Research Article Chaotic Attractor Generation via a Simple Linear Time-Varying System

Transcription

1 Discrete Dnamics in Nature and Societ Volume, Article ID 836, 8 pages doi:.//836 Research Article Chaotic Attractor Generation via a Simple Linear Time-Varing Sstem Baiu Ou and Desheng Liu Department of Engine, Arm Aviation Institute, Beijing 3, China State Ke Laborator for Turbulence and Comple Sstem and College of Engineering, Peking Universit, Beijing 87, China Correspondence should be addressed to Desheng Liu, dsliupku@gmail.com Received 8 March ; Revised 9 June ; Accepted 7 September Academic Editor: Recai Kilic Copright q B. Ou and D. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. A novel generation method of chaotic attractor is introduced in this paper. The underling mechanism involves a simple three-dimensional time-varing sstem with simple time functions as control inputs. Moreover, it is demonstrated b simulation that various attractor patterns are generated convenientl b adjusting suitable sstem parameters. The largest Lapunov eponent of the sstem has been obtained.. Introduction Chaos and Chaotic sstems, which have been etensivel investigated during the last four decades, have been found to be ver useful in a variet of applications such as science, mathematics, engineering communities, and various techniques such as identification and snchroniation. This provides a strong motivation for the current research on eploiting new chaotic sstems. Man chaotic attractors in dnamical sstems have been found numericall and eperimentall, such as Loren attractor 3 and Rössler attractor. The Chua s circuit sstem that has double scroll attractor is probabl the best known and the simplest chaotic sstem. Sukens and Vandewalle 6 proposed some effective methods for generating n- scroll attractors. Moreover, multiscroll attractors are also found in some simple sstems; it stimulates the current research on generating various comple multiscroll chaotic attractors b using some simple electronic circuits and devices. In general case, it is relativel eas to generate chaotic sstems numericall, but it is usuall ver hard to anale or verif the dnamical characteristics of nonsmooth sstems, even for the switched sstems with low dimensions 7, 8. To deal with the stabilit of the equilibrium of switched linear sstems, man efforts have been made and strict analsis

2 Discrete Dnamics in Nature and Societ has been carried out 9. In, a chaotic attractor in a new funnel shape is introduced, simpl b designing a switched sstem with hsteresis-switching signal. Motivated b previous works of chaotic attractor generation, we have made further effort to generate more chaotic behaviors, b introducing time functions, that is, a timevaring sstem with various time functions is investigated. To our happiness, chaotic attractors are observed. And moreover, the shape of the created attractors can be changed easil b changing parameters, and various comple patterns can be obtained. The statistic behavior is also discussed, which reveals the regularities in the comple dnamics. The rest of this paper is organied as follows. Section presents the analsis of chaotic attractor generation. Section 3 introduces a simple three-dimensional time-varing sstem to generate a new chaotic attractor; then Section concentrates on the pattern changes of the generated attractors with parameters variation. A brief conclusion is given in Section.. The Analsis of Chaotic Attractor Generation There are two major researching methods of the chaotic attractor, analtical method and numerical analsis. The analtical methods had two general methods, the Melnikov method and the Kalashnikov method. However, in general case, we used the numerical method due to the difficulties of the analtical method. Numerical method bases on theoretical achievements and computer simulation. Recentl, there were a lot of those achievements, such as the Poincare cross-section, power spectrum, subsampling frequenc, and continuous feedback control. In addition, the spectrum analsis, Lapunov inde, fractal dimension and topological entrop, and so forth, were common methods to describe the statistical properties of chaotic. Some analsis results of chaotic attractor generation were given 3 6, such as horseshoe map 3, Shil nikov theorem, and Li-Yorke theorem. In this section, we studied the conditions of chaotic attractor generation. We first concern general linear sstem as follows: ẋ A, R 3,. where A is the Jacobian J. We assume that A has three eigenvalues γ and σ ± jω, where γ, σ, andω are real numbers, ω /. Then, the equilibrium is stable if and onl if γ <, σ < ; otherwise the equilibrium is unstable. The range of motion of chaotic attractor can be defined, and there should be both of convergent motions and divergent motions of chaotic attractor in the certain region. In consideration of the above, it can be simulated in the case of the equilibrium being stable where convergent or the equilibrium is unstable where divergent. Therefore, a necessar condition of chaotic attractor generation for sstem. is obtained. At the equilibrium of the linear sstem., the three eigenvalues γ and σ ± jω which change with time switch between γ<, σ <, and other conditions. 3. Generating Chaotic Attractor From the analsis in Section, the three eigenvalues of the linearied sstem must be variables for generating chaotic attractor, and the time function is introduced for the

