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]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[2]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[3]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[4]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012
[5]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015
[6]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[7]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031
[8]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[9]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
[10]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[11]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453
[12]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[13]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[14]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[15]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[16]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[17]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[18]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[19]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379
[20]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences