American Institute of Mathematical Sciences

Advanced
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 - A, 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, (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. 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. 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. 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. 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. 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. 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. doi: 10.1080/00036810701556136. Google Scholar

[9]

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

[10]

S. Coombes, R. Thul and K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models,, Phys. D, 241 (2012), 2042. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0218127404010874. Google Scholar

[20]

W. S. Loud, Periodic solutions of a perturbed autonomous system,, Ann. of Math., 70 (1959), 490. 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. doi: 10.1109/81.873871. Google Scholar

[22]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Springer, (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, (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. 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. 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. 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. 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. 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. 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. doi: 10.1080/00036810701556136. Google Scholar

[9]

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

[10]

S. Coombes, R. Thul and K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models,, Phys. D, 241 (2012), 2042. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0218127404010874. Google Scholar

[20]

W. S. Loud, Periodic solutions of a perturbed autonomous system,, Ann. of Math., 70 (1959), 490. 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. doi: 10.1109/81.873871. Google Scholar

[22]

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

[1]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
[2]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[3]

Miguel Ângelo De Sousa Mendes. Quasi-invariant attractors of piecewise isometric systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 323-338. doi: 10.3934/dcds.2003.9.323
[4]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[5]

I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295
[6]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[7]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[8]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[9]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[10]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[11]

Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261
[12]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[13]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[14]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[15]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[16]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[17]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[18]

M. R. S. Kulenović, Orlando Merino. A global attractivity result for maps with invariant boxes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 97-110. doi: 10.3934/dcdsb.2006.6.97
[19]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885
[20]

Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences