# Existence and Uniqueness of the Solution of Lorentz-Rössler Systems with Random Perturbations

Authors: Xiaoying Wang; Fei Jiang; Junping Yin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider a new chaotic system based on merging two well-known systems (the Lorentz and Rössler systems). Meanwhile, taking into account the effect of environmental noise, we incorporate whit-enoise in each equation. We prove the existence, uniqueness, and the moments estimations of the Lorentz-Rössler systems. Numerical experiments show the applications of our systems and illustrate the results.


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## Body

## 1. Introduction

The Lorentz system is a well-known model(1)x˙1=σ(x2-x1),x˙2=rx1-x2-x1x3,x˙3=x1x2-βx3.
This model was introduced in 1963 by Lorentz [1]. For the meaning of the Lorentz system the reader can refer to [2] (Chaos), [3] (laser), [4] (thermospheres), [5] (brushless DC motors), [6] (electric circuits), and [7] (chemical reactions). The original Rössler system only contains one quadratic nonlinear term x1x3(2)x˙1=x2-x3,x˙2=x1+ax2,x˙3=b+x3(x1-c).
The system (2) was introduced by Rössler [8]. This model has received increasing attention due to its great potential applications in secure communication [9–12],   chemical reaction, biological systems, and so on [13].Furthermore, the general chaotic systems have many applications, especially in complex genetic networks [14–17]. Of course, since the general chaotic systems are nonlinear and have stochastic noise terms, lots of mathematical experts pay still their attentions to the mathematical theory for these nonlinear chaotic systems [18–22] and so on.In this paper, we consider a new model including the Lorentz system and the Rössler system, which has been slightly adjusted (the nonlinear terms of the Rössler systems becomex1(x3-c)):
(3)x˙1=σ(x2-x1)-γ(t)(x2-x3),x˙2=rx1-x2-α1(t)x1x3+x1+ax2,x˙3=α2(t)x1x2-βx3+b+α3(t)x1(x3-c).
Actually, if we take some especial coefficients, the system (3) would become the system (1) or (2). Therefore, the main properties of the Lorentz and Rössler systems can be included by this model. In Figure 1, we take that α3(t)=0, b=0, α1(t)=α2(t)=1, a=-1, and that our problem becomes the well-known Lorentz system with the initial value X0=(8,5,30).The attractor and the time series of the Lorentz system.
(a)
(b)Letγ(t)=1, α1(t)=α2(t)=0, and σ=0, and take the correspondingly appropriate other parameters, then the track similar with the Rössler systems can be obtained.If the coefficients of our problem areσ=9, γ(t)=-1, r=27, a=b=c=0, α1(t)=α2(t)=2, α3(t)=1, and β=8/3, then the system (3) becomes the following Lorentz-Rössler system.Remark 1.
From the structure of system (3), it is obvious that there is a great diversification of attractors in the system inner with different parameters. Since the structure of our systems is more complex than the Lorentz system and Rössler system, especially, from Figure 3, the system (3) can be used in secure communications, to design the more complex hop-frequency communications time series, which make the communications content more secure. If we use the system (3) to design the hop-frequency time series in communications, it becomes much more difficult to disturb our communications than the time series of the well-known Lorentz system and Rössler system.To obtain better applications about model (3), we must take into account the effect of environment noise, especially in secure outer communications (complex electric circumstance), convulsed circuits communications, multi-level chemical reactions, and so on. Thus we incorporate white noise in each equation of system (3)
(4)dx1=[σ(x2-x1)-γ(t)(x2-x3)]dtaaaaa+∑i=13‍u1i(x1,x2,x3)dBi(t),dx2=[rx1-x2-α1(t)x1x3+x1+ax2]dtaaaaa+∑i=13‍u2i(x1,x2,x3)dBi(t),dx3=[α2(t)x1x2-βx3+b+α3(t)x1(x3-c)]dtaaaaa+∑i=13‍u3i(x1,x2,x3)dBi(t),
where all the uij represent the intensity of the noise at time t and all the Bi(t) are standard white noise; namely, each Bi(t) is a Brownian motion defined on a complete probability space (Ω,ℱ,ℙ). If we consider the Figures 1, 2, and 3 with environment noise, the respective stochastic system can be shown by Figures 4, 5, and 6, respectively.The behavior similar with the Ro¨ssler system (initial value X0=(3,-4,2)).
