Publications
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Doan TS, Engel M, Lamb J, et al., 2018, Hopf bifurcation with additive noise, Nonlinearity, Vol: 31, ISSN: 0951-7715
We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).
Callaway M, Doan TS, Lamb JSW, et al., 2017, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, Annales de l’Institut Henri Poincaré Probabilités et Statistiques, Vol: 53, Pages: 1548-1574, ISSN: 0246-0203
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 Equations10 (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 Equations10 (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.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 Equations10 (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 t
Doan TS, 2017, ON ANALYTICITY FOR LYAPUNOV EXPONENTS OF GENERIC BOUNDED LINEAR RANDOM DYNAMICAL SYSTEMS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, Vol: 22, Pages: 3113-3126, ISSN: 1531-3492
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Doan TS, Palmer KJ, Rasmussen M, 2016, The Bohl spectrum for nonautonomous differential equations, Journal of Dynamics and Differential Equations, Vol: 29, Pages: 1459-1485, ISSN: 1572-9222
We develop the Bohl spectrum for nonautonomous lineardifferential equation on a half line, which is a spectral concept that liesbetween the Lyapunov and the Sacker–Sell spectrum. We prove thatthe Bohl spectrum is given by the union of finitely many intervals, andwe show by means of an explicit example that the Bohl spectrum doesnot coincide with the Sacker–Sell spectrum in general even for boundedsystems. We demonstrate for this example that any higher-order nonlinearperturbation is exponentially stable (which is not evident from theSacker–Sell spectrum), but we show that in general this is not true. Wealso analyze in detail situations in which the Bohl spectrum is identicalto the Sacker–Sell spectrum.
Doan TS, Rasmussen M, Kloeden PE, 2015, The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor, Discrete and Continuous Dynamical Systems - Series B, Vol: 20, Pages: 875-887, ISSN: 1531-3492
The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
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