The ninth homework

leve1, level2

物基一班
陈天懿 2013301020146

Abstract

This article uses Euler-Cromer method to study the chaos phenomena under some specific situations, then solve the problem 3.12, 3.16.

Background

Last time we discussed the motion of the simple pendulum under external force, energy dissipation and none linear condition respectively, this time we will add these three factors togather, that is the so called "physical pendulum". The physical pendulum share some commonn properties with the simple pendulum, but it has its own amazing properties, among which the chaos phenomena is the most important.

Main

The motion of a physical pendulum under different kinds of propelling force
Considering the external force, the energy dissipation and the none linear condition, the kinetic equation of the physical pendulum can be written as follows:

$$\frac{d^2\theta}{dt^2}=\frac{g}{l}\sin(\theta)-q\frac{d\theta}{dt}+F_{ex}\sin(\omega_{ex}t)$$
Set the the acceleration of gravity to $9.8m/s^2$, damping coefficient $q=0.5$ ,the frequency of the external force $f_{ex}=\frac{2}{3}$, draw the $\theta-t$ diagram of three kinds of the external force with amplitude 0.2, 0.4, 1.0 respectively.

We can see when the amplitude of the external force is relatively small, 0.2 and 0.4 for example, the motion of the pendulum is pretty much the same as the simple pendulum's motion under damping and external force. However, when the amplitude grows to 1.0, the periodic motion disappear, instead the motion becomes very complex and unpredicable. This is one of the feature of chaos.
The picture below shows the $\omega-t$ relation, which the variation tendency is similar to the $\theta-t$ diagram.

The relation between angular velocity and angle displacement
Draw the picture as below

This is the none chaos situation, we can see there relation math the properties of simple harmonic vibration.(each angle has two angle velocity corresponding with same magnitude)

This picture shows the situation when chaos occurs. It is much more complicate then the last one. But the diagram is not completely random, there is still some kind of rule in it, which can often be seen in chaos system.

Problem 3.12 The strange attractor's changing tendency of different phases of external force
Let's pick out the points of the last diagram where the corresponding time is $k$ ($k$ is positive integer) times of the period of the external force to form another diagram.

Phase $\varphi_{ex}=0$

It is called the Poincare diagram, the diagram shows the fractal structure and is called the strange attractor, which is one of the main feature of the chaos phenomena, for only one point will appear in the diagram in none chaos situation.

Then we set different phase of the external force to see the changing tendency of the strange attractor:

Phase $\varphi_{ex}=\frac{\pi}{4}$

Phase $\varphi_{ex}=\frac{\pi}{2}$

We can see from the two pictures that as the phase change from $0$ to $\frac{\pi}{4}$ then to $\frac{\pi}{2}$ the strange attractor goes right-up, then to right-down. This indicates that the strange attractor moves with the change of phase.

Problem 3.16 the change of strange attractor of under small changes of parameters of external force

Fix the frequency, change the ampilitude, we get:

We cann see that when amplitude grows larger, the position of strange attractor doesn't change, but the points becomes fewer, means the system is gradually getting out of chaos status.

Fix the amplitude, change the frequency of the external force, we get:

When the frequency slightly increases, the position of the strange attractor still remain the same, but the points becomes more intensive, implies that its attracting ability is decreasing.
see codes here(all codes are in the same folder)

Conclusion

The behavior of chaos system is very complex and even impossible to predict theoritically. But using Euler-Cromer method, we can analyze the chaos phenomena numerically. And the analysis shows that the parameters and initial conditions have large influences on status of chaos.

Reference

Computational Physics-Nicholas J. Giordano, Hisao Nakanishi