Distributed Delays Facilitate Amplitude Death of Coupled Oscillators

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.Comment: 4 pages, 5 figure

Symbolic Synchronization and the Detection of Global Properties of Coupled Dynamics from Local Information

We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the {\it homogeneous unsynchronized phase} where the coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost) identical symbolic dynamics at all the nodes in the network. We refer to this dynamical behaviour as {\it symbolic synchronization}. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behaviour of the largest Lyapunov exponent and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local chaotic dynamical function. We present results for two model dynamics, iterations of the one-dimensional logistic map and the two-dimensional H\'enon map, as local dynamical function.Comment: 21 pages, 7 figure

Local pinning of networks of multi-agent systems with transmission and pinning delays

We study the stability of networks of multi-agent systems with local pinning strategies and two types of time delays, namely the transmission delay in the network and the pinning delay of the controllers. Sufficient conditions for stability are derived under specific scenarios by computing or estimating the dominant eigenvalue of the characteristic equation. In addition, controlling the network by pinning a single node is studied. Moreover, perturbation methods are employed to derive conditions in the limit of small and large pinning strengths.Numerical algorithms are proposed to verify stability, and simulation examples are presented to confirm the efficiency of analytic results.Comment: 6 pages, 3 figure

Balancing the inverted pendulum using position feedback

It is shown how to obtain asymptotic stability in second-order undamped systems using time-delay action in the feedback of position. The effect of the delay is similar to derivative feedback in modifying the behavior of the system. Results are given on the selection of the controller parameters both in the absence and the presence of additional delay in the feedback path. The time-lag position feedback is shown to compare favorably with the conventional PD controller in terms of stability

Synchronization in discrete-time networks with general pairwise coupling

We consider complete synchronization of identical maps coupled through a general interaction function and in a general network topology where the edges may be directed and may carry both positive and negative weights. We define mixed transverse exponents and derive sufficient conditions for local complete synchronization. The general non-diffusive coupling scheme can lead to new synchronous behavior, in networks of identical units, that cannot be produced by single units in isolation. In particular, we show that synchronous chaos can emerge in networks of simple units. Conversely, in networks of chaotic units simple synchronous dynamics can emerge; that is, chaos can be suppressed through synchrony

On the symmetry of the Laplacian spectra of signed graphs

We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph

Synchronization of networks with prescribed degree distributions

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building non-synchronizing networks having a prescribed degree distribution.Comment: v2: A new theorem and a numerical example added. To appear in IEEE Trans. Circuits and Systems I: Fundamental Theory and Application