• 作业L1 3.12
• 作业L2 3.16 3.21

Homework 09 - Chaos(Strange Attractor)

ComputationalPhysics_HW 岳绍圣2013301020033

---

CONTENT:

• Homework 09 - Chaos(Strange Attractor)
  • Abstract
  • Background
  • The Driven Nonlinear Pendulum
    • 1. Oscillation Diagram
    • 2. Poincaré Section
    • 3. Bifurcation Diagram
  • Codes
  • Reference

Abstract

This article call a model for a nonlinear, damped, driven pendulum. And shows the particular properties that not exists in simple pendulum——chaos. we also changed a small amount of initial conditions, and found a exponential change afterwards. Poincare section is plotted to identify the chaos state. Also, bifurcation diagram is given to show the transistion from nonchaos to chaos.

Background

In the last homework, we analyzed properties of simple pendulum with damping and small driven force. The real fun begins when we put all three of these effects —— damping, a driving forcce, and the nonlinearity —— together at the same time.

First off, we need to figure out a seemingly impossible thing: A behavior can be deterministic and unpredicatable AT THE SAME TIME! and this behavior is exactly chaos.

Chaos phenomenon is a seemingly random irregular movement which occurs in the deterministic system, and the system is described by a deterministic theory. Its behavior is a non deterministic one. It is an inherent characteristic of nonlinear dynamical systems, and it is a common phenomenon in nonlinear systems. And in the physical pendulum case, if we increase the driven force, $F_D=1.2$ for example, we will see the chaos, that is, seemingly random oscillation.

The second concept we shall introduce is Poincaré Section. The basic idea in multi-dimensional phase space (x, DX, LDT, X2, D L x / dt L. DR / DT) choosing proper cross section, in this section for a pair of conjugate variables such as xdx, LDT take a fixed value, said this section of the Poincare section.Observation trajectory and the cross-section of the cross section (Poincare points), set them in turn as P1, P2, P3... . original phase space continuous trajectory in the Poincare section has some discrete mapping PN by them can get information about the motion characteristics. If you do not consider the initial stage of the transient process, only consider the Poincare sections of the steady state image. When the Poincare section only has a fixed point and a small number of discrete points, determine the motion is periodic; when the Poincare section is a closed curve, it can determine motion is quasi periodic; when on the Poincare section is patches of dense point and hierarchical structure, whether the motion is in a chaotic state.

The third concept we shall introduce is bifurcation diagram. Atypical bifurcation diagram on wikipedia is shown below" :Bifurcation Diagram

The Driven Nonlinear Pendulum

1. Oscillation Diagram

Putting three effects: damping, a driving force, and the nonlinearity together.we have the equation of motion:

$$\frac{{{d^2}\theta }}{{d{t^2}}} = - \frac{g}{l}\sin \theta - q\frac{{d\theta }}{{dt}} + {F_D}\sin \left( {{\Omega _D}t} \right)$$

we use Euler-Cromer method for our calculation.

Some typical results for $\theta$ and $\omega$ as functions of time, are shown in Figure 1.

　　　　　　　　　　　　　　　fig.1 Oscillation of different drive

Analysis of Fig.1:
when $F_D=0$, the motion is damped and the pendulum comes to rest after at most a few oscillations;
when $F_D=0.5$, the pendulum then moves at the driving frequency .
when $F_D=1.2$, the motion is no longer simple, even at longtimes. and it seems random.

Fig2 is a plot shows with and without resets:

　　　　　　　　　　　　　　　fig.2 With and without resets

Next we want to erase the vertical line in the fig with resets, for the vertical is in fact a brust "jump", and without them, it looks more real.

if we don't erase the vertical lines, we will see next, in Poincare section, we will see many horizontal lines which is unexpected.

2. Poincaré Section

Instead of plotting $\theta$ as a function of $t$, let us plot the angular velocity $\omega$ as a function of $\theta$. It is called Phase-space plot.

　　　　　　　　　　　　　　　fig.3 Phase-space plot

If we examine these tracjectories in a slightly different manner we find a very striking result. Here we plot $\omega$ versus $\theta$ only at times that are in phase with the driving force. that is, we plot $t$ that satisfies:
$\Omega_D t=2n\pi +\phi$, and $\phi=0$.

　　　　　　　　　　　　　　　fig.4 Poincaré Section of Fig.3

Analysis of Fig.4:

• The single point in the second subplot is the result of $t=0$ always satisfies $\Omega_D t=2n\pi$.

　　　　　　　　　　　　　　　fig.5 Amplify part of Poincaré Section

Analysis of Fig.5:

• This is an amplification of part of Poincaré Section, is there any similarity??? I don't know yet.

we can also plot when $\phi \neq 0$, and $\phi=\pi/4,\pi/2,\pi$.

　　　　　　　　　　　　　　　fig.6 Different Poincaré Sections

3. Bifurcation Diagram

Now let's turn to different values of $F_D$.

Let's firstly look the animation of the oscillations of different $F_D$.
In this case, we choose $F_D$ from 0.9 to 1.8, and the step is 0.005.

　　　　　　　　　　　　　　　fig.7 Oscillation of different F_D

Analysis of Fig.6:

• Around 1.35, it shows the same period as the driving period.
• from 0.9 to 1.35, it changes from multiple periods of driving to the same period of driving period.
• From 1.35 to 1.8, the period of oscillation increase from 1 time to 2 times to 3 times... the period of driving force.

Now we want to figure out how the transistion from nonchaos to chaos happened.
So we plot the bifurcation diagrams, which is a $\theta-F_D$ diagram.

　　　　　　　　　　　　　　　fig.7.1 Bifurcation Diagram

　　　　　　　　　　　　　　　fig.7.2 Bifurcation Diagram

Analysis of Fig.7:

• From Fig.7.1 to Fig.7.2, we erase the initial points $t=0$ which always satisfies $\Omega_D t=2n\pi$.

Which is compatible with

Codes

All these codes used can be found in my github.

Reference

[1]: Cai Hao. https://github.com/caihao/computational_physics_whu.
[2]: Nicholas J.Giordano, Hisao Nakanishi.Computational physics.清华大学出版社
[3]: YUE Shaosheng. https://www.zybuluo.com/mdeditor#352185.
[4]: Wikipedia contributors, "Bifurcation diagram," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Bifurcation_diagram&oldid=680690451.