Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic R\"ossler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.Comment: 12 pages, 13 figure

Geometric characterization of nodal domains: the area-to-perimeter ratio

In an attempt to characterize the distribution of forms and shapes of nodal domains in wave functions, we define a geometric parameter - the ratio $\rho$ between the area of a domain and its perimeter, measured in units of the wavelength $1/\sqrt{E}$. We show that the distribution function $P(\rho)$ can distinguish between domains in which the classical dynamics is regular or chaotic. For separable surfaces, we compute the limiting distribution, and show that it is supported by an interval, which is independent of the properties of the surface. In systems which are chaotic, or in random-waves, the area-to-perimeter distribution has substantially different features which we study numerically. We compare the features of the distribution for chaotic wave functions with the predictions of the percolation model to find agreement, but only for nodal domains which are big with respect to the wavelength scale. This work is also closely related to, and provides a new point of view on isoperimetric inequalities.Comment: 22 pages, 11 figure