Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 368150, 3 pages http://dx.doi.org/10.1155/2013/368150

Research Article Infinite-Scroll Attractor Generated by the Complex Pendulum Model

Sachin Bhalekar

Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India

Correspondence should be addressed to Sachin Bhalekar; [email protected]

Received 20 November 2012; Revised 4 February 2013; Accepted 12 February 2013

Academic Editor: Rodica Costin

Copyright © 2013 Sachin Bhalekar. 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 report the finding of the simple nonlinear autonomous system exhibiting infinite-scroll attractor. The system is generated from the pendulum equation with complex-valued function. The proposed system is having infinitely many saddle points of index two which are responsible for the infinite-scroll attractor.

1. Introduction where 𝑎>0is constant and 𝑥 is a real valued function. We propose a complex version of (1)givenby A variety of natural systems show a chaotic (aperiodic) behaviour. Such systems depend sensitively on initial data, 𝑧=−𝑎̈ sin (𝑧) , (2) and one cannot predict the future of the solutions. There are various chaotic systems such as the Lorenz system [1], the where 𝑧(𝑡) = 𝑥(𝑡) + 𝚤𝑦(𝑡) is a complex-valued function. The Rossler system [2], the Chen system [3], and the Lusystem[¨ 4] system (2) gives rise to a coupled nonlinear system where the dependent variables are the real-valued functions. Though the chaos has been intensively studied over the past 𝑥=−𝑎̈ sin (𝑥) cosh (𝑦) , several decades, very few articles are devoted to study the (3) complex dynamical systems. Ning and Haken [5]proposed 𝑦=−𝑎̈ sinh (𝑦) cos (𝑥) . a complex Lorenz system arising in lasers. Wang et al. [6] discussed the applications in genetic networks. Mahmoud Using the new variables 𝑥1 =𝑥,𝑥2 = 𝑥̇, 𝑥3 =𝑦,and𝑥4 = 𝑦̇, and coworkers have studied complex Van der Pol oscillator the system (3) can be written as the autonomous system of [7], new complex system [8], complex Duffing oscillator9 [ ], first-order ordinary differential equations given by and so forth. Complex multiscroll attractors have a close relationshipwithcomplexnetworksalso[10–12]. 𝑥̇1 =𝑥2, In this work, we propose a complex pendulum equation 𝑥̇ =−𝑎 (𝑥 ) (𝑥 ), exhibiting infinite-scroll attractor. The chaotic phase portraits 2 sin 1 cosh 3 (4) are plotted, and maximum Lyapunov exponents are given for 𝑥̇3 =𝑥4, the different values of the parameter. 𝑥̇4 =−𝑎sinh (𝑥3) cos (𝑥1). 2. The Model 2.1. Symmetry. Symmetry about the 𝑥1, 𝑥2-axes (or 𝑥3, The real pendulum equation is given by [13] 𝑥4 axes), since (𝑥1,𝑥2,𝑥3,𝑥4)→(𝑥1,𝑥2,−𝑥3,−𝑥4) (or (𝑥1,𝑥2,𝑥3,𝑥4) → (−𝑥1,−𝑥2,𝑥3,𝑥4))donotchangethe 𝑥=−𝑎̈ (𝑥) , sin (1) equations. 2 International Journal of Analysis

0.5 30 0.4

20 0.3 1 𝑎 𝑥 𝐿 10 0.2

0 0.1 200 400 600 800 1000 𝑡 −10 0 0 0.2 0.4 0.6 0.8 1 (a) 𝑎

Figure 1: Maximum Lyapunov exponents. 4

2.2. Conservation. Consider the following: 2 4 𝜕𝑥̇𝑖 ∇⋅𝐹=∑ =0. (5) 𝜕𝑥 2 0 𝑖=1 𝑖 𝑥 750 800 850 900 950 1000 System is conservative. 𝑡 −2 2.3. Equilibrium Points and Their Stability. It can be checked that the system (4) has infinitely many real equilibrium points −4 𝐸 = (𝑘𝜋,0,0,0) 𝑘 = 0,±1,±2,... given by 𝑘 ,where . Jacobian (b) matrix corresponding to the system (4)is 0100 −𝑎 (𝑥 ) (𝑥 )0−𝑎 (𝑥 ) (𝑥 )0 10 𝐽=( cos 1 cosh 3 sin 1 sinh 3 ). 0001 𝑎 (𝑥 ) (𝑥 )0−𝑎 (𝑥 ) (𝑥 )0 sin 1 sinh 3 cos 1 cosh 3 5 (6) 3

