Global Existence,Uniqueness and Pathwise Property of Solutions to a Stochastic R¨ossler-Lorentz System∗

Song JIANG Junping YIN

1 Introduction

The original R¨ossler system reads as

which contains only one quadratic nonlinear termx1x3and was introduced by R¨ossler in 1976(see[18]).In the recent years,this model has received increasing attention due to its theoretical challenges and great potential applications in secure communications(see[2,5,9,15]),chemical reaction,biological systems and so on(see[1]).The Lorentz system,introduced by Lorentz[12]in 1963,is a well-known model,which has found a lot of applications in different fields,and for example,we refer the reader to[16]on chaos,[7]on instabilities and lasers,[6]on thermospheres,[8]on brushless DC motors,[4]on electric circuits,and[17]on chemical reactions.

Due to their wide applications,the systems(1.1)–(1.2),and in particular,the related nonlinear stochastic systems,have been intensively studied in the last decades in the literature(see[3,10–11,13,19–20]).

In this paper,we consider a more general model which includes the R¨ossler and Lorentz systems and study the well-poseness of its corresponding stochastic system:

In fact,if we suitably take the coefficients in(1.3),then the system(1.3)reduces to(1.1)or(1.2).Therefore,the main properties of the R¨ossler and Lorentz systems can be included in this model.The following Figure 1 shows the attractor and the time series of(1.3)with the initial dataX0=(8,5,30),where we have takenα3(t)=0,b=0,α1(t)=α2(t)=1,a=−1,such that(1.3)reduces to the Lorentz system(1.2).

Figure 1 The attractor and the time series of the Lorentz system.

Letand appropriately choose other coefficients in(1.3).Then one can obtain the attractor of the R¨ossler system and the time series of it are shown in Figure 2.

Figure 2 The behavior similar to the R¨ossler system(the initial value X0=(3,−4,2)).

If we takein(1.3),then we obtain a Lorentz-R¨ossler system,the solution behavior of which is presented in Figure 3 below.

Figure 3 A R¨ossler-Lorentz system with initial value X0=(1,0.1,1).

We can also consider the case of time-dependent coefficients in(1.3),and for example,if we take1,then the behavior of the solution is shown in Figure 4.

Figure 4 The R¨ossler-Lorentz with time-dependent coefficients.

Remark 1.1From Figures 1–4,we clearly see that upon different choices of the coefficients,there are many diversi fi ed attractors given by the system(1.3).Since the structure of solutions to our new system(1.3)is more complex than that of both Lorentz and R¨ossler systems,shown for example in Figure 3,which have many good properties including nonlinear complexity,non-periodicity,well-proportioned character and so on,the system(1.3)can be used in more secure applications to design a more complex and more secure hop-frequency time series.If we use these nonlinear time series to control the frequency hopping pattern,the communications would become much more difficult to be disturbed.

In order to make the model(1.3)more widely applicable,we should take into account the effect of environment noise,particularly,in secure outer communications(complex electric circumstances),convulsed circuits communications,multi-level chemical reactions and so on.

Thus we incorporate white noise in each equation of the system(1.3):

whereuijrepresents the intensity of the noise at timetandBi(t)is a standard white noise,namely,Bi(t)is a Brownian motion defined on a complete probability space(Ω,F,P).If we consider the case of Figures 1–4 with environment noise,then the corresponding stochastic system(1.4)can be illustrated by Figures 5–8,respectively.

Figure 5 The attractor and time series of the Stochastic Lorentz system with a standard irrelated white noise(uii=1,uij,i?=j=0,i,j=1,2,3).

Figure 6 The stochastic R¨ossler system,and the intensity of the noise:uii=0.5,uij,i?=j=0,i,j=1,2,3.

Figure 7 The stochastic R¨ossler-Lorentz system with the noise uii=0.001,uij,i?=j=0,i,j=1,2,3.

Figure 8 The time-dependent stochastic R¨ossler-Lorentz system with the noise uii=0.01,uij,i?=j=0,i,j=1,2,3.

In the current paper,we shall mainly investigate the R¨ossler-Lorentz system(1.4),and consider the case thatγ(t)is not a constant but depends on the third variable and the coefficient of the third interactive termα3(t),namely

Throughout this article,unless otherwise specified,we assumeto be a complete probability space with filtrationsatisfying the usual conditions(i.e.it is right continuous and increasing whileF0contains all P-null sets).denotes a three-dimensional Brownian motion defined on this probability space.

The rest of this paper is arranged as follows.In Section 2,we describe some fundamental conditions and notations,while in Section 3,the global existence and uniqueness for the stochastic system(1.4)are established.Moreover,the path wise property of solutions is obtained.In Section 4,some numerical examples with inner random perturbations are presented,which demonstrate the results of our theoretical analysis,and exhibit the various behaviors with different inner perturbations.By the end of Section 4,we give some conclusions and discussions.

