Slow parameter drift is common in many systems (e.g., the amount of greenhouse gases in the terrestrial atmosphere is increasing). In such situations, the attractor on which the system trajectory lies can be destroyed, and the trajectory will then go
to another attractor of the system. We consider the case where there are more than one of these possible final attractors, and we ask whether we can control the outcome (i.e., the attractor that ultimately captures the trajectory) using only small controlling perturbations. Specifically, we consider the problem of controlling a noisy system whose parameter slowly drifts through a saddle-node bifurcation taking place on a fractal boundary between the basins of multiple attractors. We show that, when the noise level is low, a small perturbation of size comparable to the noise amplitude applied at a single point in time can ensure that the system will evolve toward a target attracting state with high probability. For a range of noise levels, we find that the minimum size of perturbation required for control is much smaller within a time period that starts some time after the bifurcation, providing a window of opportunity for driving the system toward a desirable state. We refer to this procedure as tipping point control.
The collective dynamics of a network of excitable nodes changes dramatically when inhibitory nodes are introduced. We consider inhibitory nodes which may be activated just like excitatory nodes but, upon activating, decrease the probability of activa
tion of network neighbors. We show that, although the direct effect of inhibitory nodes is to decrease activity, the collective dynamics becomes self-sustaining. We explain this counterintuitive result by defining and analyzing a branching function which may be thought of as an activity-dependent branching ratio. The shape of the branching function implies that for a range of global coupling parameters dynamics are self-sustaining. Within the self-sustaining region of parameter space lies a critical line along which dynamics take the form of avalanches with universal scaling of size and duration, embedded in ceaseless timeseries of activity. Our analyses, confirmed by numerical simulation, suggest that inhibition may play a counterintuitive role in excitable networks.
We characterize the distributions of size and duration of avalanches propagating in complex networks. By an avalanche we mean the sequence of events initiated by the externally stimulated `excitation of a network node, which may, with some probabilit
y, then stimulate subsequent firings of the nodes to which it is connected, resulting in a cascade of firings. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix which encodes the structure of the network. By using mean-field analyses, previous studies of avalanches in networks have not considered the effect of network structure on the distribution of size and duration of avalanches. Our results apply to individual networks (rather than network ensembles) and provide expressions for the distributions of size and duration of avalanches starting at particular nodes in the network. These findings might find application in the analysis of branching processes in networks, such as cascading power grid failures and critical brain dynamics. In particular, our results show that some experimental signatures of critical brain dynamics (i.e., power-law distributions of size and duration of neuronal avalanches), are robust to complex underlying network topologies.
The ensemble averaged power scattered in and out of lossless chaotic cavities decays as a power law in time for large times. In the case of a pulse with a finite duration, the power scattered from a single realization of a cavity closely tracks the p
ower law ensemble decay initially, but eventually transitions to an exponential decay. In this paper, we explore the nature of this transition in the case of coupling to a single port. We find that for a given pulse shape, the properties of the transition are universal if time is properly normalized. We define the crossover time to be the time at which the deviations from the mean of the reflected power in individual realizations become comparable to the mean reflected power. We demonstrate numerically that, for randomly chosen cavity realizations and given pulse shapes, the probability distribution function of reflected power depends only on time, normalized to this crossover time.
The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given populati
on are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.
