Related papers: Probability of local bifurcation type from a fixed…
The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the…
The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and…
We present a phenomenological description of the critical slowing down associated with period-doubling bifurcations in discrete dynamical systems. Starting from a local Taylor expansion around the fixed point and the bifurcation parameter,…
In order to investigate the evolutionary process of many deterministic Dynamical systems with unfixed parameter, a set of dynamical models with parameter changing continuously and the accumulation of this change might be large is introduced…
We introduce the concept of $\epsilon$-uncontrollability for random linear systems, i.e. linear system in which the usual matrices have been replaced by random matrices. We also estimate the $\epsilon$-uncontrollability in the case where…
Moment systems arise in a wide range of contexts and applications, e.g. in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a…
Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a…
Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and…
Novel method of reconstructing dynamical networks from empirically measured time series is proposed. By examining the variable--derivative correlation of network node pairs, we derive a simple equation that directly yields the adjacency…
We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…
By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the…
The paper gives a systematic analysis of singularities of transition processes in dynamical systems. General dynamical systems with dependence on parameter are studied. A system of relaxation times is constructed. Each relaxation time…
We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can…
Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the…
A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in…
We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…
We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths…
The map x -> x^x modulo p is related to a variation of the digital signature scheme in a similar way to the discrete exponentiation map, but it has received much less study. We explore the number of fixed points of this map by a statistical…
It was shown roughly thirty years ago that the density correlations of eigenvalues of large random matrices display a universal form, independent of most of the details of the distribution of the random matrix itself. We show that when the…