Stabilization of an unstable equilibrium by means of a heat supply with parametric oscillations (Q2879966)

This paper is devoted to the study of a lumped parameter model of the Rijke tube. To derive the dynamical equations, the authors use balance relations for momentum, mass, and energy. A formula for the thermal resistance is obtained by assuming that the process is isobaric. By using this assumption, the dynamics is represented by a pair of nonlinear ordinary differential equations. It is shown that the equilibrium of this ODE system is unstable under linear approximation. To stabilize this equilibrium, a periodic function \(C_a(t)\) is introduced into the model. From the physical viewpoint, the above \(C_a(t)\) can be implemented by means of a piston moving in an additional volume. A discontinuous form of \(C_a(t)\) is proposed, and domains of stability are constructed.