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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 and 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.

[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.

show all references

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.

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