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Simple harmonic motion is the solution to the differential equation:

$$\frac{\partial^2}{\partial t^2} x = − \omega^2 x$$

Interested in chaos I wanted to make the angular frequency $\omega$ be cross-coupled in a pair of oscillators, and I came up with this differential equation:

$$ \frac{\partial^2}{\partial t^2} x = − e^{y^2} x \\ \frac{\partial^2}{\partial t^2} y = − e^{x^2} y \\ $$

with initial conditions:

$$ \begin{aligned} x_0 &= 0 & y_0 &= 1 \\ \frac{\partial}{\partial t} x_0 &= 1 & \frac{\partial}{\partial t} y_0 &= 1 \end{aligned} $$

First I experimented with Runge-Kutta 4th/5th order integration (GNU Octave's ode45(), ported to C for speed), but found it exploded after some time with ever-increasing frequencies. Then I tried Velocity Verlet integration, which gave a stable result.

Changing the initial conditions to ones with higher energy (approximated by the energy computations for simple harmonic motion) resulted in explosions even with the Velocity Verlet method.

So my questions are:

  • Are the explosions (or stability) an artifact of numeric integration techniques or a true behaviour of the differential equations at the given initial conditions?
  • Is the explosivity of the equations dependant on the initial conditions?
  • Is there a simple function of the initial conditions (for example, an energy threshold) that reveals whether an explosion is inevitable or the system is stable?
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    You would get energy conservation if you started with the coupled potential $\frac12e^{x^2+y^2}$ resulting in $\ddot x+e^{x^2+y^2}x=0$ and $\ddot y+e^{x^2+y^2}y=0$. It is not obvious if or if not there exists a potential or Hamiltonian, or even a Lyapunov function, that is compatible with your system. – 2017-02-09
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    @LutzL thanks, yes your version seems stable across many initial conditions, but it misses the chaotic wandering behaviour of my original equation – 2017-02-10
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    Yes, that was to be expected as the potential and thus the Hamiltionian is rotation invariant, I think by the Noether theorems this results in the preservation of angular momentum. Break that invariance, for instance using $V(x,y)=\frac12e^{x^2+3y^2+xy}$ to hopefully get a more interesting system. – 2017-02-10
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    @LutzL thanks again - do you want to write your comments up as an answer? I ended up using $V = \frac{1}{2}e^{x^2+2y^2}$ and have a system that exhibits chaotic behaviour at some energies, and also seems stable (though I get explosions at large energies, I suspect this is due to too-large fixed step size in the Velocity Verlet integrator). I don't use an $e^{xy}$ term, because it might reach a fixed point. – 2017-02-13

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