(a)
(b)The Lorentz-Rössler combining system with initial valueX0=(1,0.1,1).
(a)
(b)The attractor and the time series of the stochastic Lorentz system with a standard irrelated white noise (uii=1, uij,i≠j=1, i,j=1,2,3).
(a)
(b)The behavior similar with the Rössler system (intensity of the noiseuii=0.5, uij,i≠j=0, i,j=1,2,3).
(a)
(b)The Lorentz-Rössler combining system with the noiseuii=0.001, uij,i≠j=0, i,j=1,2,3.
(a)
(b)In this paper, we consider thatγ(t) is not a constant and dependent on the third variable and the coefficient of the third interactive term α3(t), let γ(t)=α3(t)x3(t). Throughout this paper, unless otherwise specified, we let (Ω,ℱ,{ℱt}t≥0,ℙ) be a complete probability with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ0 contains all ℙ-null sets). B(t)∶=(B1(t),B2(t),B3(t))T denotes an 3-dimensional Brownian motion defined on this probability space.The rest of this paper is arranged as follows. In Section2, we introduce some fundamental conditions of our problem. In Section 3, the existence and uniqueness of the stochastic system (4) are established. Meanwhile, in our main results, the moments estimations of solutions are obtained. Section 4 shows some numerical simulations with inner random perturbations and outer random perturbations, which can support our results and exhibit diverse behaviors with different inner perturbations.


## 2. Fundamental Assumptions and Notations

We firstly split the system (4) into different parts and give some fundamental conditions. In this paper, we consider the generalized system (4) only forward in time t∈[0,∞). Let X=(x1,x2,x3)∈ℝ3. The Lorentz-Rössler system can be rewritten as
(5)dX=-[AX+C(X)-F]dt+U(X,t)dBt,(6)X(0)=X0,0≤t≤T<+∞,
where the initial condition X0=(x10,x20,x30)T is fixed point and independent of ℱt for all t>0. The four parts of the drift are given by
(7)A=(σ-σ0-(r+1)1-a0-α3(t)c0β),C(X)=(-α3(t)x3x2+α3(t)x32α1(t)x1x3-α2(t)x1x2-α3(t)x1x3),F=(00b),U(X,t)=(u11(t)u12(t)u13(t)u21(t)u22(t)u23(t)u31(t)u32(t)u33(t)).U(X,t):ℝ3×[0,∞)→3×3 matrices is a noise term.For the sake of brevity, we introduce some notations.(i)
For any real matrixR=[rij]∈ℝd×m, define ∥R∥22=trace(RTR)=trace(RRT)=∑i,jrij2.(ii)
For anyp∈ℕ is even, X∈ℝ3, ∥X∥2p=(x12+x22+x32)p/2.(iii)
For any two variablesX,Y∈ℝ3, (X,Y) denote the usual inner product.(iv)
Ci (i=1,2,…) denote the constants which are dependent on some parameters.(A1)  The noise termU(X,t) satisfies a Lipschitz condition and a linear growth condition
(8)∥U(X,t)∥22∶=trace(U(X,t)UT(X,t))≤C1(1+∥X∥22).The condition (A1) can be easily satisfied, for example,Ui(X,t)=xi(t)∑j=13σij(t)dBj(t), when all the σij(t) are bounded on ℝ+.For the other coefficients, we suppose the following conditions.(A2)  The matrixA satisfies (AX,X)≥λ∥X∥22, where λ>0.(A3)  The constantb and the coefficients α1(t), α2(t), and α3(t) are bounded. In addition, the interactive terms' coefficients satisfy α1(t)=α2(t)+α3(t).Remark 2.
IfU(X,t)≡0, our Lorentz-Rössler systems become a deterministic system. As for the deterministic equation, there are many methods to study. In this paper, we mainly consider the stochastic system. In any case, it is important to study the properties of the system in a weak noise environment.


