An Adaptive Algorithm for Synchronization in Diffusively Coupled Systems

We present an adaptive algorithm that guarantees synchronization in diffusively coupled systems. We first consider compartmental systems of ODEs, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments. Each set of like components may have its own weighted undirected graph describing the topology of the interconnection between compartments. The link weights are updated adaptively according to the magnitude of the difference between neighboring agents connected by the link. We next consider reaction-diffusion PDEs with Neumann boundary conditions, and derive an analogous algorithm guaranteeing spatial homogenization of solutions. We provide a numerical example demonstrating the results

Cluster synchronization of diffusively-coupled nonlinear systems: A contraction based approach

Finding the conditions that foster synchronization in networked oscillatory systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with neuronal spiking dynamics, we show that our new sufficient condition is tighter than those found in previous analyses which used nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex oscillatory systems

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the $\ell_1$ and $\ell_\infty$ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as matrix measure and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system

Critical scaling limits and singular SPDES

In this thesis, we study scaling limits of idealised models arising in statistical physics. We present three works that, in different directions, explore critical behaviour of such systems. In the first part, we study the 2D Allen–Cahn equation with white noise initial datum. This differential equation falls into the class of critical singular stochastic partial differential equations (SPDEs), for which no solution theory exists due to the roughness of the data. To give meaning to the equation, we consider the weak coupling scaling and establish non-trivial Gaussian fluctu- ations of the solution, by treating an infinite perturbative series expansion in terms of iterated stochastic integrals. To our best knowledge, this is the first time such an approach has been implemented for a (non-linear) critical SPDE and possibly a first step towards a general theory of SPDEs in this regime. Closely connected, the second part of the thesis comprises the large-scale behaviour of the 2D directed random polymer model, describing the trajec- tory of a random walk in a random potential. In the weak disorder limit, we derive an invariance principle for the polymer paths. As a consequence, the random potential has no effect on large scales, which is due to a self- averaging effect. Similar results were previously only obtained for all but the (critical) two dimensional case. Last, we study a system of particles with attractive interactions. Cluster- ing of particles occurs when tuning the system’s parameters, with growing system size. This describes a simplified model of (Bose–Einstein) condensa- tion. A careful analysis of the infinitesimal dynamics, using the Trotter–Kurtz approximation theorem, allows us to identify the limiting evolution in terms of a measure-valued Markov process. Moreover, we establish a link between the derived dynamics and the infinitely-many-neutral-alleles model in popu- lation genetics. The presented approach covers all interesting scaling regimes of the system parameters

Kinetic Theory and Renormalization Group Methods for Time Dependent Stochastic Systems

By time dependent stochastic systems we indicate efffective models for physical phenomena where the stochasticity takes into account some features whose analytic control is unattainable and/or unnecessary. In particular, we consider two classes of models which are characterized by the different role of randomness: (1) deterministic evolution with random initial data; (2) truly stochastic evolution, namely driven by some sort of random force, with either deterministic or random initial data. As an example of the setting (1) in this thesis we will deal with the discrete nonlinear Schrödinger equation (DNLS) with random initial data and we will mainly focus on its applications concerning the study of transport coefficients in lattice systems. Since the seminal work by Green and Kubo in the mid 50 s, when they discovered that transport coefficients for simple fluids can be obtained through a time integral over the respective total current correlation function, the mathematical physics community has been trying to rigorously validate these predictions and extend them also to solids. In particular, the main technical difficulty is to obtain at least a reliable asymptotic form of the time behaviour of the Green-Kubo correlation. To do this, one of the possible approaches is kinetic theory, a branch of the modern mathematical physics stemmed from the challenge of deriving the classical laws of thermodynamics from microscopic systems. Nowadays kinetic theory deals with models whose dynamics is transport dominated in the sense that typically the solutions to the kinetic equations, whose prototype is the Boltzmann equation, correspond to ballistic motion intercepted by collisions whose frequency is order one on the kinetic space-time scale. Referring to the articles in the thesis by Roman numerals [I]-[V], in [I] and [II] we build some technical tools, namely Wick polynomials and their connection with cumulants, to pave the way towards the rigorous derivation of a kinetic equation called Boltzmann-Peierls equation from the DNLS model. The paper [III] can be contextualized in the same framework of kinetic predictions for transport coefficients. In particular, we consider the velocity flip model which belongs to the family (2) of our previous classification, since it consists of a particle chain with harmonic interaction and a stochastic term which flips the velocity of the particles. In [III] we perform a detailed study of the position-momentum correlation matrix via two diffeerent methods and we get an explicit formula for the thermal conductivity. Moreover, in [IV] we consider the Lorentz model perturbed by an external magnetic field which can be categorized in the class (1): it is a gas of non interacting particles colliding with obstacles located at random positions in the plane. Here we show that under a suitable scaling limit the system is described by a kinetic equation where the magnetic field affects only the transport term, but not the collisions. Finally, in [IV] we studied a generalization of the famous Kardar-Parisi-Zhang (KPZ) equation which falls into the category (2) being a nonlinear stochastic partial differential equation driven by a space-time white noise. Spohn has recently introduced a generalized vector valued KPZ equation in the framework of nonlinear fluctuating hydrodynamics for anharmonic particle chains, a research field which is again strictly connected to the investigation of transport coefficients. The problem with the KPZ equation is that it is ill-posed. However, in 2013 Hairer succeded to give a rigorous mathematical meaning to the solution of the KPZ via an approximation scheme involving the renormalization of the nonlinear term by a formally infinite constant. In [V] we tackle a vector valued generalization of the KPZ and we prove local in time wellposedness by using a technique inspired by the so-called Wilsonian Renormalization Group

Non-equilibrium quantum dynamics : interplay of disorder, interactions and confinement

The study of collective behaviour in many-body systems often explores fundamentally new ideas absent from the mere constituents of such a system. A paradigmatic model for these studies is the spin-1/2 XXZ chain and its fermionic equivalent. This thesis can be broadly divided into the study of two fundamental aspects of this model. Firstly, we discuss localisation phenomena in one dimensional lattices as often experimentally realised in cold atom systems. Secondly, we investigate how disorder and symmetry influence heat transport in spin chains. More specifically, in the first part we consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of a strong optical lattice, a harmonic trap, and uncorrelated on-site disorder. We investigate a global inhomogeneous quantum quench and present numerical and analytical results for static and dynamical properties. We show that the approach to the non-thermal equilibrium state is extremely slow and that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localisation. We also consider the above system in a strong non-uniform electric field. In the non-interacting case, due to Wannier-Stark localisation, the single-particle wave functions are exponentially localised without quenched disorder. We show that this system remains localised in the presence of nearest-neighbour interactions and exhibits physics analogous to models of conventional many-body localisation. The second part explores the hydrodynamics of the disordered XYZ spin chain. Using time-evolving block decimation on open chains of up to 400 spins attached to thermal baths, we probe the energy transport of this system. Our principal findings are as follows. For weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Then, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive.Funded by CM-CDT and EPSRC (UK) under grants EP/G03673X/1 and EP/L015110/1