The $\gamma$-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 $\gamma$-basin of attraction of the equilibrium.
Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function / Björnsson, H.; Hafstein, S.; Giesl, P.; Scalas, E.; Gudmundsson, S.. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES B.. - ISSN 1531-3492. - 24:8(2019), pp. 4247-4269. [10.3934/dcdsb.2019080]
Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function
Scalas, E.;
2019
Abstract
The $\gamma$-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 $\gamma$-basin of attraction of the equilibrium.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.