 Energy balance for the weakly perturbed double-well oscillator

ID: energy-balance-for-the-weakly-perturbed-double-well-oscillator

For

H = \frac{1}{2} y^{2} - \frac{1}{2} x^{2} + \frac{1}{4} x^{4},

the perturbation gives

\dot{H} = ε (1 - α x^{2}) y^{2}

. Along an unperturbed orbit

H = H_{0}

x_{1}, x_{2}

Δ H = 2 ε \int_{x_{1}}^{x_{2}} (1 - α x^{2}) 2 H_{0} + x^{2} - \frac{1}{2} x^{4} d x + O (ε^{2}) .

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