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On the limit cycles of the Floquet differential equation
Multistability and localized attractors in a dissipative discrete NLS equation
1. | Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. |
2. | Departamento de Matemática y Mecánica, I.I.M.A.S - U.N.A.M., Apdo. Postal 20-726, 01000 México D. F. |
References:
[1] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
doi: 10.1016/j.na.2009.09.037. |
[2] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like nonautonomous evolution process, Int. J. of Bifurcation and Chaos 20, 9 (2010), 2751-2760.
doi: 10.1142/S0218127410027337. |
[3] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and stable manifolds, J. Diff. Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[4] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[5] |
D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett. 18 (1988), 794-796.
doi: 10.1364/OL.13.000794. |
[6] |
S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74-754. |
[7] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 1989. |
[8] |
P. Hartman, Ordinary Differential Equations, SIAM, 2002.
doi: 10.1137/1.9780898719222. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. |
[10] |
Y. V. Kartashov, V. V. Konotop and V. A. Visloukh, Two-dimensional dissipative solitons supported by localized gain, Opt. Lett., 36 (2011), 82-84.
doi: 10.1364/OL.36.000082. |
[11] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 2011. |
[12] |
C. K. Lam, B. A. Malomed, K. W. Chow and P. K. A. Wai, Spatial solitons supported by localized gain in nonlinear optical waveguides, Eur. Phys. J. Special Topics, 173 (2009), 233-243.
doi: 10.1140/epjst/e2009-01076-8. |
[13] |
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[14] |
P. Panayotaros, Continuation of normal modes in finite NLS lattices, Phys. Lett. A, 374 (2010), 3912-3919.
doi: 10.1016/j.physleta.2010.07.022. |
[15] |
P. Panayotaros and A. Aceves, Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators, Phys. Lett. A, 375 (2011), 3964-3972.
doi: 10.1016/j.physleta.2011.09.019. |
[16] |
Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Euro. Math. Soc., 2004.
doi: 10.4171/003. |
[17] |
M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lect. Notes Math. 1907, Springer, New York, 2007.
doi: 10.1007/978-3-540-71189-6. |
[18] |
M. O. Williams, C. W. McGrath and J. N. Kutz, Light-bullet routing and control with planar waveguide arrays, Opt. Express, 18 (2010), 11671-11682.
doi: 10.1364/OE.18.011671. |
[19] |
D. V. Zezyulin and V. V. Konotop, Nonlinear modes in finite-dimensional PT-symmetric systems, Phys. Rev. Lett, 108 (2012), 213906.
doi: 10.1103/PhysRevLett.108.213906. |
show all references
References:
[1] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
doi: 10.1016/j.na.2009.09.037. |
[2] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like nonautonomous evolution process, Int. J. of Bifurcation and Chaos 20, 9 (2010), 2751-2760.
doi: 10.1142/S0218127410027337. |
[3] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and stable manifolds, J. Diff. Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[4] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[5] |
D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett. 18 (1988), 794-796.
doi: 10.1364/OL.13.000794. |
[6] |
S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74-754. |
[7] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 1989. |
[8] |
P. Hartman, Ordinary Differential Equations, SIAM, 2002.
doi: 10.1137/1.9780898719222. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. |
[10] |
Y. V. Kartashov, V. V. Konotop and V. A. Visloukh, Two-dimensional dissipative solitons supported by localized gain, Opt. Lett., 36 (2011), 82-84.
doi: 10.1364/OL.36.000082. |
[11] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 2011. |
[12] |
C. K. Lam, B. A. Malomed, K. W. Chow and P. K. A. Wai, Spatial solitons supported by localized gain in nonlinear optical waveguides, Eur. Phys. J. Special Topics, 173 (2009), 233-243.
doi: 10.1140/epjst/e2009-01076-8. |
[13] |
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[14] |
P. Panayotaros, Continuation of normal modes in finite NLS lattices, Phys. Lett. A, 374 (2010), 3912-3919.
doi: 10.1016/j.physleta.2010.07.022. |
[15] |
P. Panayotaros and A. Aceves, Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators, Phys. Lett. A, 375 (2011), 3964-3972.
doi: 10.1016/j.physleta.2011.09.019. |
[16] |
Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Euro. Math. Soc., 2004.
doi: 10.4171/003. |
[17] |
M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lect. Notes Math. 1907, Springer, New York, 2007.
doi: 10.1007/978-3-540-71189-6. |
[18] |
M. O. Williams, C. W. McGrath and J. N. Kutz, Light-bullet routing and control with planar waveguide arrays, Opt. Express, 18 (2010), 11671-11682.
doi: 10.1364/OE.18.011671. |
[19] |
D. V. Zezyulin and V. V. Konotop, Nonlinear modes in finite-dimensional PT-symmetric systems, Phys. Rev. Lett, 108 (2012), 213906.
doi: 10.1103/PhysRevLett.108.213906. |
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