𝐽(𝐸 ) 𝚤√𝑎 𝚤√𝑎 −𝚤√𝑎 −𝚤√𝑎 𝑥 Since the eigenvalues of 2𝑘 are , , ,and , 0 the points 𝐸2𝑘 are stable equilibrium points. The eigenvalues 750 800 850 900 950 1000 of 𝐽(𝐸2𝑘+1) are √𝑎, √𝑎, −√𝑎,and−√𝑎. 𝑡 𝑝 An equilibrium point is called a saddle point if the −5 Jacobian matrix at 𝑝 has at least one eigenvalue with negative real part (stable) and one eigenvalue with nonnegative real part (unstable). A saddle point is said to have index one (/two) if there is exactly one (/two) unstable eigenvalue/s. It is (c) establishedintheliterature[14–17] that scrolls are generated only around the saddle points of index two. It is now clear that the system (4)hasinfinitelymany 4 saddle equilibrium points 𝐸2𝑘+1, 𝑘 = 0,±1,±2,... of index twowhichgivesrisetoaninfinite-scrollattractor. 2 4

2.4. Chaos. Maximum Lyapunov exponents (MLEs) for the 𝑥 0 system (4)areplottedinFigure 1.ThepositiveMLEsindicate 750 800 850 900 950 1000 𝑡 that the system is chaotic for 𝑎>0.Figures2(a)–2(d) show −2 the chaotic time series for 𝑎 = 0.3.Forthesamevalues 𝑎 of ,Figures3(a) and 3(b) represent multiscroll attractor in −4 (𝑥1,𝑥3) plane and in (𝑥1,𝑥2,𝑥3) space, respectively.

3. Conclusions (d)

We have generalized the real function in the pendulum Figure 2: (a) Time series 𝑥1,(b)timeseries𝑥2, (c) time series 𝑥3, equation to a complex one and studied the chaotic behaviour. and (d) time series 𝑥4. International Journal of Analysis 3

10 Duffing’s oscillators,” Physica A,vol.292,no.1–4,pp.193–206, 2001. 5 [10] J. Lu¨ and G. Chen, “Atime-varying complex dynamical network

3 model and its controlled synchronization criteria,” IEEE Trans- 𝑥 0 actions on Automatic Control, vol. 50, no. 6, pp. 841–846, 2005. 10 20 30 [11] J. Lu,X.Yu,G.Chen,andD.Cheng,“Characterizingthe¨ −5 synchronizability of small-world dynamical networks,” IEEE Transactions on Circuits and Systems,vol.51,no.4,pp.787–796, 𝑥1 2004. [12] J. Zhu, J. Lu, and X. Yu, “Flocking of multi-agent non-holonomic (a) systems with proximity graphs,” IEEE Transactions on Circuits 4 𝑥1 2 2 0 and Systems I, vol. 60, no. 1, pp. 199–210, 2013. 𝑥 0 10 −2 20 −4 30 [13] H. Goldstein, Classical Mechanics, Addison-Wesley, New York, 10 NY, USA, 1980. [14] L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll 5 family,” IEEE Transactions on Circuits and Systems,vol.33,no. 3

0 𝑥 11, pp. 1072–1097, 1986. [15] C. P. Silva, “Shil’cprime nikov’s theorem—a tutorial,” IEEE −5 Transactions on Circuits and Systems,vol.40,no.10,pp.675– 682, 1993. [16] D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyper- chaotic Chua’s circuits forming a ring,” International Journal of (b) Bifurcation and Chaos in Applied Sciences and Engineering,vol. Figure 3: (a) Phase portrait, and (b) phase portrait in space. 13, no. 10, pp. 2889–2903, 2003. [17] J. Lu,G.Chen,X.Yu,andH.Leung,“Designandanalysisof¨ multiscroll chaotic attractors from saturated function series,” The new system is equivalent to a system of four first- IEEE Transactions on Circuits and Systems,vol.51,no.12,pp. 2476–2490, 2004. order ordinary differential equations. There are infinitely many saddle points of index two for this system which are responsible for the infinite-scroll chaotic attractor. Such example of infinite-scroll attractor will help researchers in the fieldofchaostostudythepropertiesofsuchsystemsindetail.

References

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