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### Chaos and Quasiperiodicity

Some projects on chaos and quasiperiodicity from my second and third year applied mathematics courses in 2003 and 2004.

### The Lorentz Equations

The Lorentz equations are a standard example of a system exhibiting chaos and strange attractors. Here we investigate through numerical simulation the behaviour of the equations as the parameter \$r\$ is varied and find several qualitatively different behaviours, including stable foci followed by homoclinic bifurcation, limit cycles followed by Hopf bifurcation, chaos and strange attractors and regions of quasiperiodic orbits.

### The AC-Driven Pendulum

This paper explores the behaviour of the AC-driven pendulum by the method of multiple timescales for weak driving and weak or very weak damping, where the period settles slowly to that of the driver. It's behaviour for stronger damping and driving is modelled numerically showing quasiperiodic, rotation and chaotic regimes. Quasiperiodic waves of the Nonlinear Schrodinger Equation are studied in a two-parameter form, exploring the relation between the orbits and parameters and locating a periodic trajectory.

### The Van der Pol equation

We investigate the limit cycle of the Van der Pol equation and calculate its crawling and relaxation times, finding the ratio to be approximately the damping coefficient. We then study the damped pendulum driven by constant torque, which also describe a Josephson junction. The stable node is found using trajectories in its basin of attraction, and we study the hysteresis effect by following how the average angular velocity of the rotation changes when the driving torque is gradually reduced, eventually causing the trajectory to fall into the stable node. The overdamped driven pendulum is also simulated for comparison, and the effect of increasing damping is evaluated.

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The Lorentz Equations  1277k v. 1 18 Mar 2010, 13:26 Graham Poulter
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The AC-driven Pendulum  559k v. 1 18 Mar 2010, 13:21 Graham Poulter
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The Van der Pol equation  443k v. 2 18 Mar 2010, 13:27 Graham Poulter
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