PDFAbstract: We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcation.
Keywords: Piecewise linear system, structural stability, bifurcation diagram REFERENCES [1] J. Artes, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simul. 95, 2013, pp. 13–22. [1] J. Artes, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simul. 95, 2013, pp. 13–22. [2] K. Camlibel, M. Heemels and H. Schumacher, Stability and controllability of planar bimodal linear complementarity systems, Proc. IEEE Conf. Decis. Control, 2003, pp. 1651–1656. [3] M. Di Bernardo, D.J. Pagano and E. Ponce, Nonhyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach, Int. J. Bifurcation and Chaos 18, 2008, pp. 1377–1392. [4] J. Ferrer, D. Magret and M. Pena, Bimodal ˜ piecewise linear systems. Reduced forms, Int. J. Bifurcation and Chaos 20, 2010, pp. 2795–2808. [5] J. Ferrer, D. Magret and M. Pena, Differentiable ˜ Families of Planar Bimodal Linear Control Systems, Math. Probl. Eng., 292813, 2014, pp. 1–9. [6] J. Ferrer, M. Pena and A. Susin, Structural sta- ˜ bility of planar bimodal linear systems, Math. Probl. Eng., 892948, 2014, pp. 1–8. [7] J. Ferrer, M. Pena and A. Susin, Bifurcation di- ˜ agram of saddle/spiral bimodal linear systems, submitted to Int. J. Bifurcation and Chaos. [8] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcation and Chaos 8, 1998, pp. 2073–2097. [9] E. Freire, L. Pizarro and A.J. Rodriguez-Luis, Numerical continuation of homoclinic orbits to non-hyperbolic equilibria in planar systems, Nonlin. Dyn. 23, 2000, pp. 353–375. [10] J. Llibre, M. Ordonez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlin. Anal. 14, 2013, pp. 2002– 2012. [11] J. Sotomayor and R. Garcia, Structural stability of piecewise-linear vector fields, J. Diff. Eqs. 192, 2003, pp. 553–565. [12] B. Xu, F. Yang, Y. Tang and M. Lin, Homoclinic bifurcations in planar piecewise-linear systems, Discr. Dyn. Nature and Society, 732321, 2013, pp. 1–9.
Cite this paper
Josep Ferrer, Marta Pena, Antoni Susin. (2016) Bifurcation Diagram for Saddle/Source Bimodal Linear Dynamical Systems. International Journal of Mathematical and Computational Methods, 1 , 345-350
Copyright © 2016 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0