1.7.1. Outline
We start with a technical chapter, where we provide basic notations and results needed through-out this thesis. This chapter is not intended to offer substantially new insights, but rather focuses on translating well-known, real-valued results like Itˆo’s formula into a complex-valued notation,
102Note that aCn,n-valued outer-product matrix, such as the complex outer product, can of course be identified with aCn
2-valued vector, c.f. Definition2.2.
103The nonlinear perturbations furthermore violate the global Lipschitz assumption.
which will prove more suitable for modeling a system of oscillators. Chapter2in particular aims at studying Cn,n-valued stochastic processes, which will be of central interest to us throughout this thesis. For this purpose, we start with an identification of Cn with R2n and employ this identification in order to relate linear maps, complex product constructions and complex deriva-tives to their real-valued counterparts. Subsequently, we recall basic properties of the discrete Fourier transform (DFT), focusing on its relation to circulant matrices. We then introduce the complex Brownian motion and show its invariance under a DFT. Next we introduce complex-valued stochastic differential equations (SDEs) and provide a version of Itˆo’s formula applicable to complex-valued stochastic processes. Finally, we study matrix-valued processes with a particular focus on the complex outer-product process.
In Chapter 3 we specify the main model of weakly coupled oscillators which we want to inves-tigate in this thesis. We first introduce a system of uncoupled oscillators, study its symmetries and identify the conserved quantities related to these symmetries. We show that these conserved quantities correspond to the components of the systems complex outer product. In the following sections, we introduce weak coupling and perturbation terms, encompassing deterministic inter-actions as well as multiplicative, regularizing and additive-noise terms. Each of these terms is required to exhibit a circulant structure, which we employ to simplify the system’s description by performing a discrete Fourier transform. Finally, we speed up the evolution by means of a time rescaling and study the complex outer-product process. This process turns out to be no longer constant, resulting from the fact that some of the uncoupled systems symmetries were broken by the weak interaction terms.
In Chapter 4 we derive an averaging result for the complex outer-product process of a weakly coupled oscillator system under some general assumptions on the coupling terms. This result will in particular be applicable to the system described in the previous section. The weakness of the coupling, represented by a parameter ε, implies a scale hierarchy between the fast evolution of the oscillators and the slow influence of the interactions between the oscillators. In the scaling limit of ε → 0, the averaging result allows us to approximate the outer-product process by an effective process, which is governed byaverageddrift and diffusion terms. We prove this averaging theorem by adapting the strategy of [BR14], i.e. by establishing a convergence of the associated Dirichlet forms. We first state the general setup and then show that the bilinear form related to the complex outer-product process is a Dirichlet form. In following sections we examine the equivalence relation induced by the complex outer-product mapping. In particular, we study the corresponding quotient space and the so-called projected Dirichlet form on this space. Finally, we prove that the Dirichlet forms related to the weakly coupled system converge to this projected Dirichlet form, which can be associated to an effective process. Verifying the tightness of the involved processes, subsequently allows us to infer a weak convergence of the associated stochastic processes.
Since the averaging result of Chapter 4 is applicable to the oscillator system of Chapter 3, we are interested in understanding the evolution of the effective limiting process. Chapter 5 is consequently devoted to explicitly determining the corresponding SDE. We therefore calculate the averaged drift term as well as the averaged diffusion matrices corresponding to multiplicative, regularizing and additive noise. The averaging calculations can be identified with line integrals that can be calculated by means of the residue theorem. After having identified the averaged diffusion matrices, we also determine corresponding dispersion matrices which give rise to these
diffusion matrices.
In Chapter6we solve the effective SDE stated in the previous chapter. In the deterministic case, the Baker-Campbell-Hausdorff formula allows us to explicitly solve this matrix-valued equation.
For the full stochastic system, we subsequently focus on the evolution of the diagonal elements of the outer-product process, which can be interpreted as the systems eigenmode amplitudes.
We first study their evolution in the homogeneous case, i.e. in the absence of additive noise, where we obtain asymptoticsynchronization results. In particular, we apply a number theoretic result in order to relate noise-coupling topologies to their induced synchronization states. The synchronization results are then shown to persist in the inhomogeneous case, provided that we have a sufficiently small additive-noise perturbation. Finally, we apply the averaging theorem in order to obtain a synchronization statement for the original, unaveraged system.
1.7.2. Interdependence of chapters
The chapters are designed in a modular fashion, i.e. subsequent chapters mainly refer to the final results of the previous chapters:
Chapter 2 provides basic technical tools needed in all of the following chapters. The complex outer-product process, which is the central object of interest, is introduced and motivated in Chapter 3. Its evolution equation can be found in Proposition 3.39. Chapter 4 subsequently develops an averaging theory which applies to this process and which is summarized in The-orem 4.71. This theorem characterizes a matrix-valued limiting process by means of an SDE, which is given in terms of certain averaging integrals. In Chapter5 these integrals are evaluated and the resulting explicit SDE is stated in Theorem5.26and Corollary5.28. Chapter6is finally devoted to solving this SDE and characterizing its solutions.