## 3. Main Results

In this section, we will consider the stochastic equation (4) with initial value X(0)=X0=(x10,x20,x30)T∈ℝ3 and U(X,t)≠0. In order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the terms of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition. In (5), the terms -A(t)X, F, U(X,t) satisfy these two conditions. However, the term C(X,t) of (5) does not satisfy the linear growth condition, though they are locally Lipschitz continuous.At first, we introduce the modified system. Let us study a modified system obtained by truncating the term, when it is too large. Some a priori moment estimates of the solution of the modified equation enable us to prove that the modified system converges to a solution of the original problem as the truncation level goes to infinity.Lemma 3.
LetχN∈C1(ℝ3,ℝ) with χN(X)=1 for ∥X∥2≤N and χN(X)=0 for ∥X∥2≥N+1. CN(X)∶=χN(X)C(X), the modified system
(9)dXN=-(AXN+CN(XN)-F)dt+U(XN)dBt,aaaaaaaaaaaaaaaat∈[0,T],XN(0)=X0,
where XN=(xN1,xN2,xN3) and T is a finite time. And let the initial value be still independent of {ℱt}t>0 and satisfy E∥X0∥22<∞. Then the modified system (9) possesses a continuous almost sure unique solution that is {ℱt} measurable.Remark 4.
C
N
(
X
N
) is bounded and satisfies a linear growth condition and a Lipschitz condition (it can be easily obtained by the insert-value technique and primary inequalities). All other coefficients obviously satisfy a linear growth as well as a Lipschitz condition. Since the truncation function χN∈C1(ℝ3,ℝ), the modified nonlinear term CN(XN) remains differentiable, and its derivative is continuous and has a compact support. Then the assertions of Lemma 3 follow by the usual existence and uniqueness theorem.To get the uniform estimations of the system (9), we deal with the Itŏ derivative of the Lyapunov functions.Lemma 5.
Let (A2) and (A3) hold. Then one can get the estimation(10)d∥XN∥2p=-pλ2∥XN∥2pdt+∥XN∥2p-2pb22λdt+p2(p-1)∥XN∥2p-2∥U(XN)∥22dt+p∥XN∥2p-2XNTU(XN)dBt+ϕ(t)dt,
where ϕ(t)≤0 is an adapted process.Proof.
Define the Lyapunov function(11)V(X)=(x12+x22+x32)p/2=∥X∥2p.
Using the Itŏ formula with respect to V(XN) for p∈ℕ is even, we compute the Itŏ derivative of the Lyapunov function of the solution of the modified system. We consider(12)dU(XN)=∑i=13‍p2(xN12+xN22+xN32)p/2-12xNidxNi+12∑i,j=13‍p2(p2-1)(xN12+xN22+xN32)p/2-2×2xNi2xNjdxNidxNj+12∑i=13‍p2(xN12+xN22+xN32)p/2-12dxNidxNi=p∥XN∥2p-2(dXN,XN)+p(p2-1)∥XN∥2p-4∑i,j=13‍xNidxNixNjdxNj+p2∥XN∥2p-2∑i=13‍dxNidxNi=p∥XN∥2p-2{-[(AXN,XN)aaaaaaaaaaaaaa+(χN(XN)C(XN),XN)aaaaaaaaaaaaaa-(F,XN)]}dt+p(p2-1)×∥XN∥2p-4trace(XNXNTU(XN)UT(XN))dt+p2∥XN∥2p-2trace(U(XN)UT(XN))dt+p∥XN∥2p-2XNTU(XN)dBt.
We individually deal with all the right terms of (12). Firstly, from the definition of trace and Hölder inequality, we have the following.
Lemma 6.For any real matrixR∈ℝm×n, Q∈ℝn×m, the following inequalities hold:
(13)∥RQ∥2≤∥R∥2∥Q∥2,|trace(RQ)|≤∥R∥2∥Q∥2.
Proof. From Hölder inequality, it follows that
(14)∥RQ∥22=∑ij‍(∑k‍rikqkj)2≤∑i,j‍(∑k‍rik2∑l‍qlj2)=∑i,j,k,l‍(rik2qlj2)=∑i,k‍rik2∑j,l‍qlj2=∥R∥22∥Q∥22,|trace(RQ)|2=(∑i,j‍rijqji)2≤∑i,j‍rij2∑i,j‍qji2=∥R∥22∥Q∥22.
