Noise-induced reentrant transition of the stochastic Duffing oscillator

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise belongs to a well-defined interval. Noisy oscillations are found outside that range, i.e., for both weaker and stronger noise. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) – 05.45.-a Nonlinear dynamics and nonlinear dynamical systems In a classical calculation, Kapitza (1951) has shown that the unstable upright position of an inverted pendulum can be stabilised if its suspension axis undergoes sinusoidal vibrations of high enough frequency [1]. More generally, stabilisation can also be obtained with random forcing [2,3,4]. In both cases, analytical derivations of the stability limit are based on perturbative approaches, i.e., in the limit of small forcing or noise amplitudes. In a recent work [5], we studied the Duffing oscillator with parametric white noise when the fixed point of the underlying deterministic equation is stable: a purely noise-induced transition [6] occurs when stochastic forcing is strong enough compared to dissipation so as to ‘lift’ the system away from the absolute minimum of the potential well. We showed, using a factorisation argument, that the noise-induced transition occurs precisely when the Lyapunov exponent of the linearised stochastic equation changes sign. An analytical calculation of the Lyapunov exponent allowed us to deduce for all parameter values the bifurcation diagram, that was previously known only in the small noise (perturbative) limit. The relation between stochastic transitions in a nonlinear Langevin equation and the sign of the Lyapunov exponent is in fact mathematically rigorous and has been proved under fairly general conditions [7]. In [5], we also made the following striking observation: close to the bifurcation, the averaged observables of the oscillator (energy, amplitude square and velocity square), as well as all their non-zero higher-order moments, scale linearly with the distance from the threshold. This multifractal behaviour may provide a generic criterion to distinguish noise-induced transitions from ordinary deterministic bifurcations. In the present work, we reformulate the stochastic stabilisation of an unstable fixed point of the underlying deterministic system within the framework of noise-induced transitions. We extend the analysis of [5] and derive the full phase diagram of an inverted Duffing oscillator. We show in particular that a reentrant transition occurs in this zero-dimensional system. (Reentrant transitions induced by noise have been studied in the more complex setting of spatially extended systems [8]). Moreover, the inverted Duffing oscillator also exhibits a multifractal behaviour close to the transition point. The dissipative stochastic system considered here is defined by the equation: dx dt2 + γ dx dt + √ Dξ(t)x = −∂U ∂x , (1) where x(t) is the position of the oscillator at time t, γ the dissipation rate and ξ(t) denotes a stochastic process. The confining, anharmonic potential U(x) is defined as: U(x) = − 2 μx + 1 4 x , (2) where μ is a real parameter. Without noise, the corresponding deterministic system undergoes a forward pitchfork bifurcation when the origin becomes unstable as μ 2 Kirone Mallick, Philippe Marcq: Noise-induced reentrant transition of the stochastic Duffing oscillator changes sign from negative to positive values. In [5], we only considered the case μ < 0 and showed that for strong enough noise, the origin becomes unstable. Here, we study the case μ > 0 (‘inverted’ Duffing oscillator) and show that for a finite range of positive values of μ, a reentrant transition is observed when the noise amplitude is varied: the noisy oscillations obtained for weak and strong noise are suppressed for noise of intermediate amplitude. Our results are non-perturbative: they are based on an exact calculation of the Lyapunov exponent, performed for arbitrary parameter values, ξ(t) being a Gaussian white noise process. Furthermore, our numerical simulations indicate that the phenomenology described above is unchanged for coloured Ornstein-Uhlenbeck noise. We first rescale the time variable by taking the dissipative scale γ−1 as the new time unit. Equation (1) then becomes dx dt2 + dx dt − α x+ x + √ ∆ ξ(t) x = 0 , (3) where we have defined the dimensionless parameters α = μ γ2 and ∆ = D γ3 , (4) and rescaled the amplitude x(t) by a factor γ. Linearising equation (3) about the origin, we obtain the following stochastic differential equation: dx dt2 + dx dt − α x+ √ ∆ ξ(t) x = 0 . (5) The Lyapunov exponent of a stochastic dynamical system is generally defined as the long-time average of the local divergence rate from a given orbit [9]. In the case discussed here, deviations from the (trivial) orbit defined by the origin in phase space, (x(t), ẋ(t)) = (0, 0), satisfy equation (5). In practice, we use the (equivalent) definition for the (maximal) Lyapunov exponent Λ Λ = lim t→∞ 1 2 t 〈log x〉 , (6) where the brackets denote ensemble averaging. Let z(t) = ẋ(t)/x(t). From equation (5) we find that the new variable z(t) obeys: ż = α− z − z − √ ∆ ξ(t) . (7) The Lyapunov exponent Λ is equal to [5]