To reveal the physical mechanism of the thermoacoustic oscillation is an important task of basic research on thermoacoustics. The startup process of a thermoacoustic oscillation is a parametric excitation process. The so-called "parameter exciting" refers to that the oscillation process is realized by the periodic change of parameter in the system. It is generated by the capacitance change in a real thermoacoustic system. In this paper, the thermoacoustic system is simulated as a nonlinear fluid network. It consists of a variable capacitor, an inductor, a nonlinear resistor and a pumping source. It is in line with an equation of nonlinear parametric vibration. The equation is deduced by following mass, momentum, energy and state equations of the working gas parcel simultaneously. The solution of the equation of nonlinear parametric vibration has been obtained using "averaging method". The analysis shows that the inductance capacitance resonance for the thermoacoustic system is one of the parameter oscillations with a limit cycle. The condition of a homeostatic periodic motion for the thermoacoustic system has been discussed. The stability of the thermoacoustic oscillation described by the limit cycle has been analyzed using "Lyapunov function method". This theory gives a better understanding of the mechanism of thermoacoustic oscillation. (C) 2014 Elsevier Ltd. All rights reserved.