Dzhalladova et al. Advances in Difference Equations (2015) 2015:143 DOI 10.1186/s13662-015-0482-y

R E S E A R C H Open Access

Stability of the zero solution of nonlineardifferential equations under the influence ofwhite noiseIrada Dzhalladova1, Miroslava Ružicková2* and Viera Štoudková Ružicková3

*Correspondence:miroslava.ruzickova@fpv.uniza.sk2University of Žilina, Žilina, SlovakiaFull list of author information isavailable at the end of the article

AbstractThe paper deals with a system of nonlinear differential equations under the influenceof white noise. This system can be used as a mathematical model of various realproblems in finance, mathematical biology, climatology, signal theory and others.Necessary and sufficient conditions for the asymptotic mean square stability of thezero solution of this system are derived in the paper. The paper introduces a newapproach to studying such problems - construction of a suitable deterministic systemwith the use of Lyapunov function.

MSC: 34K50; 60H10; 60H30; 65C30

Keywords: stochastic systems; white noise; mean square stability; Lyapunovfunction

1 IntroductionWe can come across stochastic behavior while examining many important problems of aglobal 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 ournormal expectations, is a very important topic in climatology, meteorology and relatedfields. Common methods of studying extreme events, such as the statistical approach, theempirical-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 equa-tion was introduced by Hasselmann in his work [] on stochastic climate models that ap-peared in . There he proposed to improve deterministic models for the ‘climate’ (slowvariables) by incorporating the influence of the ‘weather’ (fast variables) in the form ofrandom noise. The univariate linear systems that appear in the work have been successfulin describing various modes of climate variability. Success of these models has inspiredresearchers to consider the stochastic forcing as a possible source of more complex dy-namics, for example, in []. The direction of stochastic parametrizations in which the de-velopment of the climate models will be possible in the coming years is formulated, forexample, in [].

Hasselmann’s works can be seen as the beginning of describing extreme events in cli-mate by a stochastic system of differential equations in which random weather changes

© 2015 Dzhalladova et al.; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduc-tion in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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are expressed by a nonlinear stochastic perturbation in the form of white noise. Stochas-tic characteristics of white noise in the system are either known or are determined by theclimate’s median value.

Many works, dedicated to the study of evolution models, use white noise to expressthe amount of ice surface of the Arctic Ocean and then estimate the likelihood that itwould spread on the entire surface of the Earth. Part of the mystery of the Earth’s peri-odic ice ages was uncovered in connection with the so-called stochastic resonance thatwas first discovered while studying the periodic recurrence of Earth’s ice ages, see [, ].Since its discovery, the stochastic resonance has been used in various experimental andtheoretical studies. For example, it occurs in bi-stable systems, where the input-outputrelationship is nonlinear, the periodic input signal is weak and there is random, uncorre-lated variation added to the signal of interest. In climatic models, the stochastic resonancemanifests at its best when regular ‘weak’ weather changes are in certain proportion to therandom weather fluctuations. By the term ‘weak’ we mean changes, that are not capableof causing climate changes on their own, but the presence of random weather changescauses disturbances in periodicity of the system, which then transforms into a differentmode.

Our work deals with a stochastic model with nonlinear member. Some related prob-lems regarding such systems were studied in []. In this paper we focus on the deter-mination of conditions that are necessary and sufficient for stable behavior of the abovementioned processes. The present paper contains sufficient conditions for the asymptoticmean square stability of the zero solution of systems with white noise. Further, sufficientconditions for instability of the zero solution of this system and sufficient conditions fora more general system are derived in the paper. The results are followed by several exam-ples. We construct Lyapunov functions as a tool to study stability of a stochastic systemthat works under the influence of white noise.

