Modification of the adiabatic invariants method in the studies of resonant dissipative systems

We study the system of equations for the canonically conjugate variables p and q specified by the one-dimensional Hamiltonian H=H(p,q,Λ₁, ...,Λ_N) dependent on Nself-consistent slightly changing parameters obeying the equations: Λ_n=f_n(Λ ₁,...,Λ_N,p,q). A broad range of oscillatory and wave processes with weak dissipation is described by analogous systems. The general method of adiabatic invariant construction for this system is proposed. Self-consistent averaged equations for the evolution of the action integral and the parameters Λ_n are obtained. The constructed theory is applied to a generalized model of the nonlinear resonance. The autoresonance (phase locking) regime of decay parametric instability in a dissipative medium is revealed.