- Research
- Open Access
Stability of the zero solution of nonlinear differential equations under the influence of white noise
- Irada Dzhalladova^{1},
- Miroslava Růžičková^{2}Email author and
- Viera Štoudková Růžičková^{3}
https://doi.org/10.1186/s13662-015-0482-y
© Dzhalladova et al.; licensee Springer. 2015
Received: 11 February 2015
Accepted: 22 April 2015
Published: 7 May 2015
Abstract
The paper deals with a system of nonlinear differential equations under the influence of white noise. This system can be used as a mathematical model of various real problems in finance, mathematical biology, climatology, signal theory and others. Necessary and sufficient conditions for the asymptotic mean square stability of the zero solution of this system are derived in the paper. The paper introduces a new approach to studying such problems - construction of a suitable deterministic system with the use of Lyapunov function.
Keywords
- stochastic systems
- white noise
- mean square stability
- Lyapunov function
MSC
- 34K50
- 60H10
- 60H30
- 65C30
1 Introduction
We can come across stochastic behavior while examining many important problems of a global character in various fields of research, for example, in the theory of climate change. Detailed understanding of extreme events in climate, of phenomena that are beyond our normal expectations, is a very important topic in climatology, meteorology and related fields. Common methods of studying extreme events, such as the statistical approach, the empirical-physical approach or the numerical modeling approach, have some limitations, and study of them has been largely empirical.
The idea of replacing the whole deterministic system with a stochastic differential equation was introduced by Hasselmann in his work [1] on stochastic climate models that appeared in 1976. There he proposed to improve deterministic models for the ‘climate’ (slow variables) by incorporating the influence of the ‘weather’ (fast variables) in the form of random noise. The univariate linear systems that appear in the work have been successful in describing various modes of climate variability. Success of these models has inspired researchers to consider the stochastic forcing as a possible source of more complex dynamics, for example, in [2]. The direction of stochastic parametrizations in which the development of the climate models will be possible in the coming years is formulated, for example, in [3].
Hasselmann’s works can be seen as the beginning of describing extreme events in climate by a stochastic system of differential equations in which random weather changes are expressed by a nonlinear stochastic perturbation in the form of white noise. Stochastic characteristics of white noise in the system are either known or are determined by the climate’s median value.
Many works, dedicated to the study of evolution models, use white noise to express the amount of ice surface of the Arctic Ocean and then estimate the likelihood that it would spread on the entire surface of the Earth. Part of the mystery of the Earth’s periodic ice ages was uncovered in connection with the so-called stochastic resonance that was first discovered while studying the periodic recurrence of Earth’s ice ages, see [4, 5]. Since its discovery, the stochastic resonance has been used in various experimental and theoretical studies. For example, it occurs in bi-stable systems, where the input-output relationship is nonlinear, the periodic input signal is weak and there is random, uncorrelated variation added to the signal of interest. In climatic models, the stochastic resonance manifests at its best when regular ‘weak’ weather changes are in certain proportion to the random weather fluctuations. By the term ‘weak’ we mean changes, that are not capable of causing climate changes on their own, but the presence of random weather changes causes disturbances in periodicity of the system, which then transforms into a different mode.
Our work deals with a stochastic model with nonlinear member. Some related problems regarding such systems were studied in [6]. In this paper we focus on the determination of conditions that are necessary and sufficient for stable behavior of the above mentioned processes. The present paper contains sufficient conditions for the asymptotic mean square stability of the zero solution of systems with white noise. Further, sufficient conditions for instability of the zero solution of this system and sufficient conditions for a more general system are derived in the paper. The results are followed by several examples. We construct Lyapunov functions as a tool to study stability of a stochastic system that works under the influence of white noise.
Definition 1
The mean stability of the zero solution of system (3) is defined in a very similar way, with only \(\|x(t)\|^{2}\) being replaced by \(\|x(t)\|\).
Definition 2
Remark 1
Denote a neighborhood of the point \(o \in{\mathbb{R}}^{m}\) as \({\mathcal{O}}(o)\).
Definition 3
Recall, if a function \(g (t, x(t) )\) is positive definite on \({\mathcal{O}}(o)\), then the function \(- g (t, x(t) )\) is negative definite on this neighborhood.
Remark 2
The previous definition is equivalent to the following one:
2 Main results
Theorem 1
Then the trivial solution of (3) is asymptotically mean square stable on the interval \([0, \infty)\).
Proof
Remark 3
Analogous result about instability can be derived in the same way as the result of Theorem 1.
The following theorem can be proved in the same way as Theorem 1.
Theorem 2
Then the trivial solution of (9) is asymptotically mean square stable on the interval \([0, \infty)\).
Remark 4
Corollary 1
Then the trivial solution of (12) is asymptotically mean square stable on the interval \([0, \infty)\).
Proof
The proof of this theorem follows immediately from Theorem 1 if \(f(x)=A x\), \(h(x)= H x\). □
Remark 5
Remark 6
3 Examples
In the following examples, we illustrate application of Theorem 1 and Remark 3. In these examples, we also construct the corresponding Lyapunov function; however, it is not necessary for verifying the stability or instability condition.
Example 1
Example 2
Example 3
Example 4
Remark 7
4 Conclusion
Declarations
Acknowledgements
The second author was supported by Grant KEGA 004ŽU-4/2014 of Slovak Grant Agency. The third author was supported by the project FSI-S-14-2290 of Brno University of Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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