Let (�,F , F ,P) be a filtered probability space, or stochastic basis, consisting of a prob-ability space (�,F ,P) and a filtration F = {Ft ,∀t ≥ } contained in F . On the proba-bility space we consider the stochastic system of nonlinear differential equations of theform

dxi(t) = fi(x) dt + hi(x) dwi(t), i = , , . . . , m, t ≥ , ()

where the state function x(t) = (x, . . . , xm)T (the operation T denotes transposition) isa continuously differentiable m-dimensional column vector-function, f = (f, . . . , fm)T ,h = (h, . . . , hm)T are also continuously differentiable m-dimensional column vector-functions such that f (o) = o, h(o) = o, o = (, , . . . , )T ∈ R

m hold. The function f repre-sents a slow deterministic process, the products hi(x) dwi(t), i = , , . . . , m, are the stochas-tic approximations to a fast phenomenon. The m-dimensional column vector-functionw = (w, . . . , wm)T indicates a standard Wiener process. The m-dimensional Wiener pro-cess is said to be standard Wiener process if

w() = o,

E(){dw(t)}

= o, ()

E(){dw(t) dwT (t)}

= I dt

Dzhalladova et al. Advances in Difference Equations (2015) 2015:143 Page 3 of 11

hold for t ≥ , I is the identity matrix. Any realization of the Wiener process w(t) is con-tinuous but not differentiable at each point. Moreover, because E(){w(t)wT (t)} = It, theprocess w(t) is a non-stationary stochastic process.

For simplicity, we denote h(x) dw(t) = (h dw, . . . , hm dwm)T in our consideration. So, theproduct h(x) dw(t) means neither a scalar product nor a vector product, but it is a columnvector with components hi(x) dwi(t), i = , , . . . , m. Using this, system () can be rewritteninto the vector form

dx(t) = f (x) dt + h(x) dw(t), t ≥ . ()

We are interested in stability of solutions of the system. There are several various def-initions of stability that can be used. Here we recall the mean square stability and theasymptotic mean square stability of the zero solution.

Definition The trivial solution of system () is said to be mean square stable on theinterval [,∞) if, for each ε > , there exists δ > such that any solution x(t) correspondingto the initial data x() exists for all t ≥ and the mathematical expectation

E(){∥∥x(t)∥∥} < ε whenever t ≥ and

∥∥x()∥∥ < δ.

The mean stability of the zero solution of system () is defined in a very similar way, withonly ‖x(t)‖ being replaced by ‖x(t)‖.

Definition The trivial solution of system () is said to be asymptotically mean squarestable on the interval [,∞) if it is stable and, moreover,

limt→∞ E(){∥∥x(t)

∥∥} = ()

is satisfied for each solution of ().

Remark It is easy to see that () is satisfied if and only if the matrix E(){x(t)} convergesto zero matrix �,

limt→∞ E(){x(t)

}= lim

t→∞ E(){x(t)xT (t)}

= �.

Denote a neighborhood of the point o ∈Rm as O(o).

Definition The function g(t, x(t)) is said to be positive definite on O(o) if g(t, x) is con-tinuous with respect to t, t ≥ , ‖x(t)‖ < ∞ on O(o), g(t, o) = , and there exists a positivedefinite quadratic form V (x) such that

g(t, x) ≥ V (x) whenever t ≥ , x ∈O(o) and∥∥x(t)

∥∥ < ∞.

Recall, if a function g(t, x(t)) is positive definite on O(o), then the function –g(t, x(t)) isnegative definite on this neighborhood.

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Remark The previous definition is equivalent to the following one:The function g(t, x(t)) is said to be positive definite on O(o) if g(t, x) is continuous with

respect to t, t ≥ , ‖x(t)‖ < ∞ on O(o), g(t, o) = , and there exists a constant k > suchthat for any positive definite quadratic form V (x),

g(t, x) ≥ kV (x) whenever t ≥ , x ∈O(o) and∥∥x(t)

∥∥ < ∞.

2 Main resultsWe define a Lyapunov function v in the form

v(x(t)

)= xT (t)Cx(t), ()

where C is an m × m positive definite symmetric matrix. We also use the diagonal matrixCd which has the same elements on the diagonal as C.

Theorem Let there exist a neighborhood O(o), in which the function

δ(x) = xT Cf (x) + hT (x)Cdh(x) ()

is negative definite with respect to system ().Then the trivial solution of () is asymptotically mean square stable on the interval

[,∞).

Proof First we calculate the differential of Lyapunov function () with respect to trajecto-ries of system (). We have

dv(x) = v(x + dx) – v(x) =(xT + dxT)

C(x + dx) – xT Cx

=(xT + f T (x) dt +

(h(x) dw(t)

)T)C

(x + f (x) dt +

(h(x) dw(t)

))– xT Cx

= xT Cx – xT Cx + xT Cf (x) dt + xT C(h(x) dw(t)

)+ f T (x) dt Cx

+ f T (x) dt Cf (x) dt + f T (x) dt C(h(x) dw(t)

)+

(h(x) dw(t)

)T Cx

+(h(x) dw(t)

)T Cf (x) dt +(h(x) dw(t)

)T C(h(x) dw(t)

).

