Synchronization hypothesis in the Winfree model

We consider $N$ oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength $\kappa$ and the spectrum width $\gamma$ of the frequencies of each oscillator. In the uncoupled regime, $\kappa=0$, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that $N$ oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in $(\gamma,\kappa)$ of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization

Strange attractors in periodically-kicked degenerate Hopf bifurcations

We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodically-kicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a new multi-parameter version of the theory of rank one maps developed by Wang and Young.Comment: 16 page

Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

From limit cycles to strange attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.Comment: 27 page

Perturbations of Noise: The origins of Isothermal Flows

We make a detailed analysis of both phenomenological and analytic background for the "Brownian recoil principle" hypothesis (Phys. Rev. A 46, (1992), 4634). A corresponding theory of the isothermal Brownian motion of particle ensembles (Smoluchowski diffusion process approximation), gives account of the environmental recoil effects due to locally induced tiny heat flows. By means of local expectation values we elevate the individually negligible phenomena to a non-negligible (accumulated) recoil effect on the ensemble average. The main technical input is a consequent exploitation of the Hamilton-Jacobi equation as a natural substitute for the local momentum conservation law. Together with the continuity equation (alternatively, Fokker-Planck), it forms a closed system of partial differential equations which uniquely determines an associated Markovian diffusion process. The third Newton law in the mean is utilised to generate diffusion-type processes which are either anomalous (enhanced), or generically non-dispersive.Comment: Latex fil

Synchronization in Winfree model with N oscillators

We consider $N$ oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength $\kappa$ and the spectrum width $\gamma$ of the frequencies of each oscillator. In the uncoupled regime, $\kappa=0$, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that $N$ oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in $(\gamma,\kappa)$ of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization

Invariant cone and synchronization state stability of the mean field models

In this article we prove the stability of mean field systems as the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of os-cillators and the distribution of the natural frequencies. In order to prove the main result, we introduce the positive invariant cone and we start by studying the linearized system. The method can be applied to others mean field models as the Kuramoto model

Lyapunov minimizing measures for expanding maps of the circle

We consider the set of maps f є Fα+=Uβ >αC1β of the circle which are covering maps of degree D, expanding, minxєS1 f ¹(x) > 1 and orientation preserving. We are interested in characterizing the set of suchmaps f which admit a unique f -invariant probability measure _ minimizing ∫1n f ¹ d µ over all f -invariant probability measures. We show there exists a set G+ C Fα+, open and dense in the C1+α topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C1+α topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f -invariant measures µ maximizing ∫ A dµ for a fixed C1-expanding map f and a general potential A, not necessarily equal to −ln f ¹

Lyapunov minimizing measures for expanding maps of the circle

We consider the set of maps f є Fα+=Uβ >αC1β of the circle which are covering maps of degree D, expanding, minxєS1 f ¹(x) > 1 and orientation preserving. We are interested in characterizing the set of suchmaps f which admit a unique f -invariant probability measure _ minimizing ∫1n f ¹ d µ over all f -invariant probability measures. We show there exists a set G+ C Fα+, open and dense in the C1+α topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C1+α topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f -invariant measures µ maximizing ∫ A dµ for a fixed C1-expanding map f and a general potential A, not necessarily equal to −ln f ¹

Reduced dimension and Rotation vector formula of ordinary differential equation

The leader trajectory function defined in this article is an approximate solution of a differential equation. It is defined by some independent one-dimensional differential equations. The generalized main result of this article asserts that if the leader trajectory exists then it is at finite distance from the solution of the system. The application of the generalized main result is to control the trajectory of the periodic systems. We prove that for any periodic system and any initial condition there exists a leader trajectory which is a linear function of the time variable. In other words, we find an exact Rotation vector formula which is the relation between the rotation vector and the initial condition. In addition, we present a necessary and sufficient condition for the existence of a locally constant rotation vector under perturbation of the system, known by the Arnold tongue