Synchronized double-frequency oscillations in a class of weakly resonant systems. (English) Zbl 1062.34046

Krasnosel’skii, A. M.; McInerney, J.; Pokrovskii, A. V.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 57, No. 7-8, 929-949 (2004).

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 \(x\).

Reviewer: Marco Spadini (Firenze)

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

Keywords:

weak resonance; Hopf bifurcation; dynamical control systems

References:

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