Inhalt

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  Introduction
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  Synchronization of coupled oscillators
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   The notion of synchronization
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   Characterization of synchronization
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  Related work on symmetries and conserved quantities
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   Coupling topologies and their symmetries
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   Conserved quantities
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  Related work on synchronization
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   Kuramoto model
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   Noise influence
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  Related work on weakly coupled systems and averaging theory
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   Weak coupling
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   Averaging theory
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   Averaging theory via dirichlet forms
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   Synchronization of noise-coupled Kuramoto oscillators
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   Setup, objectives and methods
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   Setup
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   Objectives
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   Methods
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  Outline and interdependence
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   Outline
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   Interdependence of chapters
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  Complex structures
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  Complex linear transformations
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   Identification of Cn with R2n
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   Products
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   Complex derivatives
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   Holomorphic functions and the residue theorem
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  Discrete Fourier transform
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   Definitions and basic properties
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   Convolution theorem
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   Symmetries
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  Complex-valued SDEs
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   Complex martingales
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   Complex Itô formula
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  Matrix-valued SDEs
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   Evolution of the complex outer-product process
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   Diffusion matrix for evolution of outer-product process
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   Weakly coupled oscillators
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  Uncoupled commensurate oscillators
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   Strong linear coupling of massive particles
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   Symmetries and conserved quantities
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   Weak deterministic drift
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   Circulant linear coupling
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   Nonlinear pertubations
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  Weak stochastic perturbations
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   Multiplicative noise
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   Regularizing noise
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   Additive noise
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  Combined stochastic differential equation
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   Original system
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   Existence and uniqueness
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   Transformed system
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   Time rescaling and evolution of outer-product process
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   Time rescaling
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   Complex outer-product process
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  Averaging theory
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  Setup and assumptions
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   Assumptions
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   Dirichlet forms
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   Generators and corresponding semigroups
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   Bilinear forms
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  First integrals and induced projection
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   First integrals
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   Projection induced by first integrals
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  Decomposition of quotient space and coarea formula
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   Decomposition of quotient space
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   Coarea formula
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   Projected Dirichlet form
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   Identification of Dirchlet form
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  Identification of process associated to projected Dirichlet form
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   Process associated with projected Dirichlet form
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   Infinitesimal generator related to projected Dirichlet form
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   Generalized convergence of Dirichlet forms and associated weak convergence
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   Averaging Theorem
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  Averaged system
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   Averaging via residue theorem
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   Averaged drift
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  Averaged multiplicative noise
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   Calculation of averaged diffusion matrix
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   Representation of averaged diffusion matrix via effective dispersion matrix
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  Averaged regularizing noise
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   Invariance of regularizing noise
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   Representation of averaged diffusion matrix via effective dispersion matrix
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  Averaged additive noise
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   Invariance of additive noise
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   Representation of averaged diffusion matrix via effective dispersion matrix
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   Combined noise and Itô correction
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   Combined system
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  Effective evolution
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  Deterministic evolution
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   Solution to ODE
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   Energy- and phase evolution of two-oscillator system
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  Evolution equations for eigenmode amplitudes
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   Eigenmode amplitudes
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   Pairing of eigenmodes
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  Amplitude ratio for eigenmode pair
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   SDE for amplitude ratio
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   Asymptotic evolution of ratio process in homogeneous case
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   Evolution of homogeneous system
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   Asymptotic evolution of eigenmode amplitudes
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   Pathwise synchronization of energy-normalized effective system
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   Coupling topologies and induced synchronization modes
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   Evolution of inhomogeneous system
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   Evolution of eigenmode amplitudes and the ratio process
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   Asymptotic evolution
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  Evolution of original system
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   Convergence in C([0,),Rm)
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   Synchronization of original system
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  Conclusion
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   Results
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   Outlook
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  Nomenclature
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   Spaces, measures and metrics
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   Maps and operators
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   Parameters
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   Drift terms, dispersion and diffusion matrices
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   Random variables, processes and Dirichlet forms