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Physics & Astronomy

Materials Simulation Group

Matlab and Chaos: Solving Diff. Eqs.

Matlab has some nice built-in functions for solving differential equations numerically and can do animations quite easily, so it's a handy way to explore chaotic systems (that can be represented by non-linear differential equations) without doing a whole lot of programming.

Here are some examples of using Matlab built-in "integrators".

  1. decay.m A simple "decay" model; first-order, linear ordinary differential equation.
  2. sho.m & sho_2.m Just a simple harmonic oscillator. (The second one has adjustable parameters for the spring constant and the mass.)
    The physical system modeled here is a mass attached to an ideal spring which oscillates back and forth along the x-axis without friction.
  3. pendulum.m A simple pendulum. Nearly the same as above except the restoring force is propotional to the sine of the displacement rather than just the displacement. This makes the system non-linear (but not chaotic...yet)
  4. ddp.m A damped, driven pendulum. A pendulum where friction has been added (as a velocity-dependent term) and a periodic driving force keeps adding energy to the system. For some parameter ranges this system is chaotic. This one takes ~30 seconds to make the second plot (a Poincare plot--try zooming in!).
  5. plr.m A simple sytem described as a "piece-wise, linear, Roessler-like oscillator". It's only non-linear because of the piecewise-defined driving term in the third equation---yet, this is sufficient to make it chaotic. The three coupled equations model a simple, chaoticc electric circuit. See PRL 96 244102 (2006).
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