After modifying the obtained equation, applying operation mathematical expectationE(){dv(t)}, in regard to assumptions (), and leaving aside the member with dt, we getthe equation

E(){dv(x)}

= xT Cf (x) dt + f T (x)Cx dt + hT (x)Cdh(x) dt, ()

or, in view of (), it can be written in the more simple form

E(){dv(x)}

= δ(x) dt. ()

Because δ(x) is a negative definite deterministic function, the equalities

E(){δ(x)}

= δ(x),

δ(x) ≤ –kv(x), k = const,

Dzhalladova et al. Advances in Difference Equations (2015) 2015:143 Page 5 of 11

are true (see Remark ). By using them and equation (), we get

ddt

E(){v(x)} ≤ –kE(){v(x)

},

so

E(){v(x)} ≤ e–kt .

Therefore,

limt→∞ E(){x(t)

}= lim

t→∞ E(){x(t)xT (t)}

= �,

which implies asymptotic mean square stability of a trivial solution of the considered sys-tem. �

Remark Analogous result about instability can be derived in the same way as the resultof Theorem .

Namely, if there exists a neighborhood O(o), in which the function

δ(x) = xT Cf (x) + hT (x)Cdh(x)

is positive definite with respect to system (), then the trivial solution of () is unstable onthe interval [,∞).

Further we will discuss the stability of a system in the more general form

dx(t) = f (t, x) dt +n∑

k=

hk(t, x) dwk(t), t ≥ , ()

where f , hk , k = , . . . , n, are m-dimensional column vector-functions continuously differ-entiable in both variables, such that f (t, o) = o, hk(t, o) = o hold. Expressions hk(t, x) dwk(t),k = , . . . , n, are again column vectors with components hik(t, x) dwik(t), i = , , . . . , m,k = , , . . . , n. Functions wk = (wk , . . . , wmk)T , k = , , . . . , n, are also m-dimensional col-umn vector-functions indicating standard Wiener processes, such that each of them sat-isfies the following relationships:

wk() = o,

E(){dwk(t)}

= o, ()

E(){dwk(t) dwTk (t)

}= I dt, k = , , . . . , n

for t ≥ .The following theorem can be proved in the same way as Theorem .

Theorem Let there exist a neighborhood O(o), in which the function

δ(t, x) = xT Cf (t, x) +n∑

k=

hTk (t, x)Cdhk(t, x) ()

is negative definite with respect to system ().

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Then the trivial solution of () is asymptotically mean square stable on the interval[,∞).

Remark A similar remark as Remark can be formulated in this case. Namely, if thereexists a neighborhood O(o), in which the function

δ(t, x) = xT Cf (t, x) +n∑

k=

hTk (t, x)Cdhk(t, x)

is positive definite with respect to system (), then the trivial solution of () is unstable onthe interval [,∞).

In the last part of this section, let us consider one special case of a stochastic system ofdifferential equations, namely the linear system

dx(t) = Ax(t) dt + Hx(t) dw(t), t ≥ , ()

where A, H are m × m constant matrices and w is a standard Wiener process that satis-fies ().

Corollary Let there exist a neighborhood O(o), in which the function

δ(x) = xT AT Cx + xT CAx + xT HT CdHx ()

is negative definite with respect to system ().Then the trivial solution of () is asymptotically mean square stable on the interval

[,∞).

Proof The proof of this theorem follows immediately from Theorem if f (x) = Ax, h(x) =Hx. �

Remark If there exists a neighborhood O(o), in which the function

δ(x) = xT AT Cx + xT CAx + xT HT CdHx

is positive definite with respect to system (), then the trivial solution of () is unstableon the interval [,∞).

Remark Stability of the trivial solution of a stochastic system of linear differential equa-tions in the form () was studied in the works by Korenevskii in []. The following stabil-ity criterion was obtained there by different methods: The trivial solution of system ()is asymptotically mean square stable on the interval [,∞) if and only if there exists apositive definite matrix C satisfying the matrix equation

AT C + CA + HT CH = –G,

where matrix G is positive definite.

