Math. Models Methods Appl. Sci. Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 03/29/16. For personal use only. 2 J. Cho et al.global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.
We present a practical synchronization method for the Schrödinger-Lohe (S-L) system distinct potentials. The S-L model describes the spatial-temporal evolution of the wave functions of quantum Lohe oscillators on a network with Lohe couplings. When the potential effects are ignored, complete wave function synchronization (CWFS) can emerge in the sense that the L 2 -distance between wave functions exponentially approaches zero for a class of initial wave functions. In contrast, when the Lohe oscillators are under the effect of potential forces, CWFS cannot occur. In this study, we employ a weaker concept of quantum synchronization for discussing the asymptotic collective behavior of the S-L model. This concept leads to 'practical synchronization' of the S-L model. In practical synchronization, the L 2 -distance between wave functions can be upper bounded by the inverse power of the square root of the coupling strength; as the coupling strength increases. Thus, the L 2 -discrepancy between wave functions arbitrarily decreases as the coupling strength increases. We present a sufficient analytical framework for this practical synchronization, which is a generalization of the earlier result in (
We study the emergence of bi-cluster flocking for a generalized Justh–Krishnaprasad (J–K) model and the Cucker–Smale (C–S) model with the unit speed constraint. We provide a sufficient framework for the emergence of bi-cluster flocking in terms of initial configurations and communication weights between agents. Our analytical results on bi-cluster flocking illustrate that the spatial-decay rate of communication weights is transferred to the temporal decay rate toward the bi-cluster flocking configuration. We also provide several numerical simulations and compare them to our analytical results.
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