Back
 AJCM  Vol.3 No.3 , September 2013
Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles
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.
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

 
 
Top