Infinite-Scroll Attractor Generated by the Complex Pendulum Model

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] 𝑥̈=−𝑎sin(𝑥), (1) where 𝑎>0 is constant and 𝑥 is a real valued function. We propose a complex version of (1) given by 𝑧̈=−𝑎sin(𝑧), (2) where 𝑧(𝑡)=𝑥(𝑡)+𝚤𝑦(𝑡) is a complex-valued function. The system (2) gives rise to a coupled nonlinear system ̈(𝑥)𝑦,̈𝑦(𝑥).𝑥=−𝑎sincosh𝑦=−𝑎sinhcos (3) Using the new variables 𝑥₁=𝑥,𝑥₂=𝑥̇, 𝑥₃=𝑦, and 𝑥₄=𝑦̇, the system (3) can be written as the autonomous system of first-order ordinary differential equations given by 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑎sin𝑥₁cosh𝑥₃,𝑥̇₃=𝑥₄,𝑥̇₄=−𝑎sinh𝑥₃cos𝑥₁. (4)

2.1. Symmetry

Symmetry about the 𝑥₁, 𝑥₂-axes (or 𝑥₃, 𝑥₄ axes), since (𝑥₁,𝑥₂,𝑥₃,𝑥₄)→(𝑥₁,𝑥₂,−𝑥₃,−𝑥₄) (or (𝑥₁,𝑥₂,𝑥₃,𝑥₄)→(−𝑥₁,−𝑥₂,𝑥₃,𝑥₄)) do not change the equations.

2.2. Conservation

Consider the following: ∇⋅𝐹=⁴_𝑖=1𝜕𝑥̇_𝑖𝜕𝑥_𝑖=0. (5) 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 𝐸_𝑘=(𝑘𝜋,0,0,0), where 𝑘=0,±1,±2,…. Jacobian matrix corresponding to the system (4) is 𝑥𝐽=0100−𝑎cos₁cosh𝑥₃0−𝑎sin𝑥₁sinh𝑥₃0𝑥0001𝑎sin₁sinh𝑥₃0−𝑎cos𝑥₁cosh𝑥₃0. (6) Since the eigenvalues of 𝐽(𝐸_2𝑘) are 𝚤√𝑎, 𝚤√𝑎, −𝚤√𝑎, and −𝚤√𝑎, the points 𝐸_2𝑘 are stable equilibrium points. The eigenvalues of 𝐽(𝐸_2𝑘+1) are √𝑎, √𝑎, −√𝑎, and −√𝑎.

An equilibrium point 𝑝 is called a saddle point if the 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 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 𝐸_2𝑘+1, 𝑘=0,±1,±2,… 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 𝑎>0. Figures 2(a)–2(d) show the chaotic time series for 𝑎=0.3. For the same values of 𝑎, Figures 3(a) and 3(b) represent multiscroll attractor in (𝑥₁,𝑥₃) plane and in (𝑥₁,𝑥₂,𝑥₃) space, respectively.

Figure 1 

Maximum Lyapunov exponents.

(a)

(b)

(c)

(d)

Figure 2 

(a) Time series 𝑥1, (b) time series 𝑥2, (c) time series 𝑥3, and (d) time series 𝑥4.

(a)

(b)

Figure 3 

(a) Phase portrait, and (b) phase portrait in space.

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.