THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2

Haoyu Wang; Wenqing Hu; Xiaoliang Gan; Ping Ao

doi:10.11948/20220149

2023 Volume 13 Issue 1

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Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao. THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 359-375. doi: 10.11948/20220149

Citation:

Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao. THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 359-375. doi: 10.11948/20220149

THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2

1.
Department of Mathematics, 688 Yingbin Avenue, Zhejiang Normal University, 321004 Jinhua, China
2.
Department of Mathematics and Statistics, 202 Rolla Building, Missouri University of Science and Technology, 65401 Rolla, Missouri, USA
3.
School of Mathematics and Computing Science, 1 Jinji Road, Guilin University of Electronic Technology, 541004 Guilin, China
4.
Department of physics, 99 Shangda Road, Shanghai University, 200444 Shanghai, China

Corresponding author: Email: wanghaoyu042@163.com(H. Wang)

Abstract

Abstract

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collaborators recently. We show that this potential function agrees with the generalized Lyapunov function of the deterministic part of the stochastic dynamical system. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also be interpreted as the proof of existence of Lyapunov function for a general dynamical system.
- Stochastic dynamical system /
- the generalized Lyapunov function /
- Ao's decomposition /
- potential function
MSC: 34A05, 34F05, 37H30

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References

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