By Lemma 6, trace(XNXNTU(XN)UT(XN)) of (12) can be estimated,
(15)trace(XNXNTU(XN)UT(XN))≤∥XNXNT∥2∥U(XN)UT(XN)∥2≤∥XN∥22∥U(X)∥22.
Combining (A2) and Hölder inequality, we have
(16)-(AXN,XN)+(F,XN)≤-λ2∥XN∥22+b22λ.
From (A3), we can compute
(17)(χN(XN)C(XN),XN)=(χN(XN)(-α3(t)xN3xN2+α3(t)xN32α1(t)xN1xN3-α2(t)xN1xN2-α3(t)xN1xN3),(xN1xN2xN3))=χN(XN)(-α3(t)xN3xN2xN1+α3(t)xN32xN1aaaaaaa+α1(t)xN1xN3xN2-α2(t)xN1xN2xN3aaaaaaa-α3(t)xN1xN32)≡0.
Using Lemma 6 again and combining all the above estimations from (15)–(17), we derive that
(18)d∥XN∥2p=-pλ2∥XN∥2pdt+p∥XN∥2p-2b22λdt+0+p(p2-1)∥XN∥2p-2∥U(XN)∥22dt+p2∥XN∥2p-2∥U(XN)∥22dt+p∥XN∥2p-2XNTU(XN)dBt+ϕ(t)dt,
where ϕ(t)≤0 is an adapted process, which compensates all the estimations we made. The proof of Lemma 5 is completed.Lemma 7.
Assume that the conditions (A1)–(A3) hold and letp∈ℕ be even and fixed, the initial expectation E∥X0∥2p<∞. Then
(19)E∥XN(t)∥2p≤Cp,∀t∈[0,T],
where Cp is a constant and only dependent on T, E∥X0∥2p, p, λ, C1, and b, but independent of N.Proof.
Firstly, we introduce the stopping time. ForD∈ℕ, let
(20)τD∶=inf{t∈[0,T]:∥XN(t)∥2≥D}.
Note that, for all f(·)≥0,
(21)∫0t∧τDf(s)ds≤∫0tf(s∧τD)ds.
For any t∈[0,T], using the linear growth condition of U(X), we can integrate (10) from 0 to t∧τD and then take the expectations to obtain
(22)E∥XN(t∧τD)∥2p≤E∥X(0)∥2p+E∫0t∧τD(-pλ2)∥XN(s)∥2pds+E∫0t∧τDpb22λ∥XN(s)∥2p-2ds+p(p-1)2E∫0t∧τDC1∥XN(s)∥2p-2ds+p(p-1)2C1E∫0t∧τD∥XN(s)∥2pds+0=E∥X(0)∥2p+E∫0t∧τD(C1p(p-1)2-pλ2)∥XN(s)∥2pds+E∫0t∧τD(C1p(p-1)2+pb22λ)∥XN(s)∥2p-2ds≤E∥X(0)∥2p+∫0t|C1p(p-1)2-pλ2|E∥XN(s)∥2pds+∫0t(C1p(p-1)2+pb22λ)E∥XN(s∧τD)∥2p-2ds.
Let p=2 and use the Gronwall inequalities, then there exists a constant C2, such that
(23)supt∈[0,T]E∥XN(t∧τD)∥2p≤C2.
Computing recursively, we obtain that there exists a constant Cp, such that
(24)E∥XN(t∧τD)∥2p≤E∥X0∥2p+∫0t(|C1p(p-1)2-pλ2|E∥XN(s∧τD)∥2paaaaaaaaaaaa+(C1p(p-1)2+pb22λ)Cp-2)ds≤Cp.
It is obvious that the stopping time satisfies τD→T as D→∞. By the continuity of the solution XN(t) in t, we derive that ∥XN(t∧τD)∥2p is bounded. Therefore,
(25)∥XN(t∧τD)∥2p⟶∥XN(t)∥2p,a.s.(D⟶∞).
Combining (24), (25) and Fuatou lemma, we have
(26)E∥XN(t)∥2p=ElimD→∞∥XN(t∧τD)∥2p≤limD→∞infE∥XN(t∧τD)∥2p≤Cp.Lemma 8.
