Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

We discuss the superdiffusive motion of a random walk in a medium containing random velocity fields. For a two-dimensional layered medium with y-dependent random velocities in the x direction u(y), 〈${\mathit{x}}^2$(t)〉∼${\mathit{t}}^{2\nu }$, with 2ν=3/2, and with strong sample-to-sample fluctuations. The probability distribution of displacements, averaged over environments, takes a non-Gaussian scaling form at large time, 〈P(x,t)〉∼${\mathit{t}}^{\mathrm{-}3/4}$f(x/${\mathit{t}}^{3/4}$), where v(u)∼exp(-${\mathit{u}}^{\mathrm{\delta }}$) for u≫1, with δ=4/3. For an isotropic two-dimensional medium with ${\mathit{u}}_{\mathit{x}}$(y)=f(y) and ${\mathit{u}}_{\mathit{y}}$(x)=f(x), we find ν=2/3 and δ=1-ν${)}^{\mathrm{-}1}$=3.

The time evolution of probability densities of one-dimensional nonlinear vector fields is studied using a Liouville equation approach. It is shown that the Liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation. The associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions. On the other hand, the spectrum of the Liouville operator may become continuous at the bifurcation point, a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation. The relationship between the spectral decompositions of the Liouville and of the Fokker-Planck equations is discussed. In particular, the spectral decompositions constructed here for the Liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the Fokker-Planck equation of the associated stochastic process.

The effects of low-dimensional supports (one and two dimensions) on the steady state and the dynamics of open reactive systems capable of giving rise to oscillatory behavior are studied. A lattice Lotka–Volterra model involving reaction, adsorption, and desorption mechanisms is developed for which mean-field behavior predicts a continuum of closed trajectories around a center. It is shown that the spatial constraints of the support radically change this behavior. Specifically, while in one dimension, oscillations are suppressed, in two dimensions, the system selects a preferred oscillation frequency depending on the intrinsic parameters and the lattice geometry.