On verification of limit cycle stability in autonomous nonlinear systems

A method for verification of limit-cycle stability in autonomous nonlinear systems is proposed. The method is applicable to systems with limit cycles described by a sinusoid (main oscillation) with small addition of harmonics. The main oscillation is represented in exponential form and is substituted into the nonlinear part of the initial differential equation. The nonlinear part is linearized with respect to the amplitude perturbations and the operator equation for the perturbations is obtained. Then the terms representing derivatives higher than first order are omitted in the corresponding operators and the real and imaginary parts of the simplified operator equation are separated. This results in two first-order linear differential equations for the increments of the main oscillation amplitudes. The differential equations are used for verification of the limit-cycle stability. The case of asynchronous perturbation is considered as well