Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

Hjörtur Björnsson; Sigurdur Hafstein; Peter Giesl; Enrico Scalas; Skuli Gudmundsson

doi:10.3934/dcdsb.2019080

Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

Hjörtur Björnsson
, Sigurdur Hafstein
, Peter Giesl
, Enrico Scalas
, Skuli Gudmundsson

Raunvísindadeild

Háskóli Íslands

Rannsóknarafurð: Framlag til fræðitímarits › Grein › ritrýni

Útdráttur

The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.

Upprunalegt tungumál	Enska
Síður (frá-til)	4247-4269
Síðufjöldi	23
Fræðitímarit	Discrete and Continuous Dynamical Systems - Series B
Bindi	24
Númer tölublaðs	8
DOI	https://doi.org/10.3934/dcdsb.2019080
Útgáfustaða	Útgefið - ágú. 2019

Athugasemd

Funding Information: The research for this paper was supported by the Icelandic Research Fund (Rannís) in the project ‘Lyapunov Methods and Stochastic Stability’ (152429-051), which is gratefully acknowledged. Publisher Copyright: © 2019 American Institute of Mathematical Sciences. All rights reserved.

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10.3934/dcdsb.2019080

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title = "Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function",

abstract = "The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.",

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author = "Hj{\"o}rtur Bj{\"o}rnsson and Sigurdur Hafstein and Peter Giesl and Enrico Scalas and Skuli Gudmundsson",

year = "2019",

month = aug,

doi = "10.3934/dcdsb.2019080",

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T1 - Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

AU - Björnsson, Hjörtur

AU - Hafstein, Sigurdur

AU - Giesl, Peter

AU - Scalas, Enrico

AU - Gudmundsson, Skuli

PY - 2019/8

Y1 - 2019/8

N2 - The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.

AB - The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.

KW - Basin of attraction

KW - Lyapunov function

KW - Nonlinear stochastic differential equation

KW - Radial basis function

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DO - 10.3934/dcdsb.2019080

M3 - Article

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JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 8

ER -

Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

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