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We study emergent dynamics of the Lohe hermitian sphere (LHS) model with thesame free flows under the dynamic interplay between state evolution andadaptive couplings. The LHS model is a complex counterpart of the Lohe sphere(LS) model on the unit sphere in Euclidean space, and when particles lie in theEuclidean unit sphere embedded in $\bbc^{d+1}$, it reduces to the Lohe spheremodel. In the absence of interactions between states and coupling gains,emergent dynamics have been addressed in [22]. In this paper, we further extendearlier results in the aforementioned work to the setting in which the stateand coupling gains are dynamically interrelated via two types of coupling laws,namely anti-Hebbian and Hebbian coupling laws. In each case, we present twosufficient frameworks leading to complete aggregation depending on the couplinglaws, when the corresponding free flow is the same for all particles.

Asymptotic interplay of states and adapted coupling gains in the Lohe hermitian sphere model