Discontinuity induced bifurcations of nonhyperbolic cycles in nonsmooth systems

SIAM Journal on Applied Dynamical Systems

Volumn 9, Issue 1, 2010, Pages 62-83

  Colombo, Alessandro ^a   Dercole, Fabio ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

Bifurcation; Border collision; Codimension two; Nonhyperbolic; Nonsmooth

Indexed keywords

BORDER COLLISION; CODIMENSION TWO BIFURCATION; CODIMENSION-TWO; DISCONTINUITY-INDUCED BIFURCATION; DISCRETE TIME; GEOMETRIC FEATURE; INVARIANT SET; NON-SMOOTH; NONSMOOTH SYSTEMS; SCIENCE AND ENGINEERING; SYSTEM'S DYNAMICS;

CONTINUOUS TIME SYSTEMS;

BIFURCATION (MATHEMATICS);

EID: 74049113429     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/080732377     Document Type: Article

References (26)

1. M. S. BRANICKY, V. S. BORKAR, AND S. K. MITTER, A unified framework for hybrid control: Background, model and theory, IEEE Trans. Automat. Control, 43 (1998), pp. 352-358.
2. B. BROGLIATO, Nonsmooth Mechanics-Models, Dynamics and Control, Springer-Verlag, London, 1999.
3. A. COLOMBO AND S. RINALDI, Chaos in two-party democracies, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 2133-2140.
4. H. DANKOWICZ AND G. THAKUR, A Newton method for locating invariant tori of maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), pp. 1491-1503.
5. H. DANKOWICZ AND X. ZHAO, Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators, Phys. D, 202 (2005), pp. 238-257.
6. F. DERCOLE AND Yu. A. KUZNETSOV, Slidecont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Trans. Math. Software, 31 (2005), pp. 95-119.
7. F. DERCOLE AND S. MAGGI, Detection and continuation of a border collision bifurcation in a forest fire model, Appl. Math. Comput., 168 (2005), pp. 623-635.
8. A. DHOOGE, W. GOVAERTS, AND YU. A. KUZNETSOV, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2002), pp. 141-164.
9. M. DI BERNARDO, C. J. BUDD, A. R. CHAMPNEYS, AND P. KOWALCZYK, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, New York, Berlin, 2008.
10. E. J. DOEDEL, A. R. CHAMPNEYS, F. DERCOLE, T. F. FAIRGRIEVE, YU. A. KUZNETSOV, B. OLDEMAN, R. C. PAFFENROTH, B. SANDSTEDE, X. J. WANG, AND C. H. ZHANG, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, user manual, Department of Computer Science, Concordia University, Montreal, QC, 2007.
11. A. F. FILIPPOV, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.
12. I. G. KEVREKIDIS, R. ARIS, L. D. SCHMIDT, AND S. PELIKAN, Numerical computation of invariant circles of maps, Phys. D, 16 (1985), pp. 243-251.
13. C. KNUDSEN, R. FELDBERG, AND H. TRUE, Bifurcations and chaos in a model of a rolling railway wheelset, Proc. R. Soc. London A, 338 (1992), pp. 451-469.
14. P. KOWALCZYK, M. DI BERNARDO, A. R. CHAMPNEYS, S. J. HOGAN, M. HOMER, P. T. PIIRONINEN, YU. A. KUZNETSOV, AND A. NORDMARK, Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), pp. 601-629.
15. YU. A. KUZNETSOV, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, Berlin, 2004.
16. YU. A. KUZNETSOV, O. DIEKMANN, ANDW.-J. BEYN, Dynamical Systems Essentials, Springer-Verlag, New York, 2010, to appear.
17. J. LYGEROS, K. H. JOHANSSON, S. N. SIMIĆ , J. ZHANG, AND S. S. SASTRY, Dynamical properties of hybrid automata, IEEE Trans. Automat. Control, 48 (2003), pp. 2-17.
18. S. MAGGI AND S. RINALDI, A second-order impact model for forest fire regimes, Theor. Popul. Biol., 70 (2006), pp. 174-182.
19. H. G. E. MEIJER, F. DERCOLE, AND B. OLDEMAN, Numerical bifurcation analysis, in Encyclopedia of Complexity and Systems Science, Springer-Verlag, New York, Berlin, 2009, pp. 6329-6352.
20. A. B. NORDMARK AND P. KOWALCZYK, A codimension-two scenario of sliding solutions in grazing-sliding bifurcations, Nonlinearity, 19 (2006), pp. 1-26.
21. D. J. W. SIMPSON AND J. D. MEISS, Unfolding a codimension-two, discontinuous, Andronov-Hopf bifurcation, Chaos, 18 (2008), paper 033125.
22. D. J. W. SIMPSON AND J. D. MEISS, Shrinking point bifurcations of resonance tongues for piecewisesmooth, continuous maps, Nonlinearity, 22 (2009), pp. 1123-1144.
23. P. THOTA AND H. DANKOWICZ, TC-HAT (TĈ): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 1283-1322.
24. P. THOTA, X. ZHAO, AND H. DANKOWICZ, Co-dimension-two grazing bifurcations in single-degree-offreedom impact oscillators, J. Comp. Nonlinear Dyn., 1 (2006), pp. 328-335.
25. H. TRUE AND R. ASMUND, The dynamics of a ra ilway freight wagon wheelset with dry friction damping, Vehicle Syst. Dyn., 38 (2003), pp. 149-163.
26. X. ZHAO AND H. DANKOWICZ, Unfolding degenerate grazing dynamics in impact actuators, Nonlinearity, 19 (2006), pp. 399-418.