Abstract
Nonlinear oscillators exhibit synchronization (injection-locking) to external periodic forcings, which underlies the mutual synchronization in networks of nonlinear oscillators. Despite its history of synchronization and the practical importance of injection-locking to date, there are many important open problems of an efficient injection-locking for a given oscillator. In this work, I elucidate a hidden mechanism governing the synchronization limit under weak forcings, which is related to a widely known inequality; Hölderʼs inequality. This mechanism enables us to understand how and why the efficient injection-locking is realized; a general theory of synchronization limit is constructed where the maximization of the synchronization range or the stability of synchronization for general forcings including pulse trains, and a fundamental limit of general m : n phase locking, are clarified systematically. These synchronization limits and their utility are systematically verified in the Hodgkin–Huxley neuron model as an example.
Highlights
Entrainment, which adjusts the frequencies of oscillators to that of an external forcing above a critical forcing amplitude, is a fundamental phenomenon of wide interest with a long history and a large variety of applications [1,2,3,4,5]
Regarding P1, these conventional studies only suggest the existence of a synchronization limit by finding forcings in each particular situation, and regarding P2 and P3, methods used in these conventional studies are not applicable, and a fundamental limit of entrainability has not been clarified
In this paper, we find an underlying mechanism in the above three problems leads us to a unified, global view of synchronization limits and their constructions
Summary
Entrainment, which adjusts the frequencies of oscillators to that of an external forcing (signal) above a critical forcing amplitude, is a fundamental phenomenon of wide interest with a long history and a large variety of applications [1,2,3,4,5]. Regarding P1, these conventional studies only suggest the existence of a synchronization limit by finding (possibly local optimal) forcings in each particular situation, and regarding P2 and P3, methods used in these conventional studies are not applicable, and a fundamental limit of entrainability has not been clarified. Toward this end, in this paper, we find an underlying mechanism in the above three problems leads us to a unified, global view of synchronization limits and their constructions
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