6 research outputs found

    Collective behaviors of the Lohe hermitian sphere model with inertia

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    We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength

    Towards Almost Global Synchronization on the Stiefel Manifold

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    A graph G\mathcal{G} is referred to as S1\mathsf{S}^1-synchronizing if, roughly speaking, the Kuramoto-like model whose interaction topology is given by G\mathcal{G} synchronizes almost globally. The Kuramoto model evolves on the unit circle, \ie the 11-sphere S1\mathsf{S}^1. This paper concerns generalizations of the Kuramoto-like model and the concept of synchronizing graphs on the Stiefel manifold St(p,n)\mathsf{St}(p,n). Previous work on state-space oscillators have largely been influenced by results and techniques that pertain to the S1\mathsf{S}^1-case. It has recently been shown that all connected graphs are Sn\mathsf{S}^n-synchronizing for all n2n\geq2. The previous point of departure may thus have been overly conservative. The nn-sphere is a special case of the Stiefel manifold, namely St(1,n+1)\mathsf{St}(1,n+1). As such, it is natural to ask for the extent to which the results on Sn\mathsf{S}^{n} can be extended to the Stiefel manifold. This paper shows that all connected graphs are St(p,n)\mathsf{St}(p,n)-synchronizing provided the pair (p,n)(p,n) satisfies p2n31p\leq \tfrac{2n}{3}-1
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