On-off collective imperfect phase synchronization and bursts in wave energy in a turbulent state

Multi-spectral optical imaging of the spatiotemporal dynamics of ionospheric intermittent turbulence

Equatorial plasma depletions have significant impact on radio wave propagation in the upper atmosphere, causing rapid fluctuations in the power of radio signals used in telecommunication and GPS navigation, thus playing a crucial role in space weather impacts. Complex structuring and self-organization of equatorial plasma depletions involving bifurcation, connection, disconnection and reconnection are the signatures of nonlinear evolution of interchange instability and secondary instabilities, responsible for the generation of coherent structures and turbulence in the ionosphere. The aims of this paper are three-fold: (1) to report the first optical imaging of reconnection of equatorial plasma depletions in the South Atlantic Magnetic Anomaly, (2) to investigate the optical imaging of equatorial ionospheric intermittent turbulence, and (3) to compare nonlinear characteristics of optical imaging of equatorial plasma depletions for two different altitudes at same times. We show that the degree of spatiotemporal complexity of ionospheric intermittent turbulence can be quantified by nonlinear studies of optical images, confirming the duality of amplitude-phase synchronization in multiscale interactions. By decomposing the analyses into North-South and East-West directions we show that the degree of non-Gaussianity, intermittency and multifractality is stronger in the North-South direction, confirming the anisotropic nature of the interchange instability. In particular, by using simultaneous observation of multi-spectral all-sky emissions from two different heights we show that the degree of non-Gaussianity and intermittency in the bottomside F-region ionosphere is stronger than the peak F-region ionosphere. Our results are confirmed by two sets of observations on the nights of 28 September 2002 and 9 November 2002.

Two-parameter bifurcation study of the regularized long-wave equation

We perform a two-parameter bifurcation study of the driven-damped regularized long-wave equation by varying the amplitude and phase of the driver. Increasing the amplitude of the driver brings the system to the regime of spatiotemporal chaos (STC), a chaotic state with a large number of degrees of freedom. Several global bifurcations are found, including codimension-two bifurcations and homoclinic bifurcations involving three-tori and the manifolds of steady waves, leading to the formation of chaotic saddles in the phase space. We identify four distinct routes to STC; they depend on the phase of the driver and involve boundary and interior crises, intermittency, the Ruelle-Takens scenario, the Feigenbaum cascade, an embedded saddle-node, homoclinic, and other bifurcations. This study elucidates some of the recently reported dynamical phenomena.

Edge of chaos and genesis of turbulence

The edge of chaos is analyzed in a spatially extended system, modeled by the regularized long-wave equation, prior to the transition to permanent spatiotemporal chaos. In the presence of coexisting attractors, a chaotic saddle is born at the basin boundary due to a smooth-fractal metamorphosis. As a control parameter is varied, the chaotic transient evolves to well-developed transient turbulence via a cascade of fractal-fractal metamorphoses. The edge state responsible for the edge of chaos and the genesis of turbulence is an unstable traveling wave in the laboratory frame, corresponding to a saddle point lying at the basin boundary in the Fourier space.

Wavelet-based multifractal analysis of nonlinear time series: the earthquake-driven tsunami of 27 February 2010 in Chile

We study general multifractal properties of tidal gauge and long-wave time series which show a well defined transition between two states, as is the case of sea level when a tsunami arrives. We adopt a method based on discrete wavelets, called wavelet leaders, which has been successfully used in a wide range of applications from image analysis to biomedical signals. First, we analyze an empirical time series of tidal gauge from the tsunami event of 27 February 2010 in Chile. Then, we study a numerical solution of the driven-damped regularized long-wave equation (RLWE) which displays on-off intermittency. Both time series are characterized by a sudden change between two sharply distinct dynamical states. Our analysis suggests a correspondence between the pre- and post-tsunami states (ocean background) and the on state in the RLWE, and also between the tsunami state (disturbed ocean) and the off state in the RLWE. A qualitative similarity in their singularity spectra is observed, and since the RLWE is used to model shallow water dynamics, this result could imply an underlying dynamical similarity.

Edge state and crisis in the Pierce diode

We study the chaotic dynamics of the Pierce diode, a simple spatially extended system for collisionless bounded plasmas, focusing on the concept of edge of chaos, the boundary that separates transient from asymptotic dynamics. We fully characterize an interior crisis at the end of a periodic window, thereby showing direct evidence of the collision between a chaotic attractor, a chaotic saddle, and the edge of chaos, formed by a period-3 unstable periodic orbit and its stable manifold. The edge of chaos persists after the interior crisis, when the global attractor of the system increases its size in the phase space.

Two-state on-off intermittency caused by unstable dimension variability in periodically forced drift waves

Certain high-dimensional dynamical systems present two or more attractors characterized by different energy branches. For some parameter values the dynamics oscillates between these two branches in a seemingly random fashion, a phenomenon called two-state on-off intermittency. In this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability. We characterize the parametric evolution of this variability using statistical properties of the finite-time Lyapunov exponents. As a model system that exhibits this behavior we consider periodically forced and damped drift waves. In this spatiotemporal example there is a low-dimensional chaotic attractor that is created by an interior crisis, already presenting unstable dimension variability.

