Theoretical analysis reveals the existence of a global attractor in infinite lattice dynamical systems coupled at an interface, highlighting asymptotic stability across incomplete state spaces.
A coupled lattice dynamical system generated by two infinite lattice differential equations meeting asymptotically at an interface is studied. This can be interpreted as a special case of dynamics on graphs with the sum of inputs and outputs at each node satisfying the Kirchhoff balance law. First a suitable infinite sequence state space for the model is constructed. Then the existence and uniqueness of solutions and the generation of a dynamical system by the coupled equations is established. Finally, the existence of a global attractor of the coupled dynamical system is discussed. The analysis of the attractor is complicated by the fact that the attraction is in a weaker topology than that defining the state space of the coupled dynamics under which the state space is incomplete.
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Cui et al. (2026) studied this question.
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