Research Article | Open Access
Infinite-Scroll Attractor Generated by the Complex Pendulum Model
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.
A variety of natural systems show a chaotic (aperiodic) behaviour. Such systems depend sensitively on initial data, and one cannot predict the future of the solutions. There are various chaotic systems such as the Lorenz system , the Rossler system , the Chen system , and the Lü system  where the dependent variables are the real-valued functions. Though the chaos has been intensively studied over the past several decades, very few articles are devoted to study the complex dynamical systems. Ning and Haken  proposed a complex Lorenz system arising in lasers. Wang et al.  discussed the applications in genetic networks. Mahmoud and coworkers have studied complex Van der Pol oscillator , new complex system , complex Duffing oscillator , and so forth. Complex multiscroll attractors have a close relationship with complex networks also [10–12].
In this work, we propose a complex pendulum equation exhibiting infinite-scroll attractor. The chaotic phase portraits are plotted, and maximum Lyapunov exponents are given for the different values of the parameter.
2. The Model
The real pendulum equation is given by  where is constant and is a real valued function. We propose a complex version of (1) given by where is a complex-valued function. The system (2) gives rise to a coupled nonlinear system Using the new variables , , and , the system (3) can be written as the autonomous system of first-order ordinary differential equations given by
Symmetry about the , -axes (or , axes), since (or ) do not change the equations.
Consider the following: System is conservative.
2.3. Equilibrium Points and Their Stability
It can be checked that the system (4) has infinitely many real equilibrium points given by , where . Jacobian matrix corresponding to the system (4) is Since the eigenvalues of are , , , and , the points are stable equilibrium points. The eigenvalues of are , , , and .
An equilibrium point is called a saddle point if the Jacobian matrix at 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 established in the literature [14–17] that scrolls are generated only around the saddle points of index two.
It is now clear that the system (4) has infinitely many saddle equilibrium points , of index two which gives rise to an infinite-scroll attractor.
Maximum Lyapunov exponents (MLEs) for the system (4) are plotted in Figure 1. The positive MLEs indicate that the system is chaotic for . Figures 2(a)–2(d) show the chaotic time series for . For the same values of , Figures 3(a) and 3(b) represent multiscroll attractor in plane and in space, respectively.
We have generalized the real function in the pendulum equation to a complex one and studied the chaotic behaviour. The new system is equivalent to a system of four first-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 field of chaos to study the properties of such systems in detail.
- E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
- O. E. Rossler, “An equation for continuous chaos,” Physics Letters A, vol. 57, pp. 397–398, 1976.
- G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 7, pp. 1465–1466, 1999.
- J. Lü and G. R. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659–661, 2002.
- C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Physical Review A, vol. 41, pp. 3826–3837, 1990.
- P. Wang, J. Lu, and M. J. Ogorzalek, “Global relative parameter sensitivities of the feed-forward loops in genetic networks,” Neurocomputing, vol. 78, no. 1, pp. 155–165, 2012.
- G. M. Mahmoud and A. A. M. Farghaly, “Chaos control of chaotic limit cycles of real and complex van der Pol oscillators,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 915–924, 2004.
- G. M. Mahmoud, S. A. Aly, and M. A. AL-Kashif, “Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 171–181, 2008.
- G. M. Mahmoud, A. A. Mohamed, and S. A. Aly, “Strange attractors and chaos control in periodically forced complex Duffing's oscillators,” Physica A, vol. 292, no. 1–4, pp. 193–206, 2001.
- J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 841–846, 2005.
- J. Lü, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Transactions on Circuits and Systems, vol. 51, no. 4, pp. 787–796, 2004.
- J. Zhu, J. Lu, and X. Yu, “Flocking of multi-agent non-holonomic systems with proximity graphs,” IEEE Transactions on Circuits and Systems I, vol. 60, no. 1, pp. 199–210, 2013.
- H. Goldstein, Classical Mechanics, Addison-Wesley, New York, NY, USA, 1980.
- L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1097, 1986.
- C. P. Silva, “Shil'cprime nikov's theorem—a tutorial,” IEEE Transactions on Circuits and Systems, vol. 40, no. 10, pp. 675–682, 1993.
- D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ring,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 10, pp. 2889–2903, 2003.
- J. Lü, G. Chen, X. Yu, and H. Leung, “Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Transactions on Circuits and Systems, vol. 51, no. 12, pp. 2476–2490, 2004.
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.