Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations

Holmes, P. J., J. E. Marsden and J. Scheurle

Contemp. Math., 81, (1988), 213-244

Abstract:

Both upper and lower estimates are established for the separatrix splitting of rapidly forced systems with a homoclinic orbit. The general theory is applied to the equation

$$\ddot{{\phi}}$$ + sin$$\phi$$ = $$\delta$$sin$$\bigl($$t $$\varepsilon^{{-1}}_{}$$$$\bigr)$$

for illustration. There are two types of results. First, fix $\eta$ > 0 and let 0 < $\varepsilon$ $\leq$ 1 and 0 $\leq$ $\delta$ $\leq$ $\delta_{{0}}^{{}}$ where $\delta_{0}^{}$ is sufficiently small. If the separatrices split, they do so by an amount that is no more than

C$$\delta$$exp$$\bigl($$ - $$\varepsilon^{{-1}}_{}$$($${\textstyle\tfrac{1}{2}}$$$$\pi$$ - $$\eta$$)$$\bigr)$$

where C = C($\delta_{0}^{}$) is a constant depending on $\delta_{0}^{}$ but is uniform in $\varepsilon$ and $\delta$ . Second, if we replace $\delta$ by $\varepsilon^{p}_{}$$\delta$ , with p $\geq$ 8 , then we have the sharper estimate

C₂ $$\varepsilon^{p}_{}$$$$\delta$$ e^{-$\pi$$\varepsilon$/2} $$\leq$$ splitting distance $$\leq$$ C₁ $$\varepsilon^{p}_{}$$$$\delta$$ e^{-$\pi$$\varepsilon$/2}

for positive constants C₁ and C₂ depending on $\delta_{0}^{}$ alone. In particular, in this second case, the Melnikov criterion correctly predicts exponentially small splitting and transversal intersection of the separatrices. After developing this theory we discuss some of its applications, concentrating on a 2 : 1 resonance that occurs in a KAM (Kolmogorov, Arnold, and Moser) situation and in the forced saddle node bifurcation described by

$$\ddot{{x}}$$ + $$\mu$$x + x² + x³ = $$\delta$$ f (t) .