Onset of cooperation between layered networks.

Functionalities of a variety of complex systems involve cooperations among multiple components; for example, a transportation system provides convenient transfers among airlines, railways, roads, and shipping lines. A layered model with interacting networks can facilitate the description and analysis of such systems. In this paper we propose a model of traffic dynamics and reveal a transition at the onset of cooperation between layered networks. The cooperation strength, treated as an order parameter, changes from zero to positive at the transition point. Numerical results on artificial networks as well as two real networks, Chinese and European railway-airline transportation networks, agree well with our analysis.

Basins of attraction in piecewise smooth Hamiltonian systems.

Piecewise smooth Hamiltonian systems arise in physical and engineering applications. For such a system typically an infinite number of quasi-periodic "attractors" coexist. (Here we use the term "attractors" to indicate invariant sets to which typically initial conditions approach, as a result of the piecewise smoothness of the underlying system. These "attractors" are therefore characteristically different from the attractors in dissipative dynamical systems.) We find that the basins of attraction of different "attractors" exhibit a riddled-like feature in that they mix with each other on arbitrarily fine scales. This practically prevents prediction of "attractors" from specific initial conditions and parameters. The mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems. We demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation.

Crisis induced by an escape from a fat strange set.

This article reports a characteristic crisis observed in a two-dimensional discontinuous and noninvertible map. The discontinuity border in the definition range of the mapping oscillates as the discrete time progresses so that the forward images of the border form a fat fractal. By choosing particular parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter an elliptic periodic orbit and the elliptic islands around it suddenly appear inside the fat strange set so that the iterations on the set escape to the islands. The fat chaotic attractor thus suddenly transfers to a fat transient set. The effect of the feature of the crisis on the rule of the lifetime in the transient set is discussed. It shows that the dependence of the lifetime on the control parameter follows a universal scaling law suggested by Grebogy, Ott, and Yorke [Phys. Rev. Lett. 57, 1284 (1986)], and the scaling exponent can be approximated according to the variation rules of the elliptic islands and the measure of the fat fractal. The strange repeller, which appears after the crisis, is also a fat fractal.