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Smooth stabilizers for measures on the torus
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, Spain, Spain |
2. | MYCENAE Project-Team, Paris-Rocquencourt Centre, Inria, Domaine de Voluceau BP 105, 78153 Le Chesnay Cedex, France |
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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