Normal form and stability of periodic systems with internal resonance: PMM vol. 40, n≗ 3, 1976, pp. 431–438

Abstract

We examine the problem of the Liapunov stability of the zero solution of a multidimensional nonlinear differential equation system with periodic coefficients and holomorphic right-hand sides. We investigate the critical case when the characteristic equation of the linearized system has only complex-conjugate roots equal to unity in modulus and when definite integral relations (internal resonance) exist between the characteristic indices and the frequency of the unperturbed motion. For any type of resonance we give a normal form of the system by reducing the original problem to the problem of stability under internal resonance for autonomous systems, considered earlier in [1] ; this enables us to obtain necessary and sufficient conditions for the stability of a model system, i.e. For the most important cases of odd-order resonance. We show that in the majority of cases the instability of the model system implies the instability of the complete system. We show that an even-order resonance can lead to the asymptotic stability of the system. For a second-order system we give necessary and sufficient conditions for asymptotic stability with respect to the first nonlinear term in the normal form. We indicate the extension of the results obtained to the stability of Hamiltonian systems.