Cycle flows and multistabilty in oscillatory networks: an overview

The functions of many networked systems in physics, biology or engineering rely on a coordinated or synchronized dynamics of its constituents. In power grids for example, all generators must synchronize and run at the same frequency and their phases need to appoximately lock to guarantee a steady power flow. Here, we analyze the existence and multitude of such phase-locked states. Focusing on edge and cycle flows instead of the nodal phases we derive rigorous results on the existence and number of such states. Generally, multiple phase-locked states coexist in networks with strong edges, long elementary cycles and a homogeneous distribution of natural frequencies or power injections, respectively. We offer an algorithm to systematically compute multiple phase- locked states and demonstrate some surprising dynamical consequences of multistability

Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background

We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a "second geometrization", by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition of the Jacobi stability for scalar field cosmologies in the second order formalism. As an application of the developed methods we consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. We find that the Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. By using the standard first order formulation of the cosmological models as dynamical systems we have investigated the stability of the phantom quintessence and tachyonic scalar fields, by lifting the first order system to the tangent bundle. It turns out that in the presence of a power law potential both these models are Jacobi unstable during the entire cosmological evolution.Comment: 24 pages, 14 figures, accepted for publication in Advances in High Energy Physics, special issue "Dark Physics in the Early Universe

On the relation between mathematical and numerical relativity

The large scale binary black hole effort in numerical relativity has led to an increasing distinction between numerical and mathematical relativity. This note discusses this situation and gives some examples of succesful interactions between numerical and mathematical methods is general relativity.Comment: 12 page

Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances. Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I. This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, is essential to understand the effect of noise and the phenomenon of phase diffusion. In addition, it is shown that the spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system approach. This approach is not limited to low-dimensional systems and the reduction method presented in part I is applied to a stochastic model relevant to climate dynamics in part III

Geometric Design and Stability of Power Networks

From the perspective of the network theory, the present work illustrates how the parametric intrinsic geometric description exhibits an exact set of pair correction functions and global correlation volume with and without the inclusion of the imaginary power flow. The Gaussian fluctuations about the equilibrium basis accomplish a well-defined, non-degenerate, curved regular intrinsic Riemannian surfaces for the purely real and the purely imaginary power flows and their linear combinations. An explicit computation demonstrates that the underlying real and imaginary power correlations involve ordinary summations of the power factors, with and without their joint effects. Novel aspect of the intrinsic geometry constitutes a stable design for the power systems.Comment: 23 pages, 11 figures, Keywords: Correlation; Geometry; Power Flow; Network; Stabilit

Emergent singular solutions of non-local density-magnetization equations in one dimension

We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the system's stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.Comment: 19 pages, 13 figures. Submitted to Phys. Rev.