The Chaotic Pendulum and Its Characteristics

           张志城，２０１３３０１１１０１２１ 

Abstracts

This passage mainly discuss the behavior of a chaotic pendulum.I research the dependence of angle on time,the dependence of angular velocity on angle.Besides,I draw the attractor of a chaotic pendulum and its sensitive dependence on its parameters.At last,I present the bifurcation diagram connecting to the period doubling phenomenon.

Background

• Introduction
  Chaos is a kind of charming phenomenon that has drawn our attention for quite long a time,for which many excellent mathematicians and physicists had worked.We know that the motion of a pendulum is unpredictable,however,it is also deterministic meanwhile.
• Chaotic Pendulum
  A pendulum can behave regularly and chaoticly.Here are two gifs showing the regular pendulum and chaotic pendulum seperately.We can notice that the behavior of a chaotic pendulum is very strange and unpredictable.
  
  

Whereas,the exotic behavior of a chaotic pendulum above is dominated by an equaton:
$\frac{\mathrm{d^{2}\theta]} }{\mathrm{d} t^{2}}=-\frac{g}{l}sin\theta-q\frac{\mathrm{d\theta} }{\mathrm{d} t}+F_{D}sin(\Omega_{D}t)$
I use Euler-Cromer method solving the second-order ordinary differential equation and use Vpython to realize a 3D visual.Following is my codes.

Source Codes

codes of chaotic pendulum

RESULTS

The GIF presents the difference between a regular pendulum and a chaotic pendulum vividly.Next we will research the various characteristics of a chaotic pendulum in detail.

1.The Dependence of Theta on Time

The dependence of Theta on time varies with the change of the driving force amplitudes.

• FD=0
  

• FD=0.5
  

• FD=1.2
  

2.The Phase Space of A Chaotic Pendulum

The phase space of a regular pendulum and a chaotic pendulum show much difference.

• FD=0.5
  
• FD=1.2
  

3.The Attractor of A Pendulum And Its Dependence of Parameters

• In a phase space of a chaotic space,we could find an attractor,which exhibits a fractal structure.
  FD=1.2
  
  FD=1.2 Fractal Structure
  

• The pattern of an attractor changes sensitively with parameters.
  FD=1.257
  
  FD=1.26
  

4.Period Doubling

Period doubling is a route to chaos.When the period doubles,there will be chaos.
FD=1.35

FD=1.44

FD=1.485

5.Bifurcation Diagram

• Fractal structure is an important feature of chaos.In a pendulum system,the fractal structure is bifurcation.
  Theta-FD dependence
  

Conlusion

In this report,I realize the object to display the characteristics of a chaotic pendulum.And these characteristics agree with what they should be.I verify that :
1,The behavior of a chaotic pendulum is unpredictable,and sensitively dependent on its parameters,which differs from a stable system greatly.
2,An important feature of a chaotic system is fractal structure,which can be found in the transition to chaotic states.

References & Acknowledgements

[1] Nakanishi.Computational Physics.2008
[2]I refer to Shixing Wang's(王世兴) object-oriented programming methods and codes,which helps me a lot.
[3]Yangyao Chen(陈洋遥)　helps me to present figures on zuoyebuluo,and I appreciate his helpful nature very much.