#### Abstract

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

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 [1], the Rossler system [2], the Chen system [3], and the Lü system [4] 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 [5] proposed a complex Lorenz system arising in lasers. Wang et al. [6] discussed the applications in genetic networks. Mahmoud and coworkers have studied complex Van der Pol oscillator [7], new complex system [8], complex Duffing oscillator [9], 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 [13] 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

##### 2.1. Symmetry

Symmetry about the , -axes (or , axes), since (or ) do not change the equations.

##### 2.2. Conservation

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.

##### 2.4. Chaos

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.

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#### 3. Conclusions

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.