Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

Abstract

Abstract. We study the largest Lyapunov exponent of the response of a two dimensional non-Hamiltonian system driven by additive white noise. The specific system we consider is the third-order truncated normal form of the unfolding of a Hopf bifurcation. We show that in the small-noise limit the top Lyapunov exponent always approaches zero from below (and is thus negative for noise sufficiently small); we also show that there exist large sets of parameters for which the top Lyapunov exponent is positive. Thus the two-point motion can be either stable or unstable, while the one-point motion is always stable.

Hopf bifurcation with additive noise

“…Numerical evidence from [12] and Figure 3 suggest that large shear leads to positive top Lyapunov exponent. Unfortunately, we are not able to prove this analytically and formulate this in the following conjecture.…”

“…The main aim of this paper is to provide a precise mathematical analysis to describe and explain the observations sketched above. Numerical investigations by Lin and Young [25], Wieczorek [29] and Deville et al [12] already highlighted the fact that shear can cause Lyapunov exponents to become positive and induce chaotic behaviour. A first analytical proof of this phenomenon has been given by us in [14] in the case of a stochastically driven limit cycle on the cylinder.…”

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

“…Theorem 3. Let system (1) satisfy (2), ( 3), ( 5), ( 6), and 1 ≤ n ≤ q be an integer such that assumptions (12), ( 13) and (16)…”

“…2 ( 12), ( 13), n = q λ n < 0 polynomially stable (12), (13), n > q stable (12), ( 13), ( 16), n ≤ q, σ ≥ n q λ n > δ σ,n/q µ 2 2 unstable Th. 3 ( 12), ( 13), (16),…”

“…It is well known that even weak random disturbances can lead to significant changes in the behavior of trajectories [11]. See, for example, [12][13][14][15][16][17][18][19], where the influence of autonomous stochastic perturbations on qualitative properties of solutions is discussed. Stochastic bifurcations associated with qualitative changes in the profile of stationary probability densities, in the Lyapunov spectrum function or in the dichotomy spectrum were investigated in [20][21][22][23][24][25] for systems of stochastic differential equations with time-independent coefficients.…”