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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 480259, 14 pages
Existence and Uniqueness of the Solution of Lorentz-Rössler Systems with Random Perturbations
1School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
2Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China
4Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Received 24 December 2012; Accepted 26 January 2013
Academic Editor: Jinhu Lü
Copyright © 2013 Xiaoying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The Lorentz system is a well-known model This model was introduced in 1963 by Lorentz . For the meaning of the Lorentz system the reader can refer to  (Chaos),  (laser),  (thermospheres),  (brushless DC motors),  (electric circuits), and  (chemical reactions). The original Rössler system only contains one quadratic nonlinear term The system (2) was introduced by Rössler . This model has received increasing attention due to its great potential applications in secure communication [9–12], chemical reaction, biological systems, and so on .
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 become ): 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 , , , , and that our problem becomes the well-known Lorentz system with the initial value .
Let , , and , and take the correspondingly appropriate other parameters, then the track similar with the Rössler systems can be obtained.
If the coefficients of our problem are , , , , , , and , 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) where all the represent the intensity of the noise at time and all the are standard white noise; namely, each is a Brownian motion defined on a complete probability space . 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.
In this paper, we consider that is not a constant and dependent on the third variable and the coefficient of the third interactive term , let . Throughout this paper, unless otherwise specified, we let be a complete probability with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all -null sets). denotes an 3-dimensional Brownian motion defined on this probability space.
The rest of this paper is arranged as follows. In Section 2, 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 . Let . The Lorentz-Rössler system can be rewritten as where the initial condition is fixed point and independent of for all . The four parts of the drift are given by matrices is a noise term.
For the sake of brevity, we introduce some notations. (i)For any real matrix , define . (ii)For any is even, , . (iii)For any two variables , denote the usual inner product. (iv) () denote the constants which are dependent on some parameters.
(A1) The noise term satisfies a Lipschitz condition and a linear growth condition
The condition (A1) can be easily satisfied, for example, , when all the are bounded on .
For the other coefficients, we suppose the following conditions.
(A2) The matrix satisfies , where .
(A3) The constant and the coefficients , , and are bounded. In addition, the interactive terms' coefficients satisfy .
Remark 2. If , 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 and . 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 , , satisfy these two conditions. However, the term 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. Let with for and for . , the modified system where and is a finite time. And let the initial value be still independent of and satisfy . Then the modified system (9) possesses a continuous almost sure unique solution that is measurable.
Remark 4. 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 , the modified nonlinear term 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 where is an adapted process.
Proof. Define the Lyapunov function
Using the Itŏ formula with respect to for is even, we compute the Itŏ derivative of the Lyapunov function of the solution of the modified system. We consider
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 matrix , , the following inequalities hold:
Proof. From Hölder inequality, it follows that By Lemma 6, of (12) can be estimated, Combining (A2) and Hölder inequality, we have From (A3), we can compute Using Lemma 6 again and combining all the above estimations from (15)–(17), we derive that where 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 let be even and fixed, the initial expectation . Then where is a constant and only dependent on , , , , , and , but independent of .
Proof. Firstly, we introduce the stopping time. For , let Note that, for all , For any , using the linear growth condition of , we can integrate (10) from to and then take the expectations to obtain Let and use the Gronwall inequalities, then there exists a constant , such that Computing recursively, we obtain that there exists a constant , such that It is obvious that the stopping time satisfies as . By the continuity of the solution in , we derive that is bounded. Therefore, Combining (24), (25) and Fuatou lemma, we have
Lemma 8. Let be even, , and (A1)–(A3) hold. Then there exists a constant such that where is independent of , only dependent on , , , , , and .
Proof. Note that is the stopping time introduced in (20). Integrating (10) (we take ), To estimate the expected supremum of , we omit the nonpositive terms and , then To obtain the estimation of , we take the power to (29) and use the primary inequality for (especially, let , ), We now compute the expected supremum To deal with the term , we use Hölder inequality, the linear growth condition (A1), and the primary inequality again, Note that the solution of the modified system is both continuous in both and measurable for all . By Fubini theorem and the boundary of Lemma 7, we have We now use the Burholder-Davis-Gundy inequality ([18, Theorem 2.6]) to estimate the stochastic integral with If , we use the inequality . For , we use the Hölder inequality to remove the power outside the integral. Afterwards we proceed a similar technique as we handled the term , which leads to a constant , such that Combine (31)–(36), then it follows that To complete the proof, we mention that for . Furthermore, we note that the solution is continuous in . Thus Since all terms are non negative and the limit exists, using Fatou lemma, we have 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 possesses a unique almost sure continuous solution process, which has the following properties:
If in addition for a fixed is even, then there exist two constants and , which are only dependent on , , , , and , such that
Proof. Since the coefficients of the system (5) satisfy the local Lipschitz condition, the uniqueness follows. Furthermore, Lemma 3 exists a continuous solution . To prove the existence of Theorem 9, we need to show that for .
Let denote the stopping time introduced in (20) for an . Using Lemma 8 and Chebyshev inequality, we get We can find for almost every and an such that . Moreover, we have Thus From (43), if , then for all . Thus the set is monotonously increasing and converges to as . Combining (41), we have Moreover, because is continuous in and converges to uniformly in , is also continuous in . (Actually, note that if , then we can express for almost all the limit function by for all .)
In the following proof, we have to show that the limit function is the solution of the original Lorentz-Rössler system. When , it is obvious that for all . For , we consider the limit of (9) for . By the definition of in (20), it follows that Moreover the almost sure convergence of implies In fact, if , then . So , , and Hence is a solution of the stochastic Lorentz-Rössler system on . Finally, the boundary of the moments (40) can be obtained by the uniform convergence of to in , 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 by and all of the systems of this section satisfy our conditions (A1)–(A3). To simplify, in this section, we only consider the independent noise (, ).
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 be , , , , , and , and let nonlinear parameters be , , , which easily satisfy the conditions (A1)–(A3). The stochastic and deterministic (4) can be shown by Figures 7, 8.
Let the nonlinear terms be , , and , then we can find that the stochastic and deterministic behavior make very different trajectory (Figures 9, 10).
Example 11. Let . To satisfy the condition (A2), it is enough to take and . Thus in this example, firstly, we consider , , , , , and and , , and (Figures 11, 12).
And in this example, we consider the effect of matrix . Change the linear terms and take , , , , , and (Figures 13, 14).
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 Examples 10–12, it is obvious that for any time , Theorem 9 can be ensured. And the boundary of the moments of the solutions can be obtained. (ii)Examples 10 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 Example 12, 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.
This paper supported by NSFC (Grant no. 11101044), NSFC (Grant no. 11126257), and Special Funds for Co-construction Project of Beijing.
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