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January  2015, 35(1): 59-72. doi: 10.3934/dcds.2015.35.59

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

Victoriano Carmona 1, , Emilio Freire 1, and  Soledad Fernández-García 2,

1. 

Escuela Técnica Superior de Ingeniería, Departamento de Matemática Aplicada II, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain, Spain
2. 

MYCENAE Project-Team, Paris-Rocquencourt Centre, Inria, Domaine de Voluceau BP 105, 78153 Le Chesnay Cedex, France

Received  July 2013 Revised  May 2014 Published  August 2014

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: Piecewise linear systems, invariant manifolds, invariant cones., periodic orbits, half-Poincaré maps.
Mathematics Subject Classification: Primary: 34C25, 34C45; Secondary: 34C23, 37G1.
Citation: Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59
References:
[1]

S. Barnet and R. G. Cameron, Introduction to Mathematical Control Theory, Oxford University Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Earl A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Kunze, Lecture Notes in Mathematics, Springer, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. of Math., 70 (1959), 490-529. doi: 10.2307/1970327.  Google Scholar

[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.  Google Scholar

[22]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlín, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

show all references

References:
[1]

S. Barnet and R. G. Cameron, Introduction to Mathematical Control Theory, Oxford University Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Earl A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Kunze, Lecture Notes in Mathematics, Springer, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. of Math., 70 (1959), 490-529. doi: 10.2307/1970327.  Google Scholar

[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.  Google Scholar

[22]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlín, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

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