3 Discrete Dnamics in Nature and Societ 3 convenience of understanding and calculation. Consider the following time-varing sstem: ẋ A t, R 3, 3. where /t δ b A t c acos t e, d cos t 3. a, b, c, d are constants, e is controllable parameter, δ > is infinitesimal. We can prove the sstem 3. satisfies the conditions from Section. Solving ẋ ẏ ż ields the unique sstem equilibrium,, T. We assume that: e, a 3, b., c d, δ., t π, t π. 3.3 Then, / t δ, when t. Considering the sstem trajector nearb t π/andt π. Assume Δt > is infinitesimal and Δt.. Then, π/.. ẋ t t, t t Δt, t Δt, 3. π.. ẋ t 3 t, t t Δt, t Δt. 3. And the characteristic polnomials of 3. and 3. are λ 3.6λ.λ, λ λ.λ The characteristic polnomial in 3.6 has at least one root with positive real part, meaning that the sstem in 3. is divergent, whereas the characteristic polnomial in 3.7 is convergent, meaning that the sstem in 3. is convergent. Meanwhile, the equilibrium position of sstem 3. is unstable as t π/ kπ k Z, and the equilibrium position of sstem 3. is stable as t k π k Z. The sstem 3. switches in an alternating manner between two situations: convergence and divergence. A chaotic attractor is generated as shown in Figure. However, this is not a sufficient condition for a chaotic regime to eist.

4 Discrete Dnamics in Nature and Societ a t- projection b t- projection c t- projection d Chaotic attractor e - projection Figure : Chaotic attractor generated b sstem 3.3.

5 Discrete Dnamics in Nature and Societ a e π/ b e π/ 3 7. c e π. 8 Figure : Chaotic attractor changed b e.. Various Patterns with Parameter Changing In this section, we pa attention to the dnamical behaviors of the time-varing sstem 3.3 with parameter selected in the chaotic regions in order to show the effective generation of various patterns of attractors based on the parameter selection. At first, we do not change constant parameters from 3. but let e change in the stabilit intervals. Then, the sstem displa different patterns for different values of e, as shown in Figure. In the three cases, the largest Lapunov eponents are LE.87 e π/, LE.8 e π/, LE.73 e π. Then, we chose parameters from 3. but, respectivel, change constant parameters a, b, c, d. Various attractors are produced with different values of a, b, c, d, as shown in Figures 3 and. The largest Lapunov eponents are given as follows: LE.7 a., LE.8 a., LE. a 3., LE.8 b 6, LE.86 b, LE.3 b., LE.676 c.8, LE.87 c.3, LE.6 c.,. LE.83 c.6, LE.86 d.

6 6 Discrete Dnamics in Nature and Societ a a. b a c a 3. d b e b f b. Figure 3: Different values of parameters a, b.

7 Discrete Dnamics in Nature and Societ a c.8 b c c c. d c e d. f d g d h d Figure : Different values of parameters c, d.

