The possibility of structurally stable global oscillators occuring when dissipative forces are introduced into dynamic systems asymptotically stable in the large (Q1180873)

The authors consider a structurally stable and globally asymptotically stable dynamical system \[ \ddot x=S(x,\dot x),\quad (x,\dot x)\in R^{2n},\quad n\geq 2.\leqno(1) \] Two such systems are connected by dissipative interactive forces of the viscous fraction type into the form of the \(4n\)-th order dynamical system: \[ \ddot x_ 1=S(x_ 1,\dot x_ 1)+\mu\cdot(\dot x_ 2-\dot x_ 1),\quad \ddot x_ 2=S(x_ 2,\dot x_ 2)+\mu\cdot(\dot x_ 1-\dot x_ 2),\leqno(2) \] where \(\mu\) is a diagonal nonnegative \(n\) by \(n\) matrix. They show that there exist polynomials \(S: R^{2n}\to R^{2n}\) and matrices \(\mu\) such that (2) becomes a structurally stable global oscillator.