Periodic orbits and invariant cones in three-dimensional piecewise linear systems

Victoriano Carmona; Emilio Freire; Soledad Fernández-García

doi:10.3934/dcds.2015.35.59

Article Contents

2015, Volume 35, Issue 1: 59-72. Doi: 10.3934/dcds.2015.35.59

This issue Previous Article Smooth stabilizers for measures on the torus Next Article On the modeling of moving populations through set evolution equations

Periodic orbits and invariant cones in three-dimensional piecewise linear systems

1.
Escuela Técnica Superior de Ingeniería, Departamento de Matemática Aplicada II, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla
2.
MYCENAE Project-Team, Paris-Rocquencourt Centre, Inria, Domaine de Voluceau BP 105, 78153 Le Chesnay Cedex

Received: July 2013

Revised: May 2014

Published: January 2015

Abstract / Introduction Related Papers Cited by

Abstract

We deal with the existence of invariant cones in a family of three-dimensional non-observable piecewise linear systems with two zones of linearity. We find a subfamily of systems with one invariant cone foliated by periodic orbits. After that, we perturb the system by making it observable and non-homogeneous. Then, the periodic orbits that remain after the perturbation are analyzed.

Keywords:

Mathematics Subject Classification: Primary: 34C25, 34C45; Secondary: 34C23, 37G15.

Citation:

Related Papers

Cited by

References

[1]	S. Barnet and R. G. Cameron, Introduction to Mathematical Control Theory, Oxford University Press, New York, 1985.
[2]	T. R. Blows and L. M. Perko, Bifurcation of limit cycles from centers and separatrix cycles of Planar analityc systems, SIAM Rev., 36 (1994), 341-376.doi: 10.1137/1036094.
[3]	V. Carmona, S. Fernández-García and E. Freire, Saddle-node bifurcation of invariant cones in 3D piecewise linear systems, Phys. D, 241 (2012), 623-635.doi: 10.1016/j.physd.2011.11.020.
[4]	V. Carmona, E. Freire, E. Ponce and F. Torres, On simplifying and classifying piecewise-linear systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 49 (2002), 609-620.doi: 10.1109/TCSI.2002.1001950.
[5]	V. Carmona, E. Freire, E. Ponce, J. Ros and F. Torres, Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones. Application to Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3153-3164.doi: 10.1142/S0218127405014027.
[6]	V. Carmona, E. Freire, E. Ponce and F. Torres, Bifurcation of invariant cones in piecewise linear homogeneous systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 2469-2484.doi: 10.1142/S0218127405013423.
[7]	V. Carmona, E. Freire, E. Ponce and F. Torres, The continuous matching of two stable linear systems can be unstable, Discrete and Contin. Dyn. Syst., 16 (2006), 689-703.doi: 10.3934/dcds.2006.16.689.
[8]	A. Cima, J. Llibre and M. A. Teixeira, Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging theory, Appl. Anal., 87 (2008), 149-164.doi: 10.1080/00036810701556136.
[9]	Earl A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
[10]	S. Coombes, R. Thul and K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models, Phys. D, 241 (2012), 2042-2057.doi: 10.1016/j.physd.2011.05.012.
[11]	Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Analysis, 69 (2008), 3610-3628.doi: 10.1016/j.na.2007.09.045.
[12]	E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.doi: 10.1142/S0218127498001728.
[13]	E. Freire, E. Ponce and J. Ros, Limit cycle bifurcation from center in symmetric piecewise-linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 895-907.doi: 10.1142/S0218127499000638.
[14]	E. Freire, E. Ponce and F. Torres, Hopf-like bifurcations in planar piecewise linear systems, Publ. Mat., 41 (1997), 135-148.doi: 10.5565/PUBLMAT_41197_08.
[15]	C. Kahlert and O. E. Rössler, Anaytical properties of Poincaré halfmaps in a class of piecewise-linear dynamical systems, Z. Naturforsch. A, 40 (1985), 1011-1025.
[16]	M. Kunze, Lecture Notes in Mathematics, Springer, 2000.doi: 10.1007/BFb0103843.
[17]	T. Küpper, Invariant cones for non-smooth dynamical systems, Math. Comput. Simulation., 79 (2008), 1396-1408.doi: 10.1016/j.matcom.2008.03.010.
[18]	T. Küpper, D. Weiss and H. A. Hoshman, Invariant manifolds for nonsmooth systems, Phys. D, 241 (2012), 1895-1902.doi: 10.1016/j.physd.2011.07.012.
[19]	J. Llibre and A. E. Teruel, Existence of Poincaré maps in piecewise linear differential systems in $\mathbb R^N$, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2843-2851.doi: 10.1142/S0218127404010874.
[20]	W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. of Math., 70 (1959), 490-529.doi: 10.2307/1970327.
[21]	G. M. Maggio, M. di Bernardo and M. P. Kennedy, Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator, IEEE Trans. Circuits Systems I Fund. Theory Appl., 47 (2000), 1160-1177.doi: 10.1109/81.873871.
[22]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlín, 1996.doi: 10.1007/978-3-642-61453-8.