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Synchronized double-frequency oscillations in a class of weakly resonant systems. (English) Zbl 1062.34046
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 \(x\).
MSC:
34C25 Periodic solutions to ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
93C10 Nonlinear systems in control theory
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