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

Article Contents

2019, Volume 24, Issue 8: 4247-4269. Doi: 10.3934/dcdsb.2019080

This issue Previous Article Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary Next Article Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

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

1.
Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland
2.
Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom
3.
Svensk Exportkredit, Klarabergsviadukten 61-63, 111 64 Stockholm, Sweden

Received: March 31, 2018

Revised: September 30, 2018

Early access: April 2019

Published: July 2019

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

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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.

Keywords:

Nonlinear stochastic differential equation,
Lyapunov function,
Radial basis function,
Basin of attraction.

Mathematics Subject Classification: Primary: 37B25, 65P40, 93E03, 93E15, 93D30; Secondary: 60H35, 65C30.

Citation:

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Figure 1. Above: the computed non-local Lyapunov function $ v $ for system (5.1). Below: the function $ Lv $, approximating $ -10^{-3} $

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Figure 2. Non-local Lyapunov function for system (5.2) with $ \theta = 1 $. The non-local Lyapunov functions looks very similar to the one computed in [3]

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Table 1. The table shows the Wendland function $ \psi_0(r): = \phi_{8, 6}(cr) $ as well as the related functions $ \psi_1 $ to $ \psi_6 $, defined recursively by $ \psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r} $ for $ k = 0, 1, \ldots, 5 $

	$\phi_{8, 6}$
$\psi_0(r)$	$[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$
	$ +6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$
$\psi_1(r)$	$-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$
	$+91cr+7]\, (1- cr)^{13}_+$
$\psi_2(r)$	$12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$
	$+84cr+7]\, (1- cr)^{12}_+$
$\psi_3(r)$	$-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$
$\psi_4(r)$	$112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$
$\psi_5(r)$	$-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$
$\psi_6(r)$	$446,961,715,200\, c^{12}\, (1- cr)_+^{8}$

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Table 2. The table shows the Wendland function $ \psi_0(r): = \phi_{7, 6}(cr) $ as well as the related functions $ \psi_1 $ to $ \psi_6 $, defined recursively by $ \psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r} $ for $ k = 0, 1, \ldots, 5 $

	$\phi_{7, 6}$
$\psi_0(r)$	$[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$
	$+143cr+11]\, (1- cr)^{13}_+$
$\psi_1(r)$	$-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$
	$+108cr+9]\, (1- cr)^{12}_+$
$\psi_2(r)$	$10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$
$\psi_3(r)$	$-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$
$\psi_4(r)$	$3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$
$\psi_5(r)$	$-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$
$\psi_6(r)$	$25,540,669,440\,c^{12}\,(1- cr)_+^{7}$

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Table 3. The table shows values for $ \psi_{k, i}: = \sup_{r\in[0, \infty)} |\psi_i(r)| r^k $ for the Wendland functions $ \psi_0(r): = \phi_{8, 6}(cr) $ and $ \psi_0(r): = \phi_{7, 6}(cr) $

$\psi_{k, i}$	$\phi_{8, 6}$	$\phi_{7, 6}$

$\psi_{6, 6}$	$3.148511062 \cdot 10^7\cdot c^6$	$3.240130299 \cdot 10^6\cdot c^6$
$\psi_{5, 5}$	$2.363249538\cdot 10^6\cdot c^5$	$2.588617377\cdot 10^5\cdot c^5$
$\psi_{5, 4}$	$6.409097287\cdot 10^6\cdot c^6$	$6.534280933\cdot 10^5\cdot c^6$
$\psi_{4, 4}$	$1.947997580\cdot 10^5\cdot c^4$	$2.262550039\cdot 10^4\cdot c^4$
$\psi_{4, 3}$	$6.000016519\cdot 10^5 \cdot c^5$	$6.515237949\cdot 10^4 \cdot c^5$
$\psi_{4, 2}$	$2.215560450\cdot 10^6\cdot c^6$	$2.237953342\cdot 10^5\cdot c^6$
$\psi_{3, 3}$	$1.807542870\cdot 10^4\cdot c^3$	$2.219149087\cdot 10^3\cdot c^3$
$\psi_{3, 2}$	$6.618581621\cdot 10^4\cdot c^4$	$7.625999381\cdot 10^3\cdot c^4$
$\psi_{3, 1}$	$3.172360616 \cdot 10^5 \cdot c^5$	$3.414789975 \cdot 10^4 \cdot c^5$
$\psi_{3, 0}$	$ 3.1008\cdot 10^6\cdot c^6$	$ 3.1008\cdot 10^5\cdot c^6$
$\psi_{2, 2}$	$1.970990855\cdot 10^3\cdot c^2$	$2.550970282\cdot 10^2\cdot c^2$
$\psi_{2, 1}$	$9.418422390\cdot 10^3\cdot c^3$	$1.147899628\cdot 10^3\cdot c^3$
$\psi_{2, 0}$	$9.044\cdot 10^4\cdot c^4$	$1.0336\cdot 10^4\cdot c^4$
$\psi_{1, 1}$	$2.767275907\cdot 10^2\cdot c$	$3.766803387\cdot 10^1\cdot c$
$\psi_{1, 0}$	$2.66\cdot 10^3\cdot c^2$	$3.42\cdot 10^2\cdot c^2$

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