8 8 Discrete Dnamics in Nature and Societ From these numerical simulations, it is shown that the sstem 3. demonstrates rich comple patterns when adjusting sstem parameters. From this, we can see that the proposed time-varing sstem is quite effective in the generation of attractor with obviousl quasiperiodic or chaotic behaviors based on the change of parameters.. Conclusion This paper has given a novel chaotic attractor generation method. The generation of novel chaotic attractors via a simple three-dimensional time-varing sstem with various time functions has been introduced. The results once again support the long-accepted belief that properl designed simple sstems can perform comple dnamical behaviors. Moreover, this sstem can produce various attractor patterns within a wide range of parameter values, and the statistic behavior which reveals the regularities in the comple dnamics is also discussed. In addition, the method has been developed in this article can also be applied to nonlinear dnamical sstems and other fields. It is desirable that one could design more chaos generators b means of the method proposed in this paper. References G. Chen and X. Yu, Eds., Chaos Control: Theor and Applications, vol. 9 of Lecture Notes in Control and Information Sciences, Springer, Berlin, German, 3. Y. Al-Assaf and W. M. Ahmad, Parameter identification of chaotic sstems using wavelets and neural networks, International Bifurcation and Chaos, vol., no., pp ,. 3 E. N. Loren, Deterministic nonperiodic flow, the Atmospheric Sciences, vol.,no.,pp. 3, 963. O. E. Rössler, Continuous chaos four prototpe equations, in Bifurcation Theor and Applications in Scientific Disciplines (Papers, Conf., New York, 977), vol. 36 of Ann. New York Acad. Sci., pp , New York Acad. Sci., New York, NY, USA, 979. L. O. Chua, M. Komuro, and T. Matsumoto, The double scroll famil, IEEE Transactions on Circuits and Sstems, vol. 33, no., pp. 7 8, J. A. K. Sukens and J. Vandewalle, Generation of n-double scrolls n,, 3,,..., IEEE Transactions on Circuits and Sstems I, vol., no., pp , M. Kune, Non-Smooth Dnamical Sstems, vol. 7 of Lecture Notes in Mathematics, Springer, Berlin, German,. 8 P. C. Müller, Calculation of Lapunov eponents for dnamic sstems with discontinuities, Chaos, Solitons & Fractals, vol., no. 9, pp , G. Xie and L. Wang, Necessar and sufficient conditions for controllabilit of switched linear sstems, in Proceedings of the American Control Conference, pp , Ma. G. Xie and L. Wang, Controllabilit and stabiliabilit of switched linear-sstems, Sstems & Control Letters, vol. 8, no., pp. 3, 3. A. Filippov, Differential Equations with Discontinuous Right Hand Side, Kluwer Academic Publishers, Dordrecht, The Netherlands, 988. J. Guo, G. Xie, and L. Wang, Chaotic attractor generation and critical value analsis via switching approach, Chaos, Solitons & Fractals, vol., no., pp. 6 69, 9. 3 S. Smale, The Mathematics of Time: Essas on Dnamical Sstems, Economic Processes and Related Topics, Springer, New York, NY, USA, 98. L. P. Shil nikov, A case of the eistence of denumerable set of periodic motions, Soviet Mathematics. Doklad, vol. 6, no., pp , 96. T. Y. Li and J. A. Yorke, Period three implies chaos, The American Mathematical Monthl, vol. 8, no., pp , S. S. Qiu, Stud on periodic orbit theor of chaotic attractors I, Circuits and Sstems, vol. 8, no. 6, 3.

9 Advances in Operations Research Volume Advances in Decision Sciences Volume Applied Mathematics Algebra Volume Probabilit and Statistics Volume The Scientific World Journal Volume International Differential Equations Volume Volume Submit our manuscripts at International Advances in Combinatorics Mathematical Phsics Volume Comple Analsis Volume International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Volume Volume Volume Volume Discrete Mathematics Volume Discrete Dnamics in Nature and Societ Function Spaces Abstract and Applied Analsis Volume Volume Volume International Stochastic Analsis Optimiation Volume Volume