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3 ExamplesIn the following examples, we illustrate application of Theorem and Remark . In theseexamples, we also construct the corresponding Lyapunov function; however, it is not nec-essary for verifying the stability or instability condition.

Example We consider stochastic functional differential equation in the form

dx(t) = –x(t) dt – x(t) dw(t), ()

where f , h are scalar functions, f = –x(t), h = –x(t), x is a scalar state function and w is astandard Wiener process. Stability of the zero solution of this equation can be determinedon the basis of Theorem .

We define Lyapunov function in the form (), with C = ,

v(x) = x. ()

Then equation () is in the form

E(){dv(t)}

=(–xT x – xT x +

(x)T x)dt =

(–x + x)dt.

To apply Theorem , we have to prove that there exists a neighborhood of the zero point ,where the function δ = –x + x is negative definite. This holds if and only if the followinginequality is satisfied:

x – x < ⇔ < x <

or

– √ < x < √, x �= .

That is, δ is the function of negative values at each point of (– √, √) except zero, andδ() = . Therefore, we have the conclusion that there exists a neighborhood O() =(– √, √) in which the function δ = –x + x is negative definite. So, the trivial solutionx = of considered equation is asymptotically mean square stable on the interval [,∞).

Example We consider stochastic differential equation in the form

dx(t) = ax(t) dt + bx(t) dw(t),

where a, b ∈R, a �= and x is a scalar state function.We again define Lyapunov function in the form () with C = . The function δ, in view

of Theorem , has the form

δ(x) =(a + b)x.

Hence, if a + b < , the function δ is negative definite, in the opposite case it is positivedefinite. Therefore, in accordance with Theorem and Remark , if a + b < , then the

Dzhalladova et al. Advances in Difference Equations (2015) 2015:143 Page 8 of 11

trivial solution of considered equation is asymptotically mean square stable on the interval[,∞). In the opposite case, that is, if a + b > , then the trivial solution of consideredequation is unstable.

Example We consider stochastic functional differential equation in the form

dx(t) = –x(t) dt + a sin x(t) dw(t), ()

where f , g are scalar functions, f = –x(t), g = a sin x(t), a ∈ R, a �= and x is a scalar statefunction.

We again define Lyapunov function in the form () with C = . Therefore, a function δ

from () has the form

δ(t, x) = –xT x +(a sin x(t)

)T a sin x(t)

= –x + a sin x.

Because

–x + a sin x > ⇔ x

sin x<

a

,

in the case a >√

, there exists a neighborhood of the zero point , where the functionδ is positive definite. In the case a ≤ √

, the function δ is negative definite. So, the zerosolution of considered equation is unstable for any a >

√ and it is asymptotically mean

square stable for any a ≤ √.

Example Now we consider two-dimensional stochastic functional differential equationin the form

dx(t) = x(t) dt + h(x, x) dw(t),

dx(t) = –x(t) dt + h(x, x) dw(t),()

where x = (x, x)T is a two-dimensional vector state function, f , h are two-dimensionalvector functions, f = (x(t), –x(t))T , h = (h(x, x), h(x, x))T , h �= (, ) on a neighbor-hood O(, ) with the exception of the zero point (, ), and w is a standard Wiener pro-cess.

We again define Lyapunov function in the form () with C = I ,

v(x) = x + x

.

Then the function δ = hT h and it is positive definite on the neighborhood O(, ). Thus,by Theorem , the trivial zero solution of this equation is unstable on the interval [,∞).

Remark If we take h = in system (), then it represents the stochastic second orderdifferential equation

dx(t) = –x(t) dt + h(x′, x

)dw(t),

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and the previous example implies that its solution is unstable for any nontrivial h. But afunction can be added to the right-hand side of this equation such that the trivial solutionof the new equation will be asymptotically mean square stable. The following result holds:

If

lim(x,x)→(,)

g(x, x) · x

h(x, x)

= L >

()

for a continuously differentiable function g(x, x), then the trivial solution of the system

dx(t) = x(t) dt,

dx(t) = –x(t) dt – g(x, x) dt + h(x, x) dw(t)()

is asymptotically mean square stable.Condition () holds, for example, for g(x, x) = h(x, x) = x. Then system () is the

following:

dx(t) = x(t) dt,

dx(t) = –x(t) dt – x(t) dt + x(t) dw(t).()

Behavior of solutions x(t) of this system compared to that of solutions of the unstablesystem

dx(t) = x(t) dt,

dx(t) = –x(t) dt + x(t) dw(t)()

Figure 1 Behavior of solutions of stablesystem (20).