Letp∈ℕ be even, E∥X0∥2p<∞, and (A1)–(A3) hold. Then there exists a constant C~p such that
(27)Esupt∈[0,T]∥XN(t)∥2p≤C~p,∀t∈[0,T],
where C~p is independent of N, only dependent on T, E∥x0∥2p, p, C1, λ, and b.Proof.
Note thatτD is the stopping time introduced in (20). Integrating (10) (we take p=2),
(28)∥XN(t∧τD)∥22=∥X0∥22+∫0t∧τD(-λ∥XN(s)∥22)ds+∫0t∧τDb2λds+∫0t∧τD∥U(XN(s))∥22ds+∫0t∧τD2XNT(s)U(XN(s))dBs+∫0t∧τDϕ(s)ds.
To estimate the expected supremum of ∥XN(t∧τD)∥2p, we omit the nonpositive terms∫0t∧τD(-λ∥XN(s)∥22)ds and ∫0t∧τDϕ(s)ds, then
(29)∥XN(t∧τD)∥22≤∥X0∥22+∫0t∧τDb2λds+∫0t∧τD∥U(XN(s))∥22ds+∫0t∧τD2XNT(s)U(XN(s))dBs.
To obtain the estimation of ∥XN(t∧τD)∥2p, we take the p/2 power to (29) and use the primary inequality |∑i=1nai|k≤nk-1∑i=1n|ai|k for k≥1 (especially, let n=3, k=p/2),
(30)∥XN(t∧τD)∥2p≤3(p/2-1)∥X0∥2p+3(p/2-1)×|∫0t∧τD(b2λ+∥U(XN(s))∥22)ds|p/2+3(p/2-1)×|∫0t∧τD2XNT(s)U(XN(s))dBs|p/2.
We now compute the expected supremum
(31)Esupt∈[0,T]∥XN(t∧τD)∥2p≤3(p/2-1)E∥X0∥2p+3(p/2-1)E×|∫0T(b2λ+∥U(XN(s))∥22)ds|p/2+3(p/2-1)E×supt∈[0,T]|∫0t∧τD2XNT(s)U(XN(s))dBs|p/2.
To deal with the term E|∫0T(b2/λ+∥U(XN(s))∥22)ds|p/2, we use Hölder inequality, the linear growth condition (A1), and the primary inequality again,
(32)E|∫0T(b2λ+∥U(XN(s))∥22)ds|p/2≤E|(∫0T1p/(p-2)ds)(p-2)/p(∫0T(b2λ+C1+C1∥XN(s)∥22)p/2ds)2/paaaa×(∫0T1p/(p-2)ds)(p-2)/p|p/2≤T(p-2)/2E∫0T(b2λ+C1+C1∥XN(s)∥22)p/2ds≤3p/2-1T(p-2)/2×E∫0T((b2λ)p/2+C1p/2+C1p/2∥XN(s)∥2p)ds.
Note that the solution of the modified system is both continuous in both t and ℱt measurable for all t≥0. By Fubini theorem and the boundary of Lemma 7, we have
(33)E|∫0T(b2λ+∥U(XN(s))∥22)ds|p/2≤(3T)(p-2)/2∫0T((b2λ)p/2+C1p/2+C1p/2Cp)ds∶=C~p(1).
We now use the Burholder-Davis-Gundy inequality ([18, Theorem 2.6]) to estimate the stochastic integral
(34)Esupt∈[0,T]|∫0t∧τD2XNT(s)U(XN(s))dBs|p/2≤C~p(2)E|∫0T∥XNT(s)U(XN(s))∥22ds|p/4,
with
(35)C~p2∶={(64p)p/4,0<p<4,(p/2)p/2+12(p/2-1)p/2-1,4≥p.
If p=2, we use the inequality x≤1+x. For p>2, we use the Hölder inequality to remove the power outside the integral. Afterwards we proceed a similar technique as we handled the term ∫0T∥U(XN(s))∥22ds, which leads to a constant C~p(3), such that
(36)Esupt∈[0,T]|∫0t∧τD2XNT(s)U(XN(s))dBs|p/2≤C~p(3).
Combine (31)–(36), then it follows that
(37)Esupt∈[0,T]∥XN(t∧τD)∥2p≤3(p-2)/2E∥X0∥2p+3(p-2)/2C~p(1)+3(p-2)/2C~p(3)∶=C~p.
