Abstract: In order to find the essential features of magnetohydrodynamics, this paper investigates the dynamical behavior and its synchronization of the earth's magnetic field system. We study the stability of stationary solutions to the earth's magnetic field system by using the linear stability analysis method. The chaotic behavior of the system is simulated, the simulation results show that the system can reach the chaos through the subcritical Hopf bifurcation, and there exists the periodic window in the chaotic region, a quasi-periodic regime could be formed at the high values of the parameter. The globally attractive set and positively invariant set of the earth's magnetic field system are discussed via constructing a generalized Lyapnov function. Further, an adaptive control approach is proposed to the synchronization between two chaotic systems. Numerical simulations show the effectiveness of the method.

Key words: earth's magnetic field system, global attractive set, positively invariant set, global exponential synchronization