How Network Topology Affects the Strength of Dangerous Power Grid Perturbations

Reasonably large perturbations may push a power grid from its stable synchronous state into an undesirable state. Identifying vulnerabilities in power grids by studying power grid stability against such perturbations can aid in preventing future blackouts. We use two stability measures \unicode{x2014} stability bound, which deals with a system's asymptotic behaviour, and survivability bound, which deals with a system's transient behaviour, to provide information about the strength of perturbations that destabilize the system. Using these stability measures, we have found that certain nodes in tree-like structures have low asymptotic stability, while nodes with a high number of connections generally have low transient stability

Robust Chaos

It has been proposed to make practical use of chaos in communication, in enhancing mixing in chemical processes and in spreading the spectrum of switch-mode power suppies to avoid electromagnetic interference. It is however known that for most smooth chaotic systems, there is a dense set of periodic windows for any range of parameter values. Therefore in practical systems working in chaotic mode, slight inadvertent fluctuation of a parameter may take the system out of chaos. We say a chaotic attractor is robust if, for its parameter values there exists a neighborhood in the parameter space with no periodic attractor and the chaotic attractor is unique in that neighborhood. In this paper we show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering.Comment: 4 pages, Latex, 4 postscript figures, To appear in Phys. Rev. Let

A Probabilistic Distance-Based Stability Quantifier for Complex Dynamical Systems

For a dynamical system, an attractor of the system may represent the `desirable' state. Perturbations acting on the system may push the system out of the basin of attraction of the desirable attractor. Hence, it is important to study the stability of such systems against reasonably large perturbations. We introduce a distance-based measure of stability, called `stability bound', to characterize the stability of dynamical systems against finite perturbations. This stability measure depends on the size and shape of the basin of attraction of the desirable attractor. A probabilistic sampling-based approach is used to estimate stability bound and quantify the associated estimation error. An important feature of stability bound is that it is numerically computable for any basin of attraction, including fractal basins. We demonstrate the merit of this stability measure using an ecological model of the Amazon rainforest, a ship capsize model, and a power grid model

Dangerous bifurcation at border collision: when does it occur?

It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations". In this paper we provide explanation of this phenomenon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur