Stable manifolds of saddles in piecewise smooth systems

CMES - Computer Modeling in Engineering and Sciences

Volumn 53, Issue 3, 2009, Pages 235-254

  Colombo, A ^a   Galvanetto, U ^b  

^a POLITECNICO DI MILANO   (Italy)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Basin of attraction; Discontinuous; Dynamical system; Friction oscillator; Numerical method; Piecewise smooth; Stable manifold

Indexed keywords

BASIN OF ATTRACTION; DISCONTINUOUS; DISCONTINUOUS DYNAMICAL SYSTEMS; PIECEWISE SMOOTH; STABLE MANIFOLD;

BIPED LOCOMOTION; FRICTION; NUMBER THEORY; NUMERICAL METHODS; SLIP FORMING;

DYNAMICAL SYSTEMS;

EID: 77949851018     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (21)

1. Aubin, J. P.; Cellina, A. (1984): Differential Inclusions. Springer-Verlag, Berlin.
2. Cui, T.; Liao, W.; Yu, D. (2009): Topological approach for analyszing and modeling the aerodynamic hysteresis of an airfoil. CMES: Computer Modeling in Engineering & Sciences, vol.45, pp. 273-293.
3. di Bernardo, M.; Budd, C. J.; Champneys, A. R.; Kowalczyk, P. (2008): Piecewise-smooth Dynamical Systems: Theory and Applications. Springer-Verlag.
4. Doedel, E. J.; Champneys, A. R.; Dercole, F.; Fairgrieve, T. F.; Kuznetsov, Yu. A.; Oldeman, B.; Paffenroth, R. C.; Sandstede, B.; Wang, X. J.; Zhang, C. H. (2007): AUTO-07p: Continuation and bifurcation software for ordinary differential equations. Department of Computer Science, Concordia University, Montreal, QC, 2007.
5. Filippov, A. F. (1988): Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht.
6. Galvanetto, U. (2008): Computation of the separatrix of basins of attraction in a non-smooth dynamical system. Physica D, vol.237, pp. 2263-2271.
7. Jeffrey, M. R.; Colombo, A. (2009): The two-fold singularity of discontinuous vector fields. SIAM Journal on Applied Dynamical Systems, vol.8, pp. 624-640.
8. Jeffrey, M. R.; Hogan, S. J. (submitted): The geometry of generic sliding bifurcations. SIAM Review. Online preprint: http://seis.bris.ac.uk/ mm0093/SIREV2009
9. Krauskopf, B.; Osinga, H. M.; Doedel, E. J.; Henderson, M. E.; Guckenheimer, J.; Vladimirsky, A.; Dellnitz, M.; Junge, O. (2005): A survey of methods for computing (un)stable manifolds of vector fields. International Journal of Bifurcation and Chaos, vol.15, pp. 763-791.
10. Leine, R. I. (2000): Bifurcations in Discontinuous Mechanical Systems of Filippov-Type. Ph.D. thesis, Technical University of Eindhoven, 2000.
11. Meiss, J. D. (2007): Differential Dynamical Systems. SIAM.
12. Oestreich, M.; Hinrichs, N.; Popp, K. (1996): Bifurcation and stability analysis for a non-smooth friction oscillator. Archive of Applied Mechanics, vol.66, pp. 301-314.
13. Poston, T.; Stewart, I. (1978): Catastrophe Theory and its Applications. Pitman.
14. Soliman, M. S.; Thompson, J. M. T. (1989): Integrity measures quantifying the erosion of smooth and fractal basins of attraction. Journal of Sound and Vibration, vol.135, pp. 453-475.
15. Teixeira, M. A. (1990): Stability conditions for discontinuous vector fields. Journal of Differential Equations, vol.88, pp. 15-29.
16. Teixeira, M. A. (1993): Generic bifurcation of sliding vector fields. Journal of Mathematical Analysis and Applications, vol.176, pp. 436-457.
17. Thom, R. (1975): Structural Stability and Morphogenesis. Benjamin, Reading, MA. (translated by D. H. Fowler).
18. Thompson, J. M. T.; Soliman, M. S. (1990): Fractal control boundaries of driven oscillators and their relevance to safe engineering design. Proceedings of the Royal Society of London A, vol.428, pp. 1-13.
19. Thota, P.; Dankowicz, H. (2008): TC-HAT (cTC): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems. SIAM Journal on Applied Dynamical Systems, vol.7, pp. 1283-1322.
20. Whitney, H. (1955): On singularities of mappings of Euclidean spaces 1. mappings of the plane into the plane. Annals of Mathematics, vol.62, pp. 374-410.
21. Zhusubaliyev, Z. T.; Mosekilde, E. (2003): Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. World Scientific.