Abstract
In this article an algorithm for the numerical approximation of random attractors based on the subdivision algorithm of Dellnitz and Hohmann is presented. It is applied to the stochastic Duffing-van der Pol oscillator, for which we also prove a theoretical result on the existence of stable/unstable manifolds and attractors. This system serves as a main example for a stochastically perturbed Hopf bifurcation. The results of our computations suggest that the structure of the random Duffing-van der Pol attractor and the dynamics on it are more complicated than assumed previously.
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6 References
L. Arnold. The unfolding of dynamics in stochastic analysis. Comput. Appl. Math., 16:3–25, 1997.
L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, 1998.
L. Arnold and B. Schmalfuß. Fixed points and attractors for random dynamical systems. In A. Naess and S. Krenk, editors, IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, pages 19–28. Kluwer, Dordrecht, 1996.
L. Arnold, N. Sri Namachchivaya, and K. R. Schenk-Hoppé. Toward an understanding of stochastic Hopf bifurcation: a case study. International Journal of Bifurcation and Chaos, 6:1947–1975, 1996.
A. Carverhill. Flows of stochastic dynamical systems: ergodic theory. Stochastics, 14:273–317, 1985.
H. Crauel. Extremal exponents of random dynamical systems do not vanish. Journal of Dynamics and Differential Equations, 2(3):245–291, 1990.
H. Crauel. Global random attractors are uniquely determined by attracting deterministic compact sets. Annali di Matematica, 1998. to appear.
H. Crauel, A. Debussche, and F. Flandoli. Random attractors. Journal of Dynamics and Differential Equations, 9(2):307–341, 1997.
H. Crauel and F. Flandoli. Attractors for random dynamical systems. Probab. Theory Relat. Fields, 100:365–393, 1994.
M. Dellnitz and A. Hohmann. The computation of unstable manifolds using subdivision and continuation. In I. H. H.W. Broer, S.A. van Gils and F. Takens, editors, Nonlinear Dynamical Systems and Chaos, pages 449–459. Birkhäuser, 1996.
M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math., 75:293–317, 1997.
F. Flandoli and B. Schmalfuß. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stochastics and Stochastics Reports, 59:21–45, 1996.
J.-M. Gambaudo. Perturbation of a Hopf bifurcation by an external time-periodic forcing. J. Differ. Equation, 57:172–199, 1985.
V. M. Gundlach. Random homoclinic orbits. Random & Computational Dynamics, 3:1–33, 1995.
P. Imkeller. The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations. Potential Analysis, 1998. To appear.
P. Imkeller and C. Lederer. An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator. Preprint 1998.
H. Keller and B. Schmalfuß. Attractors for stochastic differential equations with nontrivial noise. Buletinul A.S. a R.M. Mathematica, 1(26):43–54, 1998.
P. E. Kloeden, H. Keller, and B. Schmalfuß. Towards a theory of random numerical dynamics. In Stochastic Dynamics. Springer, Berlin Heidelberg New York, 1998.
Liu Pei-Dong. Random perturbations of axiom A basic sets. J. of Stat. Phys., 90:467–490, 1998.
J. E. Marsden and M. McCracken. The Hopf Bifurcation and Its Applications, volume 19 of Applied Mathematical Sciences. Springer-Verlag, 1976.
V. I. Oseledec. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197–231, 1968.
D. Ruelle. Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, New York, 1989.
K. R. Schenk-Hoppé. The stochastic Duffing-van der Pol equation. PhD thesis, Institut für Dynamische Systeme, Universität Bremen, 1996.
K. R. Schenk-Hoppé. Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlinear Dynamics, 11:255–274, 1996.
K. R. Schenk-Hoppé. Stochastic Hopf bifurcation: an example. Int. J. Non-Linear Mechanics, 31:685–692, 1996.
K. R. Schenk-Hoppé. Random attractors — general properties, existence, and applications to stochastic bifurcation theory. Discrete and Continuous Dynamical Systems, 4:99–130, 1998.
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Keller, H., Ochs, G. (1999). Numerical Approximation of Random Attractors. In: Stochastic Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-22655-9_5
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DOI: https://doi.org/10.1007/0-387-22655-9_5
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