Synchronized double-frequency oscillations in a class of weakly resonant systems (Q1881335)

The following parameterized nonlinear ODE is considered \[ L\left({d\over dt} ,\lambda\right)x= M\left({d\over dt} ,\lambda\right)f(x,\lambda), \] where \(L(p,\lambda)=p^\ell+a_{1}(\lambda)p^{\ell-1}+\ldots+a_{\ell}(\lambda)\) and \(M(p,\lambda)=b_0(\lambda)p^m+\ldots+b_m(\lambda)\) are coprime polynomials, \(\ell> m\), and \(f\) is continuous and sublinear at \(x=0\). This equation describes the dynamics of a particular single-loop control system with the sublinear feedback \(f\). The authors discuss, in the situation of a weak Hopf resonance, the existence of solutions of the form \[ x(t)=r\sin(wmt)+r\rho^*\sin(wmt+\varphi^*)+o(r), \] where \(r\), \(\rho^*\), \(w\), \(\varphi^*\) are appropriate constants and \(m\) and \(n\) are integers. This is equivalent to the study of small approximately ``double period'' oscillations of the form \[ x(t)=r_1\sin(wmt)+r_2\rho\sin(wmt+\varphi). \] In particular, it turns out that such oscillations often exist if the main homogeneous part of the nonlinearity is not a positive integer power of \(x\).