To complete the proof, we mention that t∧τD→t for D→∞. Furthermore, we note that the solution XN(t) is continuous in t. Thus
(38)supt∈[0,T]∥XN(t∧τD)∥2p⟶supt∈[0,T]∥XN(t)∥2p,a.s.(D⟶∞).
Since all terms are non negative and the limit exists, using Fatou lemma, we have
(39)Esupt∈[0,T]∥XN(t)∥2p=ElimD→∞supt∈[0,T]∥XN(t∧τD)∥2p≤liminfD→∞Esupt∈[0,T]∥XN(t∧τD)∥2p≤C~p.
The uniform boundary of the expected supremum is obtained.In the following theorem, the main results are introduced by the estimations of the moments.Theorem 9.
Let (A1)–(A3) hold. Then the Lorentz-Rössler system given by (5) and (6) with E∥X0∥22<∞ possesses a unique almost sure continuous solution process, which has the following properties:
If in additionE∥X0∥2p<∞ for a fixed p∈ℕ is even, then there exist two constants C~(1)>0 and C~(2)>0, which are only dependent on T, E∥X0∥2p, p, λ, and b, such that
(40)E∥X(t)∥2p≤C~(1),∀t∈[0,T],Esupt∈[0,T]∥X(t)∥2p≤C~(2).Proof.
Since the coefficients of the system (5) satisfy the local Lipschitz condition, the uniqueness follows. Furthermore, Lemma 3 exists a continuous solution XN(t). To prove the existence of Theorem 9, we need to show that XN(t)→X(t) for N→∞.
LetτD denote the stopping time introduced in (20) for an N∈ℕ. Using Lemma 8 and Chebyshev inequality, we get
(41)ℙ{τN(ω)<T}≤ℙ{supt∈[0,T]∥XN(t)∥2≥N}≤Esupt∈[0,T]∥XN(t)∥22N2≤C~(T,E∥X0∥22)N2⟶0,(N⟶∞).
We can find for almost every ω∈Ω and an N0(ω) such that τN0(ω)=T. Moreover, we have
(42)CN′(X)=CN(X)=C(X),N′≥N>0,∀∥X∥2≤N.
Thus
(43)τN′≥τN,XN′X0(·,ω)=XNX0(·,ω)(almostsure),on[0,τN]∀N′≥N.
From (43), if τN=T, then τN′=T for all N′≥N. Thus the set {ω:τN(ω)=T} is monotonously increasing and converges to Ω as N→∞. Combining (41), we have
(44)τN⟶ℙT,(N⟶∞).
Moreover, because XN(t) is continuous in t and converges to X(t) uniformly in t, X(t) is also continuous in t. (Actually, note that if τN0(ω)=T, then we can express for almost all ω∈Ω the limit function by X(·,ω):=XN′(·,ω) for all N′≥N0(ω).)
In the following proof, we have to show that the limit function is the solution of the original Lorentz-Rössler system. Whent=0, it is obvious that XN(0)=X(0)=X0 for all N∈ℕ. For t∈(0,T], we consider the limit of (9) for N→∞. By the definition of τD in (20), it follows that
(45)CN(XN(t∧τN))=C(X(t∧τN)),XN(t∧τN)=X(t∧τN),∀t<T.
Moreover the almost sure convergence of τN→(N→∞)T implies
(46)ℙ{supt∈[0,T]∥∫t0t(A(XN(s)-X(s))+(F-F)aaaaaaaaaaaa+CN(XN(s))-C(X(s)))dsaaaaaaaa+∫t0t(U(XN(s))-U(X(s)))dBs∥2>0}≤ℙ{τN<t}→(N→∞)0.
In fact, if τN≥t, then τN∧t=t. So XN(t)=X(t), CN(XN(t))=C(X(t)), and
(47)supt∈[0,T]∥∫t0t(A(XN(s)-X(s))+(F-F)aaaaaaaaa+CN(XN(s))-C(X(s)))dsaaaaaa+∫t0t(U(XN(s))-U(X(s)))dBs∥2=0.
Hence X(·) is a solution of the stochastic Lorentz-Rössler system on [0,T]. Finally, the boundary of the moments (40) can be obtained by the uniform convergence of XN(t) to X(t) in t, Lemmas 7 and 8.