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Figure 2 Behavior of solutions of instablesystem (21).

is illustrated in Figures and . In both of these figures, there are displayed solutionsof the systems (), () with initial conditions x() = , x() = , respectively. Thesesolutions were computed numerically with the Milstein method on the interval [, ].

4 ConclusionOur concept of studying asymptotic mean square stability of the zero solution of stochas-tic differential equations is more effective in comparison with previously known methods,and it can also be used in various application problems. For example, by stochastic equa-tion of the type

dx(t) =[ax(t) – ax(t) ln x(t)

]dt + σx(t) dw(t)

we can describe a tumor growth, where the expected size of the tumor is contaminatedwith white noise []. Such type of equations can be used also in biomedical research [,], epidemic modeling [], in describing animal motion [], receiving signals [, ]and many others. However, the approach to the study of these models in the cited worksis different compared to our method. They use numerical or statistical methods and esti-mation methods in there.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors have made the same contribution. All authors read and approved the final manuscript.

Dzhalladova et al. Advances in Difference Equations (2015) 2015:143 Page 11 of 11

Author details1Kiev National Economics Vadym Hetman University, Kyiv, Ukraine. 2University of Žilina, Žilina, Slovakia. 3Brno Universityof Technology, Brno, Czech Republic.

AcknowledgementsThe second author was supported by Grant KEGA 004ŽU-4/2014 of Slovak Grant Agency. The third author was supportedby the project FSI-S-14-2290 of Brno University of Technology.

Received: 11 February 2015 Accepted: 22 April 2015

References1. Hasselmann, K: Stochastic climate models. Part I: Theory. Tellus 28, 473-485 (1976)2. Arnold, L: Hasselmann’s program visited: the analysis of stochasticity in deterministic climate models. In:

von Storch, J-S, Imkeller, P (eds.) Stochastic Climate Models, pp. 141-158. Birkhäuser, Boston (2001)3. Dymnikov, VP, Lykosov, NV, Volodin, EM: Problems of modeling climate and climate change. Izv., Atmos. Ocean. Phys.

42(5), 568-585 (2006)4. Benzi, R, Sutera, A, Vulpiani, A: The mechanism of stochastic resonance. J. Phys. A, Math. Gen. 14(11), L453-L457 (1981)5. Benzi, R, Parisi, G, Sutera, A, Vulpiani, A: A theory of stochastic resonance in climatic change. SIAM J. Appl. Math. 43(3),

565-578 (1983)6. Lipster, R, Shiryaev, A: Statistics of Random Processes I: General Theory, 2nd edn. Springer, Berlin (2001)7. Korenevskii, DH: The Stability of Solutions to the System of Differential Equations with Coefficients Perturbed by

White Noise and Coloured Noises. IM of NASU, Kiev (2012) (in Russian)8. Lo, CF: Stochastic nonlinear Gompertz model of tumour growth. In: Proceedings of the World Congress on

Engineering, London, UK, 1-3 July (2009)9. Donnet, S, Samson, A: A review on estimation of stochastic differential equations for

pharmacokinetic/pharmacodynamic models. Adv. Drug Deliv. Rev. 65(7), 929-939 (2013)10. Picchini, U, Ditlevsen, S, De Gaetano, A, Lansky, P: Parameters of the diffusion leaky integrate-and-fire neuronal model

for a slowly fluctuating signal. Neural Comput. 20(11), 2696-2714 (2008)11. Rao, F: Dynamics analysis of a stochastic SIR epidemic model. Abstr. Appl. Anal. 2014, Article ID 356013 (2014)12. Brillinger, DR, Preisler, HK, Ager, AA, Kie, JG, Stewart, BS: Employing stochastic differential equations to model wildlife

motion. Bull. Braz. Math. Soc. 33(3), 385-408 (2002)13. Gardiner, CW: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer,

Berlin (1985)