Exercise 7: Chaos in Driven Nonelinear Pendulum

python homework chaos

Abstruct

The nonlinear damped, driven pendulum exhibits chaotic behavior for certain values of the parameters in its equation of motion. The behavior of the nonlinear pendulum in phase space is examined for specific values of these parameters. The characteristic of the system Poincare sections is examined by Euler-cromer method.

Introduction and Background

The nonlinear pendulum is one example of a very simple system that can demonstrate chaotic behavior. Chaotic oscillations are of interest in many fields, including mechanical engineering, and the case of the nonlinear damped, driven oscillator may serve as a basis for mathematical models for other systems. Thus, it is interesting to explore the behavior of the nonlinear pendulum, first because it is an example of a very simple system that can demonstrate chaotic behavior, and secondly because it is mathematically
similar to many other problems involving vibrations.

A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.

Content

• Problem 3.12
  In constructing the Poincare section in Figure3.9 we plotted points only at times that were in phase with the drive force;that is at times $t≈2πn/\Omega_D$,where $n$ is an integer.At these values of the driving force passed through zero.However,we could just as easily have chosen to make the plot at times corresponding to a maximum of the drive force,or at times $π/4$ out-of-phase with this force, etc. Construct the Poincare sections for these cases and compare them with Figure3.9.

• The Equation of Motion

  Considering the driving force an damping, we have the equation of motion:
  

  $$\frac{d^2\theta}{dt^2}=-\frac{g}{l}sin\theta-q\frac{d\theta}{dt}+F_Dsin(\Omega_Dt)\tag{1}$$
  The paramaters for the calculation are $q=\frac{1}{2}$, $l=g=9.8$,$\Omega_D=\frac{2}{3}$.

• Numerial approach
  Use Euler-cromer method we can rewrite the equation:
  

  $$\omega_{i+1}=\omega_i-[(g/l)sin\theta_i-q\omega_i+F_Dsin(\Omega_Dt_i)]△t\tag{2}$$
  $$\theta_{i+1}=\theta{i}+\omega_{i+1}△t\tag{3}$$
  $$t_{i+1}=t_i+△t\tag{4}$$
  If $\theta_{i+1}$is out of the range $[-π,π]$, add or subtract $2π$ to keep it in this range.

• Program and Results
  Click here for codes
  Figure 1:Results for $\omega$ as a function of $\theta$ for a pendulum(Poincare sections), at times $t≈2πn/\Omega_D$.
  

Figure 2:At times corresponding to a maximum of the drive force, that means $t≈(2n+1/2)π/\Omega_D$.

Figure 3:At times $π/4$ out-of-phase with drive force.

Conclusions

From the figures we can see that at defferent times, the Poincare sections are different.For the lower $F_D$( $F_D$=0.5), the relation cetween $\omega$ and $\theta$ is approximately identified and stable. When $F_D$ is higher, this relation becomes unstable and the Chaotic phenomenon is obvious.

References

• Baker, G.L. and Gollub, J.P. Chaotic Dynamics: An Introduction. 2nd
  ed. New York: Cambridge University Press, 1996.

• Szemplinska-Stupnicka, W. and Troger, H. eds. Engineering
  Applications and Dynamics of Chaos. New York: Springer-Verlag, 1991

• Chaos theory-Wikipedia

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