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A mathematical model describing the initial stage of the capture intoautoresonance for nonlinear oscillating systems with combined parametric andexternal excitation is considered. The solutions with unboundedly growingamplitude and limited phase mismatch correspond to the autoresonant capture.The paper investigates the existence, stability and bifurcations of suchsolutions in the presence of a weak dissipation in the system. Our technique isbased on the study of particular solutions with power-law asymptotics atinfinity and the construction of suitable Lyapunov functions.

Autoresonance in oscillating systems with combined excitation and weak dissipation