OF BIOLOGY   (Russian Federation)

^b UTRECHT UNIVERSITY   (Netherlands)

^c POLITECNICO DI MILANO   (Italy)

Author keywords

Continuation techniques; Discontinuous piecewise smooth systems; Filippov systems; Sliding bifurcations

Indexed keywords

ORDINARY DIFFERENTIAL EQUATIONS; STATE SPACE METHODS; TOPOLOGY;

FILIPPOV SYSTEMS;

BIFURCATION (MATHEMATICS);

EID: 0242414775     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127403007874     Document Type: Review

References (38)

1. Ascher, U. M. & Spiteri, R. J. [1994] "Collocation software for boundary value differential-algebraic equations," SIAM J. Sci. Comput. 15, 938-952.
2. Aubin, J. P. & Cellina, A. [1984] Differential Inclusions (Springer-Verlag, Berlin).
3. Bautin, N. N. & Leontovich, E. A. [1976] Methods and Techniques for Qualitative Analysis of Dynamical Systems on the Plane (Nauka, Moscow), (in Russian).
4. Bazykin, A. D. [1998] Nonlinear Dynamics of Interacting Populations (World Scientific, Singapore).
5. Brogliato, B. [1999] Nonsmooth Mechanics (Springer-Verlag, NY).
6. Dankowitz, H. & Nordmark, A. B. [2000] "On the origin and bifurcations of stick-slip oscillations," Physica D136, 280-302.
7. di Bernardo, M., Champneys, A. R. & Budd, C. J. [1998a] "Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the DC/DC buck converter," Nonlinearity 11, 858-890.
8. di Bernardo, M., Garofalo, F., Glielmo, L. & Vasca, F. [1998b] "Switchings, bifurcations and chaos in DC/DC converters," IEEE Trans. Circuits Syst. I Fund. Th. Appl. 45, 133-143.
9. di Bernardo, M., Feigin, M. I., Hogan, S. J. & Homer, M.E. [1999] "Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems," Chaos Solit. Fract. 10, 1881-1908.
10. di Bernardo, M., Budd, C. J. & Champneys, A. R. [2001a] "Grazing and border-collision in piecewise-smooth systems: A unified analytical framework," Phys. Rev. Lett. 86, 2553-2556.
11. di Bernardo, M., Johansson, K. H. & Vasca, F. [2001b] "Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations," Int. J. Bifurcation and Chaos 11, 1121-1140.
12. di Bernardo, M., Kowalczyk, P. & Nordmark, A. [2002] "Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings," Physica D11, 175-205.
13. Doedel, E. J. & Kernévez, J. P. [1986] "AUTO: Software for continuation problems in ordinary differential equations with applications," Technical Report, California Institute of Technology, Applied Mathematics.
14. Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Yu. A., Sandstede, B. & Wang, X. J. [1997] "AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont), User's Guide," Technical Report, Concordia University, Montreal, Canada.
15. Doole, S. H. & Hogan, S. J. [1996] "A piecewise linear suspension bridge model: Nonlinear dynamics and orbit continuation," Dyn. Stab. Syst. 11, 19-29.
16. Feigin, M. I. [1994] Forced Oscillations in Systems with Discontinuous Nonlinearities (Nauka, Moscow), (in Russian).
17. Filippov, A. F. [1964] "Differential equations with discontinuous right-hand side," in American Mathematical Society Translations, Series 2, (AMS), pp 199-231.
18. Filippov, A. F. [1988] Differential Equations with Discontinuous Right-Hand Sides (Kluwer Academic, Dordrecht).
19. Flügge-Lotz, I. [1953] Discontinuous Automatic Control (Princeton University Press).
20. Freire, E., Ponce, E., Rodrigo, F. & Torres, F. [1998] "Bifurcation sets of continuous piecewise linear systems with two zones," Int. J. Bifurcation and Chaos 8, 2073-2097.
21. Galvanetto, U., Bishop, S. R. & Briseghella, L. [1995] "Mechanical stick-slip vibrations," Int. J. Bifurcation and Chaos 5, 651-673.
22. Gatto, M., Mandrioli, D. & Rinaldi, S. [1973] "Pseudoequilibrium in dynamical systems," Int. J. Syst. Sci. 4, 809-824.
23. Giannakopoulos, F. & Pliete, K. [2001] "Planar systems of piecewise linear differential equations with a line of discontinuity," Nonlinearity 14, 1-22.
24. Gubar', N. A. [1971] "Bifurcations in the vicinity of a 'fused focus'," J. Appl. Math. Mech. 35, 890-895.
25. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, NY).
26. Hasler, M. & Neirynck, J. [1985] Circuits Non Linéaires (Presses Polytechniques Romandes, Lausanne).
27. Hogan, S. J. [1989] "On the dynamics of rigid-block motion under harmonic forcing," Proc. Roy. Soc. London A425, 441-476.
28. Kowalczyk, P. & di Bernardo, M. [2001] "On a novel class of bifurcations in hybrid dynamical systems: The case of relay feedback systems," in Hybrid Systems: Computation and Control, eds. di Benedetto, M. & Sangiovanni-Vincentelli, A. (Springer-Verlag, Berlin), pp. 361-374.
29. Kunze, M. & Küpper, T. [1997] "Qualitative bifurcation analysis of a non-smooth friction oscillator model," Z. Angew. Math. Phys. 48, 87-101.
30. Kunze, M. [2000] Non-Smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744, (Springer-Verlag, Berlin).
31. Kuznetsov, Yu. A. & Levitin, V. V. [1995-1997] "CONTENT: A multiplatform environment for analyzing dynamical systems," Technical Report, Dynamical Systems Laboratory, CWI, Amsterdam, The Netherlands. ftp://ftp.cwi.nl/pub/CONTENT.
32. Kuznetsov, Yu. A. [1998] Elements of Applied Bifurcation Theory, 2nd edition (Springer-Verlag, NY).
33. Leine, R. I. [2000] "Bifurcations in discontinuous mechanical systems of Filippov-type," PhD thesis, Technical University of Eindhoven, The Netherlands.
34. McGeer, T. [1990] "Passive dynamic walking," Int. J. Robot. Res. 9, 62-82.
35. Oestreich, M., Hinrichs, N., Popp, K. & Budd, C. J. [1997] "Analytical and experimental investigation of an impact oscillator," Proc. ASME 16th Biennal Conf. Mech. Vibrations and Noise, Sacramento, California, DETC97VIB-3907, 1-11.
36. Tsypkin, Ya. Z. [1984] Relay Control Systems (Cambridge University Press).
37. Utkin, V. I. [1977] "Variable structure systems with sliding modes," IEEE Trans. Automat. Contr. 22, 212-222.
38. Van de Vrande, B. L., Van Campen, D. H. & De Kraker, A. [1999] "An approximate analysis of dry-friction-induced stick-slip vibration by a smoothing procedure," Nonlin. Dyn. 19, 157-169.