Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.