2 Assumptions and Notations

We firstly split the system(1.4)into different parts and give some fundamental conditions.In this paper,we consider the generalized system(1.4)only forward in timet∈[0,∞).LetThe R¨ossler-Lorentz system can be rewritten as

where the initial datumis a fixed point independent ofFtfor allt>0.The four parts of the drift for(2.1)are given by

wherespace of 3×3-matrices is a noise term.For the system(2.1),we assume:

(A1)The matrixAsatisfiesfor some constantλ>0.

(A2)The constantband the coefficientsare bounded.In addition,the coefficients of the interactive terms satisfy

(A3)The noise termU(X,t)satisfies a Lipschitz condition and a linear growth condition,i.e.,

(A4)(The alternative condition)Either

Remark 2.1IfU(X,t)≡0,then the Lorentz-R¨ossler system becomes a deterministic one.As for deterministic equations,there are a number of methods to be used in the study of the well-poseness.And in this paper,we mainly consider the stochastic system.It is important that the system should keep a small variety in a weak noise environment,for example,outdoor-move communications,airplane oscillating communications,and complex electromagnetic environment communications.

At the end of this section,we give some notations which will be frequently used throughout the paper.For any real matrixwe define

For anyp∈N even andwe denoteFor any two variablesstands for the usual inner product.Ci(i=1,2,···)will denote generic constants which vary from line to line and depend on some parameters.

We remark that the condition(A3)can be easily satisfied.For example,

satisfies(A3)when all theσij(t)are bounded on R+.

3 Global Existence and Path wise Property

In this section,we shall show a global existence result and a path wise property for the stochastic system(1.4)with initial dataandIn order to guarantee the existence of a unique global solution for any given initial data,we require in general that the terms of the system(1.4)satisfy the linear growth and uniform Lipschitz continuity conditions.For the system(2.1),the termssatisfy these two conditions.However,the termC(X,t)does not satisfy both the linear growth and uniform Lipschitz continuity conditions.To circumvent this difficulty,we first introduce a modified system,which is solvable,by truncatingC(X,t)appropriately.Then,uniform priori estimates for the modified system enable us to show that the modified system converges to the original one as the truncation level goes to infinity,and thus we obtain a solution to(1.4).

Our modified system is given in the following lemma.

Lemma 3.1Letwith χN(X)=1for?X?2≤N and χN(X)=0forDe fi neand consider the modified system:

where .Assume that the initial data is independent ofand satisfiesThen for any fixed N>0,the modified system(3.1)possesses a continuous almost sure unique and global solution that is{Ft}measurable.

ProofIt is easy to see thatis bounded and satisfies a linear growth condition.To show thatCN(XN)is uniformly Lipschitz continuous for any fixedN>0,we observe forthat

We consider three cases for the right-hand side of the above identity.

Case 1 If bothandthen obviously,the uniform Lipschitz continuity ofcan be ensured.

Case 2 If eitherandandwe have(without loss of generality,letandwhich implies

whereCαis dependent onN,the boundary ofα1(t),α2(t),α3(t)and the assumption of Case 2.

Case 3 Ifandwe first find that

On the other hand,all other coefficients in(1.4)obviously satisfy a linear growth as well as the uniform Lipschitz continuity condition.Thanks to the truncation functionthe modified nonlinear termCN(XN)remains differentiable,and its derivatives are continuous and have compact supports.Thus,Lemma 3.1 follows from the usual existence and uniqueness theorem.

To get the uniform a priori estimates of the solution to the system(3.1),we first deal with the It?o derivatives of the Lyapunov functions.

Lemma 3.2Let the assumptions(A1)and(A2)hold.Then we have

where φ(t)≤0is an adapted process.

ProofDenoting the Lyapunov function

we use the It?o formula with respect toV(XN)forp∈N even to evaluate the It?o derivatives of the Lyapunov function of the solution to the modified system(3.1)as follows:

We have to control every term on the right-hand side of(3.3).To this end,we first need the following estimates.

Lemma 3.3For any real matrix R∈Rm×n,Q∈Rn×m,the following inequalities hold:

In fact,recalling the definition of the trace operator and H¨older’s inequality,we easily see that Lemma 3.3 follows from the following inequalities:

Hence,by Lemma 3.3,we have that

Recalling the assumption(A1)and H¨older’s inequality,we deduce that

while from(A2)we get

Using Lemma 3.3 again,and putting together the estimates(3.4)–(3.6),we conclude that

whereφ(t)≤0 is an adapted process,which compensates all the above computations.Thus,the proof of Lemma 3.2 is complete.

