On stability of autonomous systems with internal resonance: PMM vol. 39, n≗ 6, 1975, pp. 974–984

Abstract

When investigating the stability of the trivial solution of an autonomous system of ordinary differential equations in the critical case of n pairs of pure imaginary roots an essential role can be played by the presence of integral linear dependences between the system's frequencies or, in other words, by the internal resonance. Various special cases of this problem were examined in [1–6]. Our aims are: to obtain a special (normal) form of the differential equation system with internal resonance of most general form in it; to ascertain the conditions under which the presence of internal resonance does not permit the application stability investigation methods developed for resonance-free systems; to solve the stability problem in one of the most important cases of odd-order internal resonance, generalizing the preceding investigations. In the solution of the last problem the necessary and sufficient conditions are given for the stability of the model (simplified) system. Using Chetaev's theorem we show that as a rule the instability of the original system follows from the Instability of the model system. Cases of structurally-unstable instability ( ∗ ∗ Editor's Note. In the Russian text this is called: “noncoarse instability”. ) for which the model system does not resolve the problem of stability are outlined. The results obtained are extended, in particular, to Hamiltonian systems.