Temperature Synchronization, Phase Dynamics and Oscillation Death in a Ring of Thermally-coupled Rooms

Synchronization of coupled, self-excited oscillators in complex systems is a common occurance. This report examines the effects of thermal coupling through the walls of a building on temperature oscillations due to hysteretic thermostatic control. The specific case of three rooms is studied. A system of differential equations models the dynamics of each room temperature, accounting for on-off heating, heat loss to the environment, and heat exchange between rooms. Three types of solutions are observed: one in which all room temperatures oscillate in phase, another with the oscillations equidistant in phase, and a third that is time-independent. The existence and linear stability of each solution type is investigated as a function of a parameter k that represents the thermal interaction between neighboring rooms. The in-phase behavior exists and is linearly stable for all k, the out-of-phase oscillations exist in a band of k and are stable in a smaller band, and the time-independent solution exists above a certain k where they are stable. NOMENCLATURE k nondimensional thermal coupling parameter between rooms Qi nondimensional heat input in room i QT = Q1 +Q2 +Q3 t nondimensional time ts nondimensional switching time Ti nondimensional temperature in room i ∗Address all correspondence to this author. T∞ nondimensional exterior temperature = 0 T 0 i nondimensional initial temperature in room i TT = T1 +T2 +T3 Tmax nondimensional maximum temperature Tmin nondimensional minimum temperature εi perturbation to temperature of room i ρ spectral radius τ nondimensional period of oscillation