Synchronization in populations of globally coupled oscillators with inertial effects

A model for synchronization of globally coupled phase oscillators including ``inertial'' effects is analyzed. In such a model, both oscillator frequencies and phases evolve in time. Stationary solutions include incoherent (unsynchronized) and synchronized states of the oscillator population. Assuming a Lorentzian distribution of oscillator natural frequencies, $g(\Omega)$, both larger inertia or larger frequency spread stabilize the incoherent solution, thereby making harder to synchronize the population. In the limiting case $g(\Omega)=\delta(\Omega)$, the critical coupling becomes independent of inertia. A richer phenomenology is found for bimodal distributions. For instance, inertial effects may destabilize incoherence, giving rise to bifurcating synchronized standing wave states. Inertia tends to harden the bifurcation from incoherence to synchronized states: at zero inertia, this bifurcation is supercritical (soft), but it tends to become subcritical (hard) as inertia increases. Nonlinear stability is investigated in the limit of high natural frequencies.Comment: Revtex, 36 pages, submit to Phys. Rev.

Role of Network Topology in the Synchronization of Power Systems

We study synchronization dynamics in networks of coupled oscillators with bimodal distribution of natural frequencies. This setup can be interpreted as a simple model of frequency synchronization dynamics among generators and loads working in a power network. We derive the minimum coupling strength required to ensure global frequency synchronization. This threshold value can be efficiently found by solving a binary optimization problem, even for large networks. In order to validate our procedure, we compare its results with numerical simulations on a realistic network describing the European interconnected high-voltage electricity system, finding a very good agreement. Our synchronization threshold can be used to test the stability of frequency synchronization to link removals. As the threshold value changes only in very few cases when aplied to the European realistic network, we conclude that network is resilient in this regard. Since the threshold calculation depends on the local connectivity, it can also be used to identify critical network partitions acting as synchronization bottlenecks. In our stability experiments we observe that when a link removal triggers a change in the critical partition, its limits tend to converge to national borders. This phenomenon, which can have important consequences to synchronization dynamics in case of cascading failure, signals the influence of the uncomplete topological integration of national power grids at the European scale.Comment: The final publication is available at http://www.epj.org (see http://www.springerlink.com/content/l22k574x25u6q61m/

Asymptotic description of transients and synchronized states of globally coupled oscillators

A two-time scale asymptotic method has been introduced to analyze the multimodal mean-field Kuramoto-Sakaguchi model of oscillator synchronization in the high-frequency limit. The method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution. Each component evolves toward a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them. Synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values. Our results agree very well with direct numerical simulations of the nonlinear Fokker-Planck equation for the probability density. Numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal (symmetric and asymmetric) frequency distribution with or without external field. We also recover in a very easy and intuitive way the only other known analytical results: those corresponding to reflection-symmetric bimodal frequency distributions near bifurcation points.Comment: Revtex,12 pag.,9 fig.;submitted to Physica

Dynamical aspects of mean field plane rotators and the Kuramoto model

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c.Comment: 18 pages, 1 figur

An investigation of naphthalenediimides as central building blocks in model compounds for scanning tunneling microscope induced light emission experiments and förster resonance energy transfer studies

Abstract Scanning tunnelling microscopy (STM) is a powerful technique to observe surfaces at the atomic level. The resolution of this STM technique is good enough to study the electronic properties of single molecules adsorbed onto metallic substrates. An important step towards controllable single molecular technologies is the determination of how the molecule substrate interaction changes the local molecular electronic structure. Since this electronic structure of molecules is strongly perturbed by the electrons of the underlying metallic substrate, an electronic decoupling of the molecules from the metal surface is required to isolate the electronic properties of an individual molecule. Förster resonance energy transfer (FRET) has found many applications in different fields of science, because it allows the determination of the distance between two chromophores in the 1 10 nm range. In addition to other factors, this energy transfer is also dependent on the relative orientation of donor and acceptor chromophores to each other. This thesis describes the design, synthesis and investigations of model compounds for: 1). STM induced light emission experiments from single molecules and 2). for FRET studies