# zbMATH — the first resource for mathematics

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
##### Keywords:
weak resonance; Hopf bifurcation; dynamical control systems
Full Text:
##### References:
 [1] Braverman, M.E.; Rozonoer, L.I.; Braverman, M.E.; Rozonoer, L.I., Robustness of linear dynamical systems, parts I and II, Autom. remote control, Autom. remote control, 53, 1 Part 1, 34-43, (1992) · Zbl 0805.93023 [2] F.N. Grigoriev, N.A. Kuznetsov, Optimal control in one nonlinear problem, Proceedings of International Conference on Control, Institute for Control Science, Moscow, Vol. 1, 1999, pp. 124-126. [3] J. Guckenheimer, Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, Vol. 42, Springer, New York, 1990. [4] Knobloch, E.; Proctor, M.R.E., The double Hopf bifurcation with 2:1 resonance, Proc. R. soc. London A, 415, 61-90, (1988) · Zbl 0646.34042 [5] Kozjakin, V.S., Subfurcation of periodic oscillations, Soviet math dokl., 18, 1, 18-21, (1977) · Zbl 0371.34023 [6] Krasnosel’skii, A.M.; Chernorutskii, V.V., Hopf bifurcations generated by small nonlinear terms, Autom. remote control, 60, 9 Part 1, 1237-1242, (1999) · Zbl 1059.34510 [7] Krasnosel’skiı̆, A.M.; Krasnosel’skiı̆, M.A., Large-amplitude cycles in autonomous systems with hysteresis, Soviet math. dokl., 32, 1, 14-17, (1985) · Zbl 0602.34024 [8] Krasnosel’skii, A.M.; Kuznetsov, N.A.; Rachinskii, D.I., On resonant equations with unbounded nonlinearities, Doklady mathematics, 62, 1, 44-48, (2000) [9] Krasnosel’skii, A.M.; Rachinskii, D.I.; Schneider, K., Hopf bifurcations in resonance 2:1, Nonlinear analysis. theory, methods & applications, 52, 3, 943-960, (2003) · Zbl 1035.34027 [10] Krasnosel’skiı̆, M.A.; Kozjakin, V.S., The method of parameter functionalization in the problem of bifurcation points, Soviet math. dokl., 22, 2, 513-517, (1980) · Zbl 0471.34031 [11] Krasnosel’skii, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer Berlin, Heidelberg, New York, Tokyo [12] Maier, A.G., Robust transformation of a circle to itself, Research notes gorky univ., 12, 215-229, (1939), (in Russian) [13] Marsden, J.; McCracken, M., Hopf bifurcation and its applications, (1982), Springer New York [14] Pokrovskii, A.; Holland, F.; Suzuki, M.; Suzuki, T.; McInerney, J., Robustness of an analog dynamic memory system to a class of information transmission channels perturbations, Funct. differential equations, 6, 3-4, 411-438, (1999) · Zbl 1030.94052 [15] Vidyasagar, M., Nonlinear system analysis, (1993), Prentice-Hall Englewood Cliffs, NJ · Zbl 0900.93132
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.