Global Dynamics to the Periodic Ferromagnetic Spin Chain System

.In this paper, we study the dynamical system driven by the ferromagnetic spin chain system defined in \( S^1\) with or without external fields. In the first part, we prove the convergence to equilibriums for arbitrary large initial data of finite energy and prove the existence of heteroclinic orbits emerging from equilibriums of nontrivial energy. Moreover, we find a topological property of these heteroclinic orbits. In the second part, we study the stabilization of these equilibriums by adding an external field. Particularly, we prove that any given finite numbers of equilibriums can be stabilized simultaneously by choosing some external field.Keywordsferromagnetic spin chaindissipative infinite dimensional systemMSC codes35K5935Q55