Dynamical behavior in the equation of Josephson junction type (Q923806)

The authors study the dynamical behavior in the equation \(\ddot x+(1+f'(x))\dot x+f(x)=\rho +bg(t),\) where f(x) and g(t) satisfy the following conditions: \(f,g\in C^ 2\); \(f(-x)=-f(x)\), \(f(x+2\pi)=f(x)\), \(f(\pi -x)=f(x)\), \(f(\pi +x)=-f(x)\), \(| f(x)| \leq 1\), \(| f'(x)| \leq 1\), \(f'(x)\geq 0\), \(x\in [0,\pi /2]\); \(g(-t)=-g(t)\), \(g(t+T)=g(t)\), \(g(T/2-t)=g(t)\), \(g((T/2)+t)=-g(t),| g(t)| \leq 1\), \(| g'(t)| \leq 1\), \(g'(t)\geq 0\), \(t\in [0,T/4],T>0\); \(\rho\geq 0\), \(b\geq 0\), \(\epsilon <1\). For the case \(\beta >2/(1+\epsilon)\), the problem is completely solved by reduction to a circle mapping. The conclusion is that no chaotic motion takes place, and every trajectory is asymptotically periodic or quasi-periodic.