Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise

Mark Callaway, Thai Son Doan, Jeroen S. W. Lamb, and Martin Rasmussen

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Abstract

We develop the dichotomy spectrum for random dynamical systems and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise.

Crauel and Flandoli (J. Dynam. Differential Equations 10 (1998) 259–274) had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random equilibrium with negative Lyapunov exponent throughout, thus “destroying” this bifurcation. Indeed, we show that in this example the dynamics before and after the underlying deterministic bifurcation point are topologically equivalent.

However, in apparent paradox to (J. Dynam. Differential Equations 10 (1998) 259–274), we show that there is after all a qualitative change in the random dynamics at the underlying deterministic bifurcation point, characterized by the transition from a hyperbolic to a non-hyperbolic dichotomy spectrum. This breakdown manifests itself also in the loss of uniform attractivity, a loss of experimental observability of the Lyapunov exponent, and a loss of equivalence under uniformly continuous topological conjugacies.

Résumé

Nous développons le spectre de dichotomie pour les systèmes dynamiques aléatoires et nous démontrons son utilité pour la caractérisation des bifurcations de fourches dans des systèmes dynamiques aléatoires avec du bruit additif.

Crauel et Flandoli (J. Dynam. Differential Equations 10 (1998) 259–274) ont précédemment montré que l’ajout de bruit additif à un système comprenant une bifurcation de fourche déterministe produit un unique équilibre aléatoire attractif avec un exposant de Lyapunov négatif partout, « détruisant » ainsi cette bifurcation. En effet, nous montrons dans cet exemple que la dynamique avant et après le point de bifurcation déterministe sous-jacent sont topologiquement équivalentes.

Cependant, dans un paradoxe apparent avec (J. Dynam. Differential Equations 10 (1998) 259–274), nous montrons qu’il y a après tout un changement qualitatif du système aléatoire au point du bifurcation déterministe sous-jacent, caractérisé par la transition du spectre de dichotomie hyperbolique à un spectre non-hyperbolique. Cette rupture se manifeste elle-même aussi dans une perte d’attractivité uniforme, une perte d’observabilité expérimentale de l’exposant de Lyapunov, et une perte d’équivalence sous conjugaisons topologiques uniformes et continues.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 53, Number 4 (2017), 1548-1574.

Dates
Received: 4 March 2015
Revised: 4 May 2016
Accepted: 6 May 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773718

Digital Object Identifier
doi:10.1214/16-AIHP763

Subjects
Primary: 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 37H20: Bifurcation theory [See also 37Gxx]

Keywords
Dichtomy spectrum Finite-time Lyapunov exponent Pitchfork bifurcation Random dynamical system Topological equivalence

Citation

Callaway, Mark; Doan, Thai Son; Lamb, Jeroen S. W.; Rasmussen, Martin. The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1548--1574. doi:10.1214/16-AIHP763. https://projecteuclid.org/euclid.aihp/1511773718


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