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Detectable canard cycles with singular slow dynamics of any order at the turning point
Time-dependent attractor for the Oscillon equation
1. | Indiana University Mathematics Department, Bloomington, IN 47405, United States |
2. | Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, FL 33149, United States |
3. | The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405 |
∂ tt $u(x,t) +H $ ∂ t$ u(x,t) -\e^{-2Ht}$ ∂ xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \R,$
with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\A=\A(t)$. The kernel sections $\A(t)$ have finite fractal dimension.
References:
[1] |
A. B. Adib, M. Gleiser and C. A. S. Almeida, Long lived oscillons from asymmetric bubbles: Existence and stability,, Phys. Rev. D, 66 (2002).
doi: doi:10.1103/PhysRevD.66.085011. |
[2] |
L. Arnold, "Random Dynamical Systems,", Springer-Verlag, (1998).
|
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).
|
[4] |
V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\R^3$,, Discrete Cont. Dynam. Syst., 7 (2001), 719.
doi: doi:10.3934/dcds.2001.7.719. |
[5] |
Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems,, Probab. Theory Related Fields, 95 (1993), 87.
doi: doi:10.1007/BF01197339. |
[6] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems,, Dynam. Contin. Discrete Impuls. Systems A, 10 (2003), 491.
|
[7] |
T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of nonautonomous partial differential equations,, ANZIAM J., 45 (2003), 207.
doi: doi:10.1017/S1446181100013274. |
[8] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.
doi: doi:10.1016/j.na.2005.03.111. |
[9] |
T. Caraballo, G. Ł ukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263.
|
[10] |
D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynam. Systems Theory, 2 (2002), 9.
|
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279.
|
[12] |
V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,'', American Mathematical Society Colloquium Publications, (2002).
|
[13] |
E. J. Copeland, M. Gleiser and H. R. Muller, Oscillons: Resonant configurations during bubble collapse,, Phys. Rev. D., 52 (1995), 1920.
doi: doi:10.1103/PhysRevD.52.1920. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
doi: doi:10.1007/BF02219225. |
[15] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365.
doi: doi:10.1007/BF01193705. |
[16] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Cont. Dyn. Systems, 10 (2004), 221.
|
[17] |
E. Farhi, N. Graham, V. Khemani, et al., An oscillon in the $SU(2)$ gauged Higgs model,, Phys. Rev. D, 72 (2005).
doi: doi:10.1103/PhysRevD.72.101701. |
[18] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, Proc. Amer. Math. Soc., 134 (2006), 117.
doi: doi:10.1090/S0002-9939-05-08340-1. |
[19] |
J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations,, J. Math. Pures Appl., 66 (1987), 273.
|
[20] |
M. Gleiser and A. Sornberger, Longlived localized field configurations in small lattices: Application to oscillons,, Phys. Rev. E, 62 (2000), 1368.
doi: doi:10.1103/PhysRevE.62.1368. |
[21] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', Mathematical Surveys and Monographs, (1988).
|
[22] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'', Recherches en Mathmatiques Appliques [Research in Applied Mathematics], (1991).
|
[23] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'', Cambridge University Press, (1991).
|
[24] |
B. B. Mandelbrot, "The Fractal Geometry of Nature,'', Schriftenreihe fr den Referenten. [Series for the Referee] W. H. Freeman and Co., (1982).
|
[25] |
A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,'', Handbook of Differential Equations: Evolutionary Equations, IV (2008), 103.
|
[26] |
P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, Nonlinear Anal., 71 (2009), 3956.
doi: doi:10.1016/j.na.2009.02.065. |
[27] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Cont. Dynam. Syst., 10 (2004), 473.
doi: doi:10.3934/dcds.2004.10.473. |
[28] |
V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, Comm. Pure Appl. Anal., 2 (2007), 481.
|
[29] |
A. Riotto, Are oscillons present during a first order electroweak phase transition?,, Phys. Lett. B, 365 (1996), 64.
doi: doi:10.1016/0370-2693(95)01239-7. |
[30] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V.\ Reitmann, 73 (1992), 185. Google Scholar |
[31] |
M. Schroeder, "Fractals, Chaos, Power Laws,'', W. H. Freeman and Company, (1991).
|
[32] |
C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645.
doi: doi:10.1088/0951-7715/19/11/008. |
[33] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition, Applied Mathematical Sciences, (1997).
