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Regulation of axisymmetric Rayleigh-Bénard convection using boundary temperature coupling of the two circular plates

Abstract:

Controlled delay of regular, chaotic, and periodic regimes of instabilities is studied in the problem of axisymmetric Rayleigh-Bénard convection in a vertical cylinder. A feedback control is assumed at the boundaries, which leads to a coupling of the two boundary temperatures. A classical type solution is impossible in such a situation. Hence, a novel series solution procedure is adopted to arrive at the generalized Lorenz model. Due to feedback control, delayed onset of regular convection is observed and the percentage of such a delay as a function of the controller gain parameter, K , is reported. The changes in the pitchfork bifurcation point, the homoclinic orbit, and the Hopf bifurcation point due to feedback control are highlighted with the help of a bifurcation diagram. This diagram shows that the influence of feedback control is to advance the onset of homoclinic bifurcation and delay the onset of Hopf bifurcation. The results indicate that feedback control shows preference for Hopf bifurcation and is antagonistic toward homoclinic bifurcation. The shortening of the time of existence of the strange attractor intermittent with a periodic/quasi-periodic state, which is preceded by the fully periodic motion as K increases is observed using the largest-Lyapunov-exponent plot, the bar-code plot, and the bifurcation diagram. The results coming out of the Kaplan-Yorke dimension reiterates the results depicted by other indicators concerning the influence of K on chaos. The practical importance of the control strategy that is used in the paper is also mentioned in the paper.