Optimal Phase Description of Chaotic Oscillators

We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincar\'e surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled of the amplitude dynamics, and provides a proper description of phase resetting of chaotic oscillations. The method is illustrated with the R\"ossler and Lorenz systems.Comment: 10 Pages, 14 Figure

Desynchronizing two oscillators while stimulating and observing only one

Synchronization of two or more self-sustained oscillators is a well-known and studied phenomenon, appearing both in natural and designed systems. In some cases, the synchronized state is undesired, and the aim is to destroy synchrony by external intervention. In this paper, we focus on desynchronizing two self-sustained oscillators by short pulses delivered to the system in a phase-specific manner. We analyze a non-trivial case when we cannot access both oscillators but stimulate only one. The following restriction is that we can monitor only one unit, be it a stimulated or non-stimulated one. First, we use a system of two coupled Rayleigh oscillators to demonstrate how a loss of synchrony can be induced by stimulating a unit once per period at a specific phase and detected by observing consecutive inter-pulse durations. Next, we exploit the phase approximation to develop a rigorous theory formulating the problem in terms of a map. We derive exact expressions for the phase -- isostable coordinates of this coupled system and show a relation between the phase and isostable response curves to the phase response curve of the uncoupled oscillator. Finally, we demonstrate how to obtain phase response information from the system using time series and discuss the differences between observing the stimulated and unstimulated oscillator

Dual -1 Hahn polynomials and perfect state transfer

We find all the $XX$ spin chains with perfect state transfer (PST) that are connected with the dual -1 Hahn polynomials $R_n(x; \alpha,\beta,N)$. For $N$ odd we recover a model that had already been identified while for $N$ even, we obtain a new system exhibiting PST.Comment: 11 page

High-order phase reduction for coupled 2D oscillators

Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. For such a reduction, the coupling should be small, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction

A new model for mixing by double-diffusive convection (semi-convection): I. The conditions for layer formation

The process referred to as "semi-convection" in astrophysics and "double-diffusive convection in the diffusive regime" in Earth and planetary sciences, occurs in stellar and planetary interiors in regions which are stable according to the Ledoux criterion but unstable according to the Schwarzschild criterion. In this series of papers, we analyze the results of an extensive suite of 3D numerical simulations of the process, and ultimately propose a new 1D prescription for heat and compositional transport in this regime which can be used in stellar or planetary structure and evolution models. In a preliminary study of the phenomenon, Rosenblum et al. (2011) showed that, after saturation of the primary instability, a system can evolve in one of two possible ways: the induced turbulence either remains homogeneous, with very weak transport properties, or transitions into a thermo-compositional staircase where the transport rate is much larger (albeit still smaller than in standard convection). In this paper, we show that this dichotomous behavior is a robust property of semi-convection across a wide region of parameter space. We propose a simple semi-analytical criterion to determine whether layer formation is expected or not, and at what rate it proceeds, as a function of the background stratification and of the diffusion parameters (viscosity, thermal diffusivity and compositional diffusivity) only. The theoretical criterion matches the outcome of our numerical simulations very adequately in the numerically accessible "planetary" parameter regime, and can easily be extrapolated to the stellar parameter regime. Subsequent papers will address more specifically the question of quantifying transport in the layered case and in the non-layered case.Comment: Submitted to Ap

Molecular and functional basis of phenotypic convergence in white lizards at White Sands

There are many striking examples of phenotypic convergence in nature, in some cases associated with changes in the same genes. But even mutations in the same gene may have different biochemical properties and thus different evolutionary consequences. Here we dissect the molecular mechanism of convergent evolution in three lizard species with blanched coloration on the gypsum dunes of White Sands, New Mexico. These White Sands forms have rapidly evolved cryptic coloration in the last few thousand years, presumably to avoid predation. We use cell-based assays to demonstrate that independent mutations in the same gene underlie the convergent blanched phenotypes in two of the three species. Although the same gene contributes to light phenotypes in these White Sands populations, the specific molecular mechanisms leading to reduced melanin production are different. In one case, mutations affect receptor signaling and in the other, the ability of the receptor to integrate into the melanocyte membrane. These functional differences have important ramifications at the organismal level. Derived alleles in the two species show opposite dominance patterns, which in turn affect their visibility to selection and the spatial distribution of alleles across habitats. Our results demonstrate that even when the same gene is responsible for phenotypic convergence, differences in molecular mechanism can have dramatic consequences on trait expression and ultimately the adaptive trajectory.Organismic and Evolutionary Biolog