Unstable periodic orbits

Spatially Localized Unstable Periodic Orbits

Using an innovative damped-Newton method, we report the first calculation of many distinct unstable periodic orbits (UPOs) of a large high-dimensional extensively chaotic system. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov fractal dimension D = 8.8 can be es...

Unstable Periodic Orbits in Turbulence

Recently both shear turbulence and isotropic turbulence have been investigated by means of unstable periodic orbits. These orbits are embedded in a high dimensional chaotic attractor that represents the turbulent flow. In both cases, the periodic motion can be shown to bear a strong similarity to the turbulent motion. Therefore we can learn about the nature of turbulence by studying the periodi...

Locating unstable periodic orbits in recurrence plots

Recurrence plots and unstable periodic orbits.

A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geo...

Unstable periodic orbits in the Lorenz attractor.

We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computatio...