Next,we derive uniform a priori estimates forXN(t).We begin with the following lemma.

Lemma 3.4Assume that the assumptions(A1)–(A4)hold,and let p∈Nbe even and fixed,and the initial expectation Then,

where the constant Cp>0depends only on,b,but not on N.

ProofFirst,we introduce the stopping time.For anyD∈N,let

In view of the assumption(A4),we consider the following two cases,respectively.

Case 1 If,then(3.2)becomes

Integrating the above inequality from 0 tot∧τDand taking the expectation,we obtain

In particular,we takep=2 and use the Gronwall inequality to infer that

whereis independent oft.Computing recursively,we obtain for any finitep∈N even that

If we apply the Gronwall inequality again,we get

whereare independent oft∧τD.

Therefore,for any finitep∈N even,we have

Case 2 If,then(3.2)becomes

Keeping in mind thatand(3.10),we derive that

We can argue similarly to(3.9)to obtain

Combining(3.9)with(3.11),we conclude

whereandare independent ofNandt,and depend on the constantsonly.

It is obvious that the stopping time satisfiesBy the continuity of the solutionXN(t)int,we see that for any fixedis bounded,as,and

Putting all the above uniform estimates fortogether,using(3.12)and the Fatou lemma,we obtain

whereCpis independent oft.Hence,

which completes the proof.

Under a weaker condition on the initial expectation,we still have Lemma 3.4,namely,the following lemma.

Lemma 3.5Let and T>0be arbitrary but fixed.Assume that the conditions(A1)–(A4)hold.Then there exists a constantsuch that

whereis independent of N,and depends possibly on T,C2,C4,C1,λ,b only.

ProofIn view of the uniform boundedness Lemma 3.4,we see that it is not necessary to use the stopping time in the following discussion.If we integrate(3.2)from 0 tot(t∈[0,T]),we obtain

Taking the expectation to the above identity,and using the Burholder-Davis-Gundy inequality to control the stochastic integral,we infer that

which implies the boundedness given in the lemma.

Making use of the above lemmas,we are able to show the main results of this paper,i.e.,Theorems 3.1–3.2 below.

Theorem 3.1Suppose that the conditions(A1)–(A4)are satisfied.Then,the R¨ossler-Lorentz systems(2.1)–(2.2)withpossesses a global unique almost sure continuous solution process,which has the following property:

If,in addition,for a fixed p∈Neven,then

ProofFirst,we show the existence.By virtue of Lemma 3.1,the truncated system(3.1)has a continuous solutionXN(t)(t∈[0,∞)).To prove the existence,we show that the sequenceXN(t)is convergentin some sense.LetτDdenote the stopping time introduced in(3.8)for anN∈N.From Lemma 3.5 and the Chebyshev inequality we get

whence

and

Thus for almost everyω∈Ω,there exists anN0(ω),such thatτN0(ω)is large enough.Moreover,one has

Hence,

By virtue of(3.17),ifthenfor allTherefore,the setis monotonously increasing and converges to Ω asN→∞.

Moreover,because for anyN∈N,XN(t)is continuous intand converges uniformly inttoX(t),X(t)is also continuous int(in fact,note that ifτN0(ω)is sufficiently large,then we can express for almost allω∈Ω the limit function asfor all

Next,we have to show that the limit functionX(t)is indeed a solution of the original R¨ossler-Lorentz system.Whent=0,it is obvious thatXN(0)=X(0)=X0for allN∈N,while fort>0 we show thatX(t)solves(2.1)–(2.2)by taking it to the limitation in(3.1)asN→∞.To this end,recalling the definition of(3.8),we have

and

where we have used the fact that ifτN≥t,thenτN∧t=t,and consequently,,and

Therefore,X(·)is a solution of the stochastic R¨ossler-Lorentz system on[0,∞).Meanwhile,the boundary of the moments(3.14)can be obtained by the uniform-in-tconvergence ofXN(t)toX(t)and Lemma 3.4.Finally,for any fixedN>0,the coefficients of the truncated system(3.1)satisfy the uniform Lipschitz continuity condition,so the uniqueness of the the system(3.1)can be ensured.Combining(3.18)with the uniform-in-tconvergence ofXN(t)toX(t),we obtain the uniqueness of solutions to the R¨ossler-Lorentz system by an almost sure means.

In Theorem 3.1 we have discussed the existence,uniqueness and the moment properties of solutions to the systems(2.1)–(2.2).Now,we study the pathwise property.

Theorem 3.2Let the conditions in Theorem3.1hold.Then for any initial data X0∈Ω,the solution X(t)of(2.1)established in Theorem3.1satisfies

ProofWe use the same notations as in the proof of Theorem 3.1.Following a procedure similar to that used for(3.2),we obtain

whereψ(t)≤0 is an adapted process.

