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Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

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DOI: 10.4236/ajcm.2013.33032    4,148 Downloads   6,141 Views   Citations

ABSTRACT

This article describes the implementation of a novel method for detection and continuation of bifurcations in non-smooth complex dynamic systems. The method is an alternative to existing ones for the follow-up of associated phenomena, precisely in the circumstances in which the traditional ones have limitations (simultaneous impact, Filippov and first derivative discontinuities and multiple discontinuous boundaries). The topology of cycles in non-smooth systems is determined by a group of ordered segments and points of different regions and their boundaries. In this article, we compare the limit cycles of non-smooth systems against the sequences of elements, in order to find patterns. To achieve this goal, a method was used, which characterizes and records the elements comprising the cycles in the order that they appear during the integration process. The characterization discriminates: a) types of points and segments; b) direction of sliding segments; and c) regions or discontinuity boundaries to which each element belongs. When a change takes place in the value of a parameter of a system, our comparison method is an alternative to determine topological changes and hence bifurcations and associated phenomena. This comparison has been tested in systems with discontinuities of three types: 1) impact; 2) Filippov and 3) first derivative discontinuities. By coding well-known cycles as sequences of elements, an initial comparison database was built. Our comparison method offers a convenient approach for large systems with more than two regions and more than two sliding segments.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

I. Arango, F. Pineda and O. Ruiz, "Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles," American Journal of Computational Mathematics, Vol. 3 No. 3, 2013, pp. 222-230. doi: 10.4236/ajcm.2013.33032.

References

[1] M. Amorin, S. Divenyi, L. Franca and H. I. Weber, “Numerical and Experimental Investigations of the Nonlinear Dynamics and Chaos in Non-Smooth Systems,” Journal of Sound and Vibration, Vol. 301, No. 1-2, 2007, pp. 59-73. doi:10.1016/j.jsv.2006.09.014
[2] I. Arango, “Singular Point Tracking: A Method for the Analysis of Sliding Bifurcations in Non-smooth Systems,” PhD Thesis, Universidad Nacional de Colombia, 2011.
[3] I. Arango and J. A. Taborda, “Integration-Free Analysis of Non-smooth Local Dynamics in Planar Filippov Systems,” International Journal of Bifurcation and Chaos, Vol. 19, No. 3, 2009, pp. 947-975. doi:10.1142/S0218127409023391
[4] M. Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, “Piecewise-Smooth Dynamical Systems: Theory and Applications,” Springer, Berlin, 2008. doi:10.1007/978-1-84628-708-4
[5] P. Casini and F. Vestroni, “Nonstandard Bifurcations in Oscillators with Multiple Discontinuity Boundaries,” Nonlinear Dynamics, Vol. 35, No. 1, 2004, pp. 41-59. doi:10.1023/B:NODY.0000017487.21283.8d
[6] A. Colombo, M. Di Bernardo, E. Fossas and M. R. Jeffrey, “Teixeira Singularities in 3D Switched Feedback Control Systems,” Systems & Control Letters, Vol. 59, No. 10, 2010, pp. 615-622. doi:10.1016/j.sysconle.2010.07.006
[7] F. Dercole and Y. Kuznetsov, “SlideCont: An Auto97 Driver for Bifurcation Analysis of Filippov Systems,” ACM Transactions on Mathematical Software (TOMS), Vol. 31, No. 1, 2005, pp. 95-119. doi:10.1145/1055531.1055536
[8] L. Dieci and L. Lopez, “Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 2023-2051. doi:10.1137/080724599
[9] A. Filippov and F. Arscott, “Differential Equations with Discontinuous Righthand Sides: Control Systems, Volume 18 of Mathematics and Its Applications: Soviet Series,” Springer, Berlin, 1988.
[10] M. Guardia, T. M. Seara and M. A. Teixeira, “Generic Bifurcations of Low Codimension of Planar Filippov Systems,” Journal of Differential Equations, Vol. 250, No. 4, 2011, pp. 1967-2023. doi:10.1016/j.jde.2010.11.016
[11] Y. Kuznetsov, S. Rinaldi and A. Gragnani, “One-Parameter Bifurcations in Planar Filippov Systems,” International Journal of Bifurcation and Chaos, Vol. 13, No. 8, 2003, pp. 2157-2188. doi:10.1142/S0218127403007874
[12] R. I. Leine, “Bifurcations in Discontinuous Mechanical Systems of Filippov-Type,” PhD Thesis, Technische Universiteit Eindhoven, 2000.
[13] W. Marszalek and Z. Trzaska, “Singular Hopf Bifurcations in DAE Models of Power Systems,” Energy and Power Engineering, Vol. 3, No. 1, 2011, pp. 1-8. doi:10.4236/epe.2011.31001
[14] I. Merillas, “Modeling and Numerical Study of Non-smooth Dynamical Systems: Applications to Mechanical and Power,” PhD Thesis, Technical University of Catalonia, 2006.
[15] I. Arango and J. A. Taborda, “Integration-Free Analysis of Non-Smooth Local Dynamics in Planar Filippov System,” International Journal of Bifurcation and Chaos, Vol. 19, No. 3, 2009, pp. 947-975.
[16] A. Nordmark, “Existence of Periodic Orbits in Grazing Bifurcations of Impacting Mechanical Oscillators,” Nonlinearity, Vol. 14, No. 6, 2001, pp. 1517-1542. doi:10.1088/0951-7715/14/6/306
[17] T. S. Parker and L. Chua, “Practical Numerical Algorithms for Chaotic Systems,” Springer Limited, London, 2011.
[18] P. Thota and H. Dankowicz, “TC-HAT (TC): A Novel Toolbox for the Continuation of Periodic Trajectories in Hybrid Dynamical Systems,” SIAM Journal on Applied Dynamical Systems, Vol. 7, No. 4, 2008, pp. 1283-1322. doi:10.1137/070703028

  
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