# 4.E: Nonlinear Systems and Chaos (Exercises)

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1. Consider the chaotic motion of the driven damped pendulum whose equation of motion is given by

${\small \ }\ddot{\phi}+\Gamma \dot{\phi}+\omega _{0}^{2}\sin \phi =\gamma \omega _{0}^{2}\cos \omega t \nonumber$

for which the Lyapunov exponent is $$\lambda =1$$ with time measured in units of the drive period.

1. Assume that you need to predict $$\phi \left( t\right)$$ with accuracy of $$\ 10^{-2}$$ $$radians$$, and that the initial value $$\phi \left( 0\right)$$ is known to within $$10^{-6}$$ $$radians$$ . What is the maximum time horizon $$t_{\max }$$ for which you can predict $$\phi \left( t\right)$$ to within the required accuracy?
2. Suppose that you manage to improve the accuracy of the initial value to $$10^{-9}$$ $$radians$$ (that is, a thousand-fold improvement). What is the time horizon now for achieving the accuracy of $$10^{-2}$$ $$radians$$?
3. By what factor has $$t_{\max }$$ improved with the $$1000-fold$$ improvement in initial measurement.
4. What does this imply regarding long-term predictions of chaotic motion?
2. A non-linear oscillator satisfies the equation $$\ddot{x}+\dot{ x}^{3}+x=0.$$ Find the polar equations for the motion in the state-space diagram. Show that any trajectory that starts within the circle $$r<1$$ encircle the origin infinitely many times in the clockwise direction. Show further that these trajectories in state space terminate at the origin.
3. Consider the system of a mass suspended between two identical springs as shown.
If each spring is stretched a distance $$d$$ to attach the mass at the equilibrium position the mass is subject to two equal and oppositely directed forces of magnitude $$\kappa d$$. Ignore gravity. Show that the potential in which the mass moves is approximately

$U(x)=\left\{ \frac{\kappa d}{l}\right\} x^{2}+\left\{ \frac{\kappa (l-d)}{ 4l^{3}}\right\} x^{4}$

Construct a state-space diagram for this potential.

4. A non-linear oscillator satisfies the equation

$\ddot{x} + (x^{2}+\dot{x}^{2}-1) \dot{x} + x = 0 \nonumber$

Find the polar equations for the motion in the state-space diagram. Show that any trajectory that starts in the domain $$1<r<\sqrt{3}$$ spirals clockwise and tends to the limit cycle $$r=1$$. [The same is true of trajectories that start in the domain $$0<r<1$$. ] What is the period of the limit cycle?

5. A mass $$m$$ moves in one direction and is subject to a constant force $$+F_{0}$$ when $$x<0$$ and to a constant force $$-F_{0}$$ when $$x>0$$. Describe the motion by constructing a state space diagram. Calculate the period of the motion in terms of $$m,F_{0}$$ and the amplitude $$A$$. Disregard damping.
6. Investigate the motion of an undamped mass subject to a force of the form $F(x)= ( \begin{array}{c} -kx\hspace{1.2in}\left\vert x\right\vert <a \\ -(k+\delta )x+\delta a\hspace{0.55in}\left\vert x\right\vert >a \end{array} \nonumber$

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