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A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range versus forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Holder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For 1:1 entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in L p -space with p > 1 , and (ii) a construction of the global optimal pulse-like forcings in L 1 -space, and for m : n entrainment ( m ≠ n ), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for 1 p ≤ ∞ and these pulse-like forcings for p = 1 . These theoretical findings are verified by systematic, extensive numerical calculations and simulations.

We analyse a piecewise-linear oscillator that consists of a three- L C resonant circuit with a hysteresis element. Three sets of two-dimensional linear equations, including a hysteresis function, represent the governing equations of the circuit, and all the Lyapunov exponents are calculated in a remarkably simple manner based on derived explicit solutions. Various dynamical phenomena such as two-torus, three-torus, and hyperchaos with four positive Lyapunov exponents are observed by Lyapunov analysis. We obtained a detailed bifurcation diagram in which novel bifurcation structure which we call a “two-torus Arnold tongue” is observed where two-torus generating regions exist in a three-torus generating region as if periodic states exist in a two-torus generating region.

The period-doubling transition to chaos along the edge of an Arnold tongue is known to display unusual organization and scaling behavior (Kuznetsov et al. (2005) [7] ). It is also known that forced period-doubling systems may be associated with the appearance of so-called period-doubled tori (Arneodo et al. (1983) [15] ). Using the Rossler system as an example, we present a detailed analysis of the bifurcation structure associated with the forcing of a three-dimensional period-doubling system. We explain how this structure is related to the recently discovered phenomenon of multi-layered tori and discuss different bifurcation scenarios that transform a resonance torus into a period-doubled ergodic torus. Similar bifurcation phenomena have recently been observed in a biologically relevant model of kidney blood flow regulation in response to fluctuations in arterial pressure.

We consider the Arnold Tongue of the Arnold family of circle maps associated to a fixed Diophantine rotation number θ . The corresponding maps of the family are analytically conjugate to a rigid rotation. This conjugation is defined on a (maximal) complex strip of the circle and, after a suitable scaling, the size of this strip is given by an analytic function of the perturbative parameter. The main purpose of this paper is to perform a numerical accurate computation of this function and of its Taylor expansion. This allows us to verify previous theoretical results. The rotation numbers we select are quadratic irrationals, mainly the Golden Mean. By introducing a nonstandard extrapolation process, especially suited for the problem, we compute all the quantities required (rotation numbers, Arnold Tongues, Fourier and Taylor coefficients) with high precision.

The phase diagram of a triangular network of overdamped Josephson Junctions driven by independent current drives is studied in terms of vortices. There are no vortices in the fixed-point region, while in the p q Arnold tongue, vortices appear in sequences which repeat themselves every q vortices. Precisely q − p vortices in each sequence are injected by the non-uniform drive. We explicitly identify the vortex sequence at infinity and find that for large input currents, all reals between 1 and 1 2 can be given a unique binary representation in terms of vortices.