Energy of a non-linearly damped oscillator

[Oerg]

I was reading Strogatz's book on nonlinear dynamics and chaos and in Example 7.2.2, he stated the energy function of the nonlinear oscillator

$$\ddot{x} + (\dot{x})^3 + x = 0$$


as

$$E(x, \dot{x}) = \frac{1}{2} (x^2 + \dot{x}^2)$$

But isn't this the energy function for the harmonic oscillator $$\ddot{x} + x = 0$$ since $$\int x \, dx = \frac{1}{2} x^2$$?

 

[haruspex]

True, but how does that prevent its being the energy function for the damped version of the equation? It still represents the energy remaining in the system, but in the damped case it is not conserved.