Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

Chemnitz, Dennis; Engel, Maximilian

doi:10.1007/s00220-023-04764-z

Cited by 3 publications

(6 citation statements)

References 34 publications

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“…Because of this symmetry we will assume without loss of generality that b ≥ 0. We note that [16] and [12] use −b where [14] and this paper use b; happily, because of (10) this is not a source of confusion.…”

Section: Evaluation Of λmentioning

confidence: 98%

“…The interpretation of µ/(σ √ a) is less clear. Adopting the argument in [12] for the case of positive µ > 0 and small noise intensity σ, we have radius = µ/a, contraction = 2µ and effective noise = noise/radius = σ a/µ. Then…”

Section: Scalingmentioning

confidence: 99%

“…For matrix calculations it is convenient to introduce the infinitesimal rotation matrix J = 0 −1 1 0 . Following [14,12], define new independent Wiener processes…”

Section: Furstenberg-khasminskii Formulamentioning

confidence: 99%

“…The stochastic system with σ > 0 has been studied from a theoretical viewpoint in DeVille, Namachchivaya and Rapti [14], Doan, Engel, Lamb and Rasmussen [16], Chemnitz and Engel [12] and Breden and Engel [11]. See also Wieczorek [31] in relation to stochastic bifurcation of noise-driven lasers, and Dijkstra, Frankcombe and von der Heydt [15] and Tantet, Chekroun, Dijkstra and Neelin [29] in relation to models in oceanography.…”

Section: Introductionmentioning

confidence: 99%

“…When µ > 0 the deterministic system underlying (1) has a stable limit cycle, and the white noise acts as random "kicking" in the two coordinate directions. Engel, Lamb and Rasmussen [17] consider a special case of stochastic kicking of a limit cycle, and their formula for the associated Lyapunov exponent plays an important role in the recent result of Chemnitz and Engel [12] for the system (1).…”

Section: Introductionmentioning

confidence: 99%

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Lyapunov exponents and shear-related chaos for a Hopf bifurcation with additive noise

Baxendale

2023

Preprint

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This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent λ associated with this random dynamical system as one or more of the parameters in the system tend to 0 or ∞. This enables the construction of a bifurcation diagram in parameter space showing stable regions where λ <0 (implying synchronization) and unstable regions where λ > 0 (implying chaotic behavior). The value of λ depends strongly on the shearing effect of the twist factor b/a of the deterministic Hopf bifurcation. If b/a is sufficiently small then λ <0$ regardless of all the other parameters in the system. But when all the parameters except b are fixed then λ grows like a positive multiple of b2/3 as b → ∞. 2020 Mathematics Subject Classification: 37H10, 37H15, 60H10, 60J60.

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Section: Evaluation Of λmentioning

confidence: 98%

Section: Scalingmentioning

confidence: 99%

“…For matrix calculations it is convenient to introduce the infinitesimal rotation matrix J = 0 −1 1 0 . Following [14,12], define new independent Wiener processes…”

Section: Furstenberg-khasminskii Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lyapunov exponents and shear-related chaos for a Hopf bifurcation with additive noise

Baxendale

2023

Preprint

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Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise

Baxendale

2024

Probab. Theory Relat. Fields

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This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent $$\lambda $$ λ associated with this random dynamical system as one or more of the parameters in the system tend to 0 or $$\infty $$ ∞ . This enables the construction of a bifurcation diagram in parameter space showing stable regions where $$\lambda <0$$ λ < 0 (implying synchronization) and unstable regions where $$\lambda > 0$$ λ > 0 (implying chaotic behavior). The value of $$\lambda $$ λ depends strongly on the shearing effect of the twist factor b/a of the deterministic Hopf bifurcation. If b/a is sufficiently small then $$\lambda <0$$ λ < 0 regardless of all the other parameters in the system. But when all the parameters except b are fixed then $$\lambda $$ λ grows like a positive multiple of $$b^{2/3}$$ b 2 / 3 as $$b \rightarrow \infty $$ b → ∞ .

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Noise-induced instabilities in a stochastic Brusselator

Engel,

Olicón-Méndez

2024

Contemporary Mathematics

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We consider a stochastic version of the so-called Brusselator - a mathematical model for a two-dimensional chemical reaction network - in which one of its parameters is assumed to vary randomly. It has been suggested via numerical explorations that the system exhibits noise-induced synchronization when time goes to infinity. Complementing this perspective, in this work we explore some of its finite-time features from a random dynamical systems perspective. In particular, we focus on the deviations that orbits of neighboring initial conditions exhibit under the influence of the same noise realization. For this, we explore its local instabilities via finite-time Lyapunov exponents. Furthermore, we present the stochastic Brusselator as a fast-slow system in the case that one of the parameters is much larger than the other one. In this framework, an apparent mechanism for generating the stochastic instabilities is revealed, being associated to the transition between the slow and fast regimes.

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Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

Cited by 3 publications

References 34 publications

Lyapunov exponents and shear-related chaos for a Hopf bifurcation with additive noise

Lyapunov exponents and shear-related chaos for a Hopf bifurcation with additive noise

Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise

Noise-induced instabilities in a stochastic Brusselator

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