Chapter 3 Problem 3.16: Strange attractor

计算物理

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Name: 陈锋
Student number: 2013301020145
April 21,2016

Abstract

This page is about how a strange attractor is altered by some small changes in one of the pendulum parameters, especially the changes of the drive amplitude and drive frequency. The changes of the strange attractor will be showed in this article.

Introduction

Chaotic systems do not mean random system. They are deterministic for the systems must obey certain deterministic laws of physics, but still exhibit behavior that is unpredictable due to an extreme sensitivity to initial conditions.

Figure 1: Normal driven pendulum, with $F_D = 0.5$ and $\Omega_D = 2/3$. The yellow arrow represents the driven forve.

Figure 2: This is a chaotic pendulum, with $F_D = 1.2$ and $\Omega_D = 2/3$. There is not a fixed period as a normal harmonic pendulum and its motion is unpredictable.

Figure 3: The phase diagram of the chaotic case, where $F_D = 1.2$, $\Omega_D = 2/3$, time is from 0 to 10000s.

The Equation of motion of driven nonlinear pendulum

In this case, we do not assume the small-angle approximation and do not expand $sin\theta$ term. Also, we include friction of the form $-q(d\theta/dt)$ and add a term of driving force $F_Dsin(\Omega_Dt)$. Putting all of these terms together, we have the equation of motion:

$$\begin{equation} \frac{d^2\theta}{dt^2} = -\frac{g}{l}sin\theta - q \frac{d\theta}{dt} + F_Dsin(\Omega_Dt) \end{equation}$$
This is the model for a nonlinear, damped, driven pendulum, the physical pendulum.
Again, we rewrite equation(1) as two first-order differential equations and obtaion:
$$\begin{equation} \frac{d\omega}{dt} = -\frac{g}{l}sin\theta - q \omega + F_Dsin(\Omega_Dt) \\ \frac{d\theta}{dt} = \omega \end{equation}$$
According to these equations, we can use Euler-Cromer method or Runge-Kutta method to solve this problem.

The influence of initial value of $\theta$

First of all, I consider the influence of the initial value of $\theta$, which is similar to procedures in the textbook. The results for $\Delta \theta = |\theta_1-\theta_2|$ are shown in Figure 4 for two different values fro the drive amplitude. We should notice that the amplitude difference has increased exponentially at the first beginning. However, it tends to fluctuate about $10^0$.

Figure 4: Results for $\Delta \theta$ from our comparison of two identical pendulums with $F_D = 1.2$ and $\Omega_D = 2/3$. The initial values for $\theta$ for the two pendulums differed by 0.001 rad.

The influence of the driven force $F_D$

In this section, I try to observe the effect on the strange attractor, if I change the drive amplitude by a small amount. The result are listed in the following diagrams. All the diagrams have been run with $g/l = 1$, $q = 0.5$, $\Omega_D = 2/3$.
Firstly, I make the driven force amplitude for the two pendulums differed by 0.01 N:

Figure 5: The left diagram and the right one show the strange attractor at diverse driven force amplitude. The middle one shows the results of $\Delta \theta$. If we also use Lyapunov exponent to describe this situation, the coefficient $\lambda$ is obviously larger than zero. In other words, the case has been totally different after the changing of driven force amplitude.

Figure 6: These diagrams show the variance of strange attractors when I change the dirven amplitude from 1.15 N to 1.25 N. It seems that all of them keep the same shape, but the quantity and the distribution of points may be different.

Figure 7: This is a comparison of the strange attractor of three diverse cases, including 1.15N, 1.20N, 1.25N. Apparently, as the increase of amplitude in such a small range, there will be a translation of strange attractor from right (blue one) to left (green one).

Figure 8: The comparison of phase diagrams between $F_D = 1.10N$, $F_D = 1.20N$ and $F_D = 1.25N$. With the increase of driven amplitude, the central part of the phase diagram becomes larger and larger.

The influence of the driven frequency $\Omega_D$

This section is to discuss the influence on the oscillatory motion of a small change in the driven frequency. I will show the difference of the corresponding strange attractor diagrams of diverse cases. All the diagrams have been run with $g/l = 1$, $q = 0.5$, $F_D = 1.2$. The results are as follows:
Firstly, I make the driven frequency for the two pendulums differed by 0.01/3 Hz:

Figure 9: The left diagram and the right one show the strange attractor at diverse driven frequency. The middle one shows the results of $\Delta \theta$. If we also use Lyapunov exponent to describe this situation, the coefficient $\lambda$ is obviously larger than zero. In other words, the case has been totally different after the changing of driven frequency.

Figure 10: These diagrams show the variance of strange attractors when I change the dirven frequency from 1.95/3 Hz to 2.05/3 Hz. It seems that all of them keep the same shape, but the quantity and the distribution of points may be different.

Figure 11: This is a comparison of the strange attractor of three diverse frequency cases, including 1.95/3 Hz, 2.00/3 Hz, 2.05/3 Hz. Apparently, as the increase of frequency in such a small range, there will be a translation of strange attractor from left (blue one) to right (green one).

Figure 12: The comparison of phase diagrams between $\Omega_D = 1.95/3 Hz$, $\Omega_D = 2.00/3 Hz$ and $\Omega_D = 2.05/3 Hz$. With the increase of driven frequency, the central part of the phase diagram contracts.

Conclusion

The small amount changes in either the driven amplitude or dirven frequency will cause a translation in the attractor. However, if the change become large enough, the chaos might be gone and return to a normal driven pendulum situation.

Reference

1.Giodano, N.J., Nakanishi, H. Computational Physics. Tsinghua University Press, December 2007.

Programme Code

• Chapter 3 Problem 3.16