Simplex analysis of dynamical systems with a perturbation. New criteria of stabilization of a homoclinic separatrix and its use - Mitropolsky , Yu. O.; Antonishin , I. O.; Prikarpatsky , A. K.; Samoilenko , V. G. (Ukrains’kyi Matematychnyi Zhurnal, 1992)
We study adiabatic invariants of nonlinear dynamical systems of autonomous and nonautonomous type on symplectic manifolds, and, in particular, give criteria for the existence of such invariants and methods for constructing them explicitly. We make a thorough analysis of the phenomena of splitting of a homoclinic separatrix from a hyperbolic singular point, and propose a new exact method for constructing an analogue of Melʹnikov's function, which gives a necessary and sufficient condition for transversality, which generalizes automatically to dynamical systems in Rn, n ∈ Z+.