Hopf bifurcation with additive noise
File(s)
Accepted version
Author(s)
Doan, Thai Son
Engel, Maximilian
Lamb, Jeroen
Rasmussen, M
Type
Journal Article
Abstract
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).
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).
Date Issued
2018-08-21
Date Acceptance
2018-07-09
Citation
Nonlinearity, 2018, 31 (10)
ISSN
0951-7715
Publisher
IOP Publishing
Journal / Book Title
Nonlinearity
Volume
31
Issue
10
Copyright Statement
© 2018 IOP Publishing Ltd & London Mathematical Society. This is an author-created, un-copyedited version of an article accepted for publication in [insert name of journal]. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at http://iopscience.iop.org/article/10.1088/1361-6544/aad208/meta
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Commission of the European Communities
Grant Number
EP/I004165/1
643073
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Physics, Mathematical
Mathematics
Physics
dichotomy spectrum
Hopf bifurcation
Lyapunov exponent
random attractor
random dynamical system
stochastic bifurcation
STOCHASTIC DUFFING-VAN
RANDOM DIFFEOMORPHISMS
DICHOTOMY SPECTRUM
DYNAMICAL-SYSTEMS
EQUILIBRIUM
ATTRACTOR
0102 Applied Mathematics
General Mathematics
Publication Status
Published