First,we discuss the behavior of the solution in the time interval[t,t+1].Letp=2.For anyt>0,we integrate(3.20)over(s,t),and then take the supremum and expectation to get

which,together with the Burkholder-Davis-Gaudy inequality and the condition(A3),gives

wheredepends onandb2only,but not ont.Thus,the inequality(3.22)implies that the constantsatisfies

Letεbe arbitrary.It follows from the Chebyshev inequality that

Applying the well-known Borel-Cantelli Lemma(see[14]),we find that for almost allω∈Ω,

Hence,there exists ak0(ω),such that for almost allω∈Ω,(3.23)holds wheneverk≥k0.Consequently,for almost allω∈Ω,ifk≥k0andk≤t≤k+1,one has

Taking,we conclude

which,by lettingε→0,gives the pathwise property(3.19).In other words,the solution would not grow faster thant1+ε,with probability one.

4 Numerical Results

In this section,we present some numerical results with different parameters which show qualitative differences between the stochastic and deterministic Lorentz-R¨ossler systems,and illustrate our theoretic analysis.We use the stochastic and deterministic Runge-Kutta schemes to carry out our numerical tests,for all of which the initial datum is taken to beX0=(1,0.1,1)and all coefficients in the systems are chosen so that the conditions(A1)–(A4)are satisfied.For simplicity,we only consider the independent noiseFirst we consider some examples to exhibit the moment estimates of the R¨ossler-Lorentz system.

Figure 9 The attractor and the time series of the stochastic R¨ossler-Lorentz system with a white noise uii=0.01,i=1,2,3.

Figure 10 The attractor and the time series of the deterministic R¨ossler-Lorentz system.

If we take the nonlinear terms to be small transformations:

then we find that the stochastic and deterministic solutions possess very different trajectories(see Figures 11–12).

Figure 11The attractor and the time series of the stochastic R¨ossler-Lorentz system with a white noise uii=0.01,i=1,2,3.

Figure 12 The attractor and the time series of the deterministic R¨ossler-Lorentz system.

Example 4.2Letc=0.We takeβ>0 anda<1 to satisfy the condition(A1).In this example,we first consider the case withσ=12,r=−10,a=−7,b=1,β=2,c=0,and

The corresponding numerical solutions are illustrated in Figures 13–14.

Figure 13 Stochastic system with a white noise uii=0.1,i=1,2,3.

Figure 14 Deterministic system.

Then,we consider the e ff ect of matrixA.Change the linear terms and assume thatσ=9,r=−10,a=−18,b=1,β=2,c=0,and the corresponding numerical solutions are shown in Figures 15–16.Example 4.3In this example,we mainly test the e ff ect of noises.Takeσ=2,r=0,a=and

Figure 15 Transformed linear terms system with a white noise uii=0.1,i=1,2,3.

Figure 16 Deterministic system with different linear terms.

Thus,the corresponding stochastic and deterministic numerical solutions are presented in Figures 17–20.

Figure 17 Stochastic system with a white noise uii=0.1,i=1,2,3.

Figure 18 Stochastic system with a white noise uii=0.01,i=1,2,3.

Figure 19 Stochastic system with a white noise uii=0.001,i=1,2,3.

Figure 20 Deterministic system with a white noise uii=0,i=1,2,3.

In this paper,we have established a sufficient condition under which the stochastic system(1.3)has a unique solution.Moreover,we have carried out a number of numerical experiments,which show some interesting qualitative behaviors of solutions,summarized as follows.

(1)From Examples 1–3 we obviously see that for any timeT,Theorem 3.1 can be ensured,and the boundary of the moments of the solutions can be obtained.

(2)Examples 1–2 clearly show the dynamical behavior of the system(1.4),which mainly depends on the nonlinear terms.In particular,Example 1 gives small transforms on nonlinear terms,but the respective paths have large di ff erences.

(3)For a more general Lorentz-R¨ossler system,if the uniqueness can be ensured,we can show,for Example 3 in particular,that the stochastic system converges toward the corresponding deterministic system when the intensity of the noise goes to zero.

(4)All the numerical results on the system(1.3)give us very abundant expressions,including the behavior of the well-known Lorentz and R¨ossler systems.Furthermore,parts of the systems could be used to make a more complex and more secure hop-frequency time series.

(5)In view of the numerical results given in Figure 4,we have clearly found that the combined stochastic Lorentz-R¨ossler system possesses better properties than the corresponding deterministic system.

AcknowledgementThe authors would like to thank the referee for reading the paper carefully and giving the valuable comments which helped to improve the presentation of the paper.

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