Behaviour of a Nonlinear Convective Oscillator as Modified by Sub- and Super-Critically Unstable Hopf Bifurcations

Abstract

A connected-vessel system, subjected to a steady thermal forcing which is slightly asymmetrical around the temperature of maximum density for water, will be examined. Criteria for self-sustained oscillatory behaviour of the temperatures in the two containers are established and the resulting oscillation is investigated, with particular emphasis on its parameter-dependent transitions to a stationary solution. The oscillation proves to be somewhat noteworthy in being fundamentally nonlinear, since it may be formally demonstrated that no periodic behaviour is conceivable without the cubic nonlinearities which occur in the governing equations. Hence the system under consideration here does not represent a nonlinear modification of an essentially linear oscillator as do e.g.. the Van der Pol and Duffing equations, however dominant an influence the nonlinear terms may exert on the behaviour of their solutions