Abstract
The paper discusses several issues related to the numerical computation of the stable manifold of saddle-like periodic cycles in piecewise smooth dynamical systems. Results are presented for a particular stick–slip system. In the second part of the paper the same mechanical model is used to briefly describe the interaction between fold and adding-sliding bifurcations.
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References
Barreiro, A., Aracil, J., Pagano, D.: Detection of attraction domains of nonlinear systems using bifurcation analysis and Lyapunov functions. Int. J. Control 75, 314–327 (2002)
Colombo, A., Galvanetto, U.: Stable manifolds of saddles in piecewise smooth systems. CMES 53, 235–254 (2009)
Cruck, E., Moitie, R., Seube, N.: Estimation of basins of attraction for uncertain systems with affine and Lipschitz dynamics. Dyn. Control 11, 211–227 (2001)
Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalzyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, New York (2007)
Galvanetto, U.: Nonlinear dynamics of multiple friction oscillators. Comp. Method Appl. Mech. Eng. 178(3–4), 291–306 (1999)
Galvanetto, U.: Numerical computation of Lyapunov exponents in discontinuous maps implicitly defined. Comput. Phys. Commun. 131, 1–9 (2000)
Galvanetto, U.: Computation of the separatrix of basins of attraction in a non-smooth dynamical system. Phys. D 237, 2263–2271 (2008)
Hsu, C.S.: Cell-to-cell mapping: a method of global analysis for nonlinear systems. Springer, New York (1987)
Krauskopf, B., Osinga, H.M., Doedel, E.J., Henderson, M.E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurc. Chaos 14, 763–791 (2005)
Merillas, I.: Modeling and numerical study of nonsmooth dynamical systems. Ph.D. thesis, Dept. Matematica Aplicada IV, Universitat Politècnica de Catalunya (2006)
Oestreich, M., Hinrichs, N., Popp, K.: Bifurcation and stability analysis for a non-smooth friction oscillator. Arch. Appl. Mech. 66, 301–314 (1996)
Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989)
Soliman, M.S., Thompson, J.M.T.: Integrity measures quantifying the erosion of smooth fractal basins of attraction. J. Sound Vib. 135, 453–475 (1989)
Thompson, J.M.T., Soliman, M.S.: Fractal control boundaries of driven oscillators and their relevance to safe engineering design. Proc. R. Soc. Lond. A 428, 1–13 (1990)
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Galvanetto, U., Colombo, A. (2013). Computation of the Basins of Attraction in Non-smooth Dynamical Systems. In: Wiercigroch, M., Rega, G. (eds) IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design. IUTAM Bookseries (closed), vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5742-4_2
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DOI: https://doi.org/10.1007/978-94-007-5742-4_2
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