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 t… Show more
“…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.…”
Section: Negativity Of Top Lyapunov Exponent and Synchronisationmentioning
confidence: 93%
“…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.…”
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 nonuniform 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 (ampitude-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).
“…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.…”
Section: Negativity Of Top Lyapunov Exponent and Synchronisationmentioning
confidence: 93%
“…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.…”
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 nonuniform 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 (ampitude-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).
“…Theorem 3. Let system (1) satisfy (2), ( 3), ( 5), ( 6), and 1 ≤ n ≤ q be an integer such that assumptions (12), ( 13) and (16)…”
Section: Resultsmentioning
confidence: 99%
“…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),…”
Section: Assumptionsmentioning
confidence: 99%
“…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.…”
The effect of multiplicative stochastic perturbations on Hamiltonian systems on the plane is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The proposed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.
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