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For example, suppose the energy E of a one-dimensional system is given by [itex]E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4[/itex], where ε is a positive constant. Now, I can easily show that all phase trajectories, regardless of energy, are closed by just plotting the various trajectories. But how do I prove it?

I can show that for any (positive) value of E, there's a value of x such that dx/dt is zero. But does this *prove* that the phase trajectories are closed? If it does, I don't see how.