|
[34] |
P. B. Umbanhower, F. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer,, Nature, 382 (1996), 793.
doi: doi:10.1038/382793a0. |
[35] |
Y. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365.
doi: doi:10.1016/j.na.2006.11.002. |
show all references
References:
[1] |
A. B. Adib, M. Gleiser and C. A. S. Almeida, Long lived oscillons from asymmetric bubbles: Existence and stability,, Phys. Rev. D, 66 (2002).
doi: doi:10.1103/PhysRevD.66.085011. |
[2] |
L. Arnold, "Random Dynamical Systems,", Springer-Verlag, (1998).
|
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).
|
[4] |
V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\R^3$,, Discrete Cont. Dynam. Syst., 7 (2001), 719.
doi: doi:10.3934/dcds.2001.7.719. |
[5] |
Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems,, Probab. Theory Related Fields, 95 (1993), 87.
doi: doi:10.1007/BF01197339. |
[6] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems,, Dynam. Contin. Discrete Impuls. Systems A, 10 (2003), 491.
|
[7] |
T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of nonautonomous partial differential equations,, ANZIAM J., 45 (2003), 207.
doi: doi:10.1017/S1446181100013274. |
[8] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.
doi: doi:10.1016/j.na.2005.03.111. |
[9] |
T. Caraballo, G. Ł ukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263.
|
[10] |
D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynam. Systems Theory, 2 (2002), 9.
|
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279.
|
[12] |
V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,'', American Mathematical Society Colloquium Publications, (2002).
|
[13] |
E. J. Copeland, M. Gleiser and H. R. Muller, Oscillons: Resonant configurations during bubble collapse,, Phys. Rev. D., 52 (1995), 1920.
doi: doi:10.1103/PhysRevD.52.1920. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
doi: doi:10.1007/BF02219225. |
[15] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365.
doi: doi:10.1007/BF01193705. |
[16] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Cont. Dyn. Systems, 10 (2004), 221.
|
[17] |
E. Farhi, N. Graham, V. Khemani, et al., An oscillon in the $SU(2)$ gauged Higgs model,, Phys. Rev. D, 72 (2005).
doi: doi:10.1103/PhysRevD.72.101701. |
[18] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, Proc. Amer. Math. Soc., 134 (2006), 117.
doi: doi:10.1090/S0002-9939-05-08340-1. |
[19] |
J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations,, J. Math. Pures Appl., 66 (1987), 273.
|
[20] |
M. Gleiser and A. Sornberger, Longlived localized field configurations in small lattices: Application to oscillons,, Phys. Rev. E, 62 (2000), 1368.
doi: doi:10.1103/PhysRevE.62.1368. |
[21] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', Mathematical Surveys and Monographs, (1988).
|
[22] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'', Recherches en Mathmatiques Appliques [Research in Applied Mathematics], (1991).
|
[23] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'', Cambridge University Press, (1991).
|
[24] |
B. B. Mandelbrot, "The Fractal Geometry of Nature,'', Schriftenreihe fr den Referenten. [Series for the Referee] W. H. Freeman and Co., (1982).
|
[25] |
A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,'', Handbook of Differential Equations: Evolutionary Equations, IV (2008), 103.
|
[26] |
P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, Nonlinear Anal., 71 (2009), 3956.
doi: doi:10.1016/j.na.2009.02.065. |
[27] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Cont. Dynam. Syst., 10 (2004), 473.
doi: doi:10.3934/dcds.2004.10.473. |
[28] |
V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, Comm. Pure Appl. Anal., 2 (2007), 481.
|
[29] |
A. Riotto, Are oscillons present during a first order electroweak phase transition?,, Phys. Lett. B, 365 (1996), 64.
doi: doi:10.1016/0370-2693(95)01239-7. |
[30] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V.\ Reitmann, 73 (1992), 185. Google Scholar |
[31] |
M. Schroeder, "Fractals, Chaos, Power Laws,'', W. H. Freeman and Company, (1991).
|
[32] |
C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645.
doi: doi:10.1088/0951-7715/19/11/008. |
[33] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition, Applied Mathematical Sciences, (1997).
|
[34] |
P. B. Umbanhower, F. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer,, Nature, 382 (1996), 793.
doi: doi:10.1038/382793a0. |
[35] |
Y. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365.
doi: doi:10.1016/j.na.2006.11.002. |
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