1.Abstract
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EXERCISES
3.13. Write a program to calculate and compare the behavior of two, nearly identical pendulums. Use it to calculate the divergence of two nearby trajectories in the chaotic regime, as in Figure 3.7, and make a qualitative estimate of the corresponding Lyapunov exponent from the slope of a plot of log(Δθ) as a function of t.
3.14. Repeat the previous problem, but give the two pendulums slightly different damping factors. How does the value of the Lyapunov exponent compare with that found in Figure 3.7?
2.Background
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Chaos Theory
Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
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Euler-Cromer Method
In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method.
3.Main
- (As usual,In SI, angles in radians)
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Formulation
In fact, I think there is no necessity to spend extra effort on solving the pendulum problem, because we have already studied it thoroughly from a simple pendulum to pendulum adding dissipation and a driving force. So I will only show the final formula without deriving:
What is new to us is adding the nonlinearity, which transforms the formula into:
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Algorithm
As the authour of our textbook urges, I'd like to use the Euler-Cormer Method to solve this problem (The Pseudocode for subroutine with it is given.)
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Thinking
In the course of computational physics, when practicing,I will try my best to insist on two princples of programming in my mind:
Simplicity
Generality
So in this program I spare some time to make it possible for users to input initial conditions. (It is easier to input one by one, but more difficult to input altogether in an array.)-
Results
Firstly, I will show all the figures that have emerged in Section 3.1, 3.2, 3.3 to review what I have learned in class.
Secondly, calculate and compare the behavior of two nearly identical pendulums. In detail, calculate the divergence of two nearby tragectories in chaotic regime.
The Figures are as follows:
**☛If you want to enjoy the codes, please click here: **
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1. Given different initial angles
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2.Given different damping factors
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Pity no curve fitting on them!
4. Conclusion
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To be on the safe side, I solved problem 3.12:
☛Code is the included in the previous file but relevant part is annotated for simplicity
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Quantative solution: Lyapunov exponent (to be continued...)
5. Acknowlegement
- Prof. Cai
- Song Gaygay (Song Fengfeng)
- Wikipedia
