We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electro-optic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized, even when the coupling between them is changing unpredictably.

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Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a network/graph, we focused on finding characterizations of k-dense communities. For each edge uv ∈ E(G), the edge multiplicity of uv in G is given byFor an integer k with k ≥ 2, a k-dense community of a graph G, denoted by DC k (G), is a maximal connected subgraph of G induced by the vertex setIn this research, we characterize which graphs are k-dense but not (k + 1)-dense for some values of k and study the minimum and maximum number of edges such graphs can have. A better understanding of k-dense sub-graphs (or communities) helps in the study of the connectivity of large complex graphs (or networks) in the real world.