Hopf bifurcation with additive noise
| Autor: | Maximilian Engel, Martin Rasmussen, Thai Son Doan, Jeroen S. W. Lamb |
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| Přispěvatelé: | Engineering & Physical Science Research Council (EPSRC), Commission of the European Communities |
| Rok vydání: | 2018 |
| Předmět: |
RANDOM DIFFEOMORPHISMS
Dynamical systems theory 37C75 37D45 37G35 37H10 37H15 random attractor General Mathematics Mathematics Applied General Physics and Astronomy dichotomy spectrum Dynamical Systems (math.DS) Lyapunov exponent 01 natural sciences symbols.namesake 0102 Applied Mathematics 0103 physical sciences Attractor FOS: Mathematics Hopf bifurcation Mathematics - Dynamical Systems 0101 mathematics STOCHASTIC DUFFING-VAN Mathematical Physics Mathematics Science & Technology Applied Mathematics Physics random dynamical system Probability (math.PR) 010102 general mathematics Mathematical analysis Statistical and Nonlinear Physics White noise Nonlinear Sciences::Chaotic Dynamics Physics Mathematical stochastic bifurcation EQUILIBRIUM Ordinary differential equation Physical Sciences symbols ATTRACTOR 010307 mathematical physics DYNAMICAL-SYSTEMS Random dynamical system Mathematics - Probability Linear stability |
| Popis: | 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). |
| Databáze: | OpenAIRE |
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