Universal scaling law for the largest Lyapunov exponent in coupled map lattices

Lyapunov exponents for temporal networks

By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instability of a temporal network and construct a measure to estimate the network maximum Lyapunov exponent (nMLE) of a temporal network trajectory. Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions and estimate the nMLE directly from a single network trajectory. We validate our method for a range of synthetic generative network models displaying low- and high-dimensional chaos and finally discuss potential applications.

Scaling behavior of nonhyperbolic coupled map lattices

Coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of Nth order which exhibit strongest possible chaotic behavior for small coupling constants a. We prove that the expectations of arbitrary observables scale with sqrt of a in the low-coupling limit, contrasting the hyperbolic case which is known to scale with a . Moreover we prove that there are log-periodic oscillations of period ln N2 modulating the sqrt of a dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant one-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.

Universal scaling of Lyapunov exponents in coupled chaotic oscillators

We have uncovered a phenomenon in coupled chaotic oscillators where a subset of Lyapunov exponents, which are originally zero in the absence of coupling, can become positive as the coupling is increased. This occurs for chaotic attractors having multiple scrolls, such as the Lorenz attractor. We argue that the phenomenon is due to the disturbance to the relative frequencies with which a trajectory visits different scrolls of the attractor. An algebraic scaling law is obtained which relates the Lyapunov exponents with the coupling strength. The scaling law appears to be universal.

Fundamental scaling laws of on-off intermittency in a stochastically driven dissipative pattern-forming system

Noise-driven electroconvection in sandwich cells of nematic liquid crystals exhibits on-off intermittent behavior at the onset of the instability. We study laser scattering of convection rolls to characterize the wavelengths and trajectories of the stochastic amplitudes of the intermittent structures. The pattern wavelengths and statistics of these trajectories are in quantitative agreement with simulations of the linearized electrohydrodynamic equations. The fundamental tau(-3/2) distribution law for the durations tau of laminar phases as well as the power law of the amplitude distribution of intermittent bursts are confirmed in the experiments. Power spectral densities of the experimental and numerically simulated trajectories are discussed.