Asymptotic formulas for Melnikov integrals with application to a sliding toggle block (Q5934264)

Here, the author considers the Melnikov integral for differential equations of the form NEWLINE\[NEWLINE \ddot x =f(x)-\mu_2 \dot x +\mu_1 \sin \omega t, \quad x\in \mathbb{R}. \tag{*}NEWLINE\]NEWLINE Suppose (i) \(f \in C^{n+2}(\mathbb{R},\mathbb{R})\); (ii) \(f(0)=0\), \(f'(0)=(2/n)^2\), \(f^{(k)}(0)=0\) for \(2\leq k \leq n\), \(f^{(n+1)}(0)<0\); (iii) \(V(0)=V(1)=0\), \(V(x)<0\) for \(0<x<1\), \(V'(1)>0\), with \(V(x)=-\int^x_0 f(s) ds\). The author shows the existence of a homoclinic solution to (*), which can be found implicitly. Asymptotic formulas for Melnikov integrals are established. An application to a sliding toggle block is detailed.