## 4. Numerical Results

In this section, we give some numerical results with different parameters for which the stochastic and deterministic Lorentz-Rössler systems show qualitatively different behaviors and illustrate our theory. The respective numerical schemes are the stochastic and deterministic Runge-Kutta numerical difference schemes. All of the initial value of our problems are given byX0=(1,0.1,1) and all of the systems of this section satisfy our conditions (A1)–(A3). To simplify, in this section, we only consider the independent noise (uij,i≠j=0, i,j=1,2,3).Firstly, we consider some examples to exhibit the moment estimations of the Lorentz-Rössler combining systems.Example 10.
Let the coefficients of the linear terms beσ=2, r=1, a=-7, b=1, β=2, and c=1, and let nonlinear parameters be α1(t)=2/(cos(t2)+2), α2(t)=1/(cos(t2)+2), α3(t)=1/(cos(t2)+2), which easily satisfy the conditions (A1)–(A3). The stochastic and deterministic (4) can be shown by Figures 7, 8.
Let the nonlinear terms beα1(t)=4/(sin(t2)+8), α2(t)=3/(sin(t2)+8), and α3(t)=1/(sin(t2)+8), then we can find that the stochastic and deterministic behavior make very different trajectory (Figures 9, 10).The attractor and the time series of the Stochastic Lorentz-Rössler combining system with a white noiseuii=0.01, i=1,2,3.
(a)
(b)The attractor and the time series of the deterministic Lorentz-Rössler system.
(a)
(b)The attractor and the time series of the Stochastic Lorentz-Rössler combining system with a white noiseuii=0.01, i=1,2,3.
(a)
(b)The attractor and the time series of the deterministic Lorentz-Rössler system.
(a)
(b)Example 11.
Letc=0. To satisfy the condition (A2), it is enough to take β>0 and a<1. Thus in this example, firstly, we consider σ=12, r=-10, a=-7, b=1, β=2, and c=0 and α1(t)=1/(sint+2)+5/(sin(t2)+2), α2(t)=5/(sin(t2)+2), and α3(t)=1/(sint+2) (Figures 11, 12).
And in this example, we consider the effect of matrixA. Change the linear terms and take σ=9, r=-10, a=-18, b=1, β=2, and c=0 (Figures 13, 14).Stochastic system with a white noiseuii=0.1, i=1,2,3.
(a)
(b)Deterministic system.
(a)
(b)Transform linear terms system with a white noiseuii=0.1, i=1,2,3.
(a)
(b)Deterministic system with different linear terms.
(a)
(b)Example 12.
In this example, we mainly consider the effect of the noise. Letσ=2, r=0, a=-18, b=1, β=|cost|, c=0 and α1(t)=t/(2t+2)-((2sint+3)+1/(cos(3t)+2)), α2(t)=-((2sint+3)+1/(cos(3t)+2)), and α3(t)=t/(2t+2) (Figures 15–18).Stochastic system with a white noiseuii=0.1, i=1,2,3.
(a)
(b)Stochastic system with a white noiseuii=0.01, i=1,2,3.
(a)
(b)Stochastic system with a white noiseuii=0.001, i=1,2,3.
(a)
(b)Deterministic system with a white noiseuii=0, i=1,2,3.
(a)
(b)In this paper, we have established a sufficient condition under which the stochastic system (3) has a unique solution. And from these numerical results, some interesting qualities can be seen.(i)
From Examples10–12, it is obvious that for any time T, Theorem 9 can be ensured. And the boundary of the moments of the solutions can be obtained.(ii)
Examples10 and 11 show that for (4), the effect of the dynamical behavior would mainly depend on the nonlinear terms. Some transformations especially about linear terms are given by Example 11, but the paths have small transformations.(iii)
For the more general Lorentz-Rössler systems, if the uniqueness can be ensured, especially in Example12, we can show that the stochastic systems converge toward the deterministic system when the intensity of noise converges to zero.(iv)
The system (3) gives us very abundant expressions including the behavior of the well-known Lorentz system and the Rössler system. Some systems with appropriate parameters in (3) can be used to make good hop-frequency time series.



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*Source: 480259-2013-04-04.xml*