Erico L. Rempel

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.

Coherent structures and the saturation of a nonlinear dynamo

Eulerian and Lagrangian tools are used to detect coherent structures in the velocity and magnetic fields of a mean-field dynamo, produced by direct numerical simulations of the three-dimensional compressible magnetohydrodynamic equations with an isotropic helical forcing and moderate Reynolds number. Two distinct stages of the dynamo are studied: the kinematic stage, where a seed magnetic field undergoes exponential growth; and the saturated regime. It is shown that the Lagrangian analysis detects structures with greater detail, in addition to providing information on the chaotic mixing properties of the flow and the magnetic fields. The traditional way of detecting Lagrangian coherent structures using finite-time Lyapunov exponents is compared with a recently developed method called function M. The latter is shown to produce clearer pictures which readily permit the identification of hyperbolic regions in the magnetic field, where chaotic transport/dispersion of magnetic field lines is highly enhanced.

DETECTION OF COHERENT STRUCTURES IN PHOTOSPHERIC TURBULENT FLOWS

We study coherent structures in solar photospheric flows in a plage in the vicinity of the active region AR 10930 using the horizontal velocity data derived from Hinode/Solar Optical Telescope magnetograms. Eulerian and Lagrangian coherent structures (LCSs) are detected by computing the Q-criterion and the finite-time Lyapunov exponents of the velocity field, respectively. Our analysis indicates that, on average, the deformation Eulerian coherent structures dominate over the vortical Eulerian coherent structures in the plage region. We demonstrate the correspondence of the network of high magnetic flux concentration to the attracting Lagrangian coherent structures (aLCSs) in the photospheric velocity based on both observations and numerical simulations. In addition, the computation of aLCS provides a measure of the local rate of contraction/expansion of the flow.

Gradient pattern analysis of short nonstationary time series: an application to Lagrangian data from satellite tracked drifters

Based on the gradient pattern analysis (GPA) technique we introduce a new methodology for analyzing short nonstationary time series. Using the asymmetric amplitude fragmentation (AAF) operator from GPA we analyze Lagrangian data observed as velocity time series for ocean flow. The results show that quasi-periodic, chaotic and turbulent regimes can be well characterized by means of this new geometrical approach.

Gradient pattern analysis of Swift–Hohenberg dynamics: phase disorder characterization

In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.

Supertransient magnetohydrodynamic turbulence in Keplerian shear flows.

A subcritical transition to turbulence in magnetized Keplerian shear flows is investigated by using a statistical approach. Three-dimensional numerical simulations of the shearing box equations with zero net magnetic flux are employed to determine the transition from decaying to sustained turbulence as a function of the magnetic Reynolds number R{m}. The results reveal no clear transition to sustained turbulence as the average lifetime of the transients grows as an exponential function of R{m}, in accordance with a type-II supertransient law.

Characterization of local self-similarity and criticality in the solar active regions

From solar radio burst data we computed wavelet transforms and frequency distribution for investigation of self-similar temporal variability and power-laws, as the fundamental conditions for characterization of dynamical criticality (self or forced) in the solar active regions. The main result indicates that, as for the global activity, the local coronal magnetic field, in millisecond time scales, can be in a critical state where the dynamics of solar active regions works as avalanches of many small intermittent particle acceleration events.

Lagrangian coherent structures at the onset of hyperchaos in the two-dimensional Navier-Stokes equations

We study a transition to hyperchaos in the two-dimensional incompressible Navier-Stokes equations with periodic boundary conditions and an external forcing term. Bifurcation diagrams are constructed by varying the Reynolds number, and a transition to hyperchaos (HC) is identified. Before the onset of HC, there is coexistence of two chaotic attractors and a hyperchaotic saddle. After the transition to HC, the two chaotic attractors merge with the hyperchaotic saddle, generating random switching between chaos and hyperchaos, which is responsible for intermittent bursts in the time series of energy and enstrophy. The chaotic mixing properties of the flow are characterized by detecting Lagrangian coherent structures. After the transition to HC, the flow displays complex Lagrangian patterns and an increase in the level of Lagrangian chaoticity during the bursty periods that can be predicted statistically by the hyperchaotic saddle prior to HC transition.

On–off intermittency and amplitude-phase synchronization in Keplerian shear flows

We study the development of coherent structures in local simulations of the magnetorotational instability in accretion discs in regimes of on-off intermittency. In a previous paper [Chian et al., Phys. Rev. Lett. 104, 254102 (2010)], we have shown that the laminar and bursty states due to the on-off spatiotemporal intermittency in a one-dimensional model of nonlinear waves correspond, respectively, to nonattracting coherent structures with higher and lower degrees of amplitude-phase synchronization. In this paper we extend these results to a three-dimensional model of magnetized Keplerian shear flows. Keeping the kinetic Reynolds number and the magnetic Prandtl number fixed, we investigate two different intermittent regimes by varying the plasma beta parameter. The first regime is characterized by turbulent patterns interrupted by the recurrent emergence of a large-scale coherent structure known as two-channel flow, where the state of the system can be described by a single Fourier mode. The second regime is dominated by the turbulence with sporadic emergence of coherent structures with shapes that are reminiscent of a perturbed channel flow. By computing the Fourier power and phase spectral entropies in three-dimensions, we show that the large-scale coherent structures are characterized by a high degree of amplitude-phase synchronization.

Route to hyperchaos in Rayleigh-Bénard convection

Transition to hyperchaotic regimes in Rayleigh-Benard convection in a square periodicity cell is studied by three-dimensional numerical simulations. By fixing the Prandtl number at P=0.3 and varying the Rayleigh number as a control parameter, a bifurcation diagram is constructed where a route to hyperchaos involving quasiperiodic regimes with two and three incommensurate frequencies, multistability, chaotic intermittent attractors and a sequence of boundary and interior crises is shown. The three largest Lyapunov exponents exhibit a linear scaling with the Rayleigh number and are positive in the final hyperchaotic attractor. Thus, a route to weak turbulence is found.