TY  - JOUR
A1  - Hoegele, Michael
A1  - Pavlyukevich, Ilya
T1  - The exit problem from a neighborhood of the global attractor for dynamical systems perturbed by heavy-tailed levy processes
JF  - Stochastic analysis and applications
N2  - We consider a finite-dimensional deterministic dynamical system with the global attractor ? which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing ?. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity E>0 and solve the asymptotic first exit time and location problem from D in the limit of E?0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of E, just as in the case when ? is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative -stable Levy noise.
KW  - alpha-stable Levy process
KW  - Canonical (Marcus) SDE
KW  - First exit location
KW  - First exit time
KW  - Global attractor
KW  - Ito SDE
KW  - Multiplicative noise
KW  - Regular variation
KW  - Stratonovich SDE
KW  - Van der Pol oscillator
Y1  - 2014
U6  - https://doi.org/10.1080/07362994.2014.858554
SN  - 0736-2994
SN  - 1532-9356
VL  - 32
IS  - 1
SP  - 163
EP  - 190
PB  - Taylor & Francis Group
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Högele, Michael
A1  - Pavlyukevich, Ilya
T1  - Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Levy type noise
JF  - Stochastics and dynamic
N2  - We consider a finite dimensional deterministic dynamical system with finitely many local attractors K-iota, each of which supports a unique ergodic probability measure P-iota, perturbed by a multiplicative non-Gaussian heavy-tailed Levy noise of small intensity epsilon > 0. We show that the random system exhibits a metastable behavior: there exists a unique epsilon-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures P-iota. In particular our approach covers the case of dynamical systems of Morse-Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative alpha-stable Levy noise in the Ito, Stratonovich and Marcus sense. As examples we consider alpha-stable Levy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.
KW  - Hyperbolic dynamical system
KW  - Morse-Smale property
KW  - physical SRB measures
KW  - stable limit cycle
KW  - small noise asymptotic
KW  - alpha-stable Levy process
KW  - multiplicative noise
KW  - Ito integral
KW  - Stratonovich integral
KW  - stochastic Marcus (canonical) differential equation
KW  - multiscale dynamics
KW  - metastability
KW  - embedded Markov chain
KW  - randomly forced Duffing equation
KW  - birhythmic behavior
Y1  - 2015
U6  - https://doi.org/10.1142/S0219493715500197
SN  - 0219-4937
SN  - 1793-6799
VL  - 15
IS  - 3
PB  - World Scientific
CY  - Singapore
ER  -