Two-state on-off intermittency and the onset of turbulence in a spatiotemporally chaotic system

We show the existence of two-state on-off intermittent behavior in spatially extended dynamical systems, using as an example the damped and forced drift wave equation. The two states are stationary solutions corresponding to different wave energies. In the language of (Fourier-mode) phase space these states are embedded in two invariant manifolds that become transversely unstable in the regime where two-state on-off intermittency sets in. The distribution of laminar duration sizes is compatible with the similar phenomenon occurring in time only in the presence of noise. In an extended system the noisy effect is provided by the spatial modes excited by the perturbation. We show that this intermittency is a precursor of the onset of strong turbulence in the system.

Amplitude-phase synchronization at the onset of permanent spatiotemporal chaos

Amplitude and phase synchronization due to multiscale interactions in chaotic saddles at the onset of permanent spatiotemporal chaos is analyzed using the Fourier-Lyapunov representation. By computing the power-phase spectral entropy and the time-averaged power-phase spectra, we show that the laminar (bursty) states in the on-off spatiotemporal intermittency correspond, respectively, to the nonattracting coherent structures with higher (lower) degrees of amplitude-phase synchronization across spatial scales.

Experimental evidence of phase coherence of magnetohydrodynamic turbulence in the solar wind: GEOTAIL satellite data

Magnetohydrodynamic (MHD) turbulence is commonly observed in the solar wind. Nonlinear interactions among MHD waves are likely to produce finite correlation of the wave phases. For discussions of various transport processes of energetic particles, it is fundamentally important to determine whether the wave phases are randomly distributed (as assumed in the quasi-linear theory) or have a finite coherence. Using a method based on the surrogate data technique, we analysed the GEOTAIL magnetic field data to evaluate the phase coherence in MHD turbulence in the Earth's foreshock region. The results demonstrate the existence of finite phase correlation, indicating that nonlinear wave-wave interactions are in progress.

Property change of unstable fixed point and phase synchronization in controlling spatiotemporal chaos by a periodic signal

Mechanisms for the suppression of spatiotemporal chaos (STC) in one-dimensional driven drift-wave system to a spatially regular state by a periodic signal are investigated. In the driving wave coordinate, by transforming the system to a set of coupled oscillators (modes) moving in a periodic potential, it is found that the modes can be enslaved one by one through phase synchronization (PS) by the control signal; for some modes frequency-locking occurs while the other modes display multilooping PS without frequency-locking. Further study of the linear behavior of the modes shows that the saddle point embedded in the STC is changed to an unstable focus, which makes it possible for the imperfect PS to change to a perfect functional one, leading to the suppression of the STC.

Onset of spatiotemporal chaos in a nonlinear system

We describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state. The onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency.

Intermittent nature of solar wind turbulence near the Earth's bow shock: phase coherence and non-Gaussianity

The link between phase coherence and non-Gaussian statistics is investigated using magnetic field data observed in the solar wind turbulence near the Earth's bow shock. The phase coherence index Cphi, which characterizes the degree of phase correlation (i.e., nonlinear wave-wave interactions) among scales, displays a behavior similar to kurtosis and reflects a departure from Gaussianity in the probability density functions of magnetic field fluctuations. This demonstrates that nonlinear interactions among scales are the origin of intermittency in the magnetic field turbulence.

Origin of transient and intermittent dynamics in spatiotemporal chaotic systems

Nonattracting chaotic sets (chaotic saddles) are shown to be responsible for transient and intermittent dynamics in an extended system exemplified by a nonlinear regularized long-wave equation, relevant to plasma and fluid studies. As the driver amplitude is increased, the system undergoes a transition from quasiperiodicity to temporal chaos, then to spatiotemporal chaos. The resulting intermittent time series of spatiotemporal chaos displays random switching between laminar and bursty phases. We identify temporally and spatiotemporally chaotic saddles which are responsible for the laminar and bursty phases, respectively. Prior to the transition to spatiotemporal chaos, a spatiotemporally chaotic saddle is responsible for chaotic transients that mimic the dynamics of the post-transition attractor.

Riddling of the orbit in a high dimensional torus and intermittent energy bursts in a nonlinear wave system

The bifurcation scenario in a practically interesting nonlinear wave system is investigated by using a new scheme that is performed in a purely nonlinear wave framework with the Doppler effect taken into account. High-dimensional tori with three or four competing frequencies are observed. With variation of parameters a riddling of the orbit may occur intermittently due to tangency in certain dimensions, which temporarily spoils the toroidal topology and induces an energy burst where spatial coherence of the wave is lost.

Intermittency induced by attractor-merging crisis in the Kuramoto-Sivashinsky equation

We characterize an attractor-merging crisis in a spatially extended system exemplified by the Kuramoto-Sivashinsky equation. The simultaneous collision of two coexisting chaotic attractors with an unstable periodic orbit and its associated stable manifold occurs in the high-dimensional phase space of the system, giving rise to a single merged chaotic attractor. The time series of the post-crisis regime displays intermittent behavior. The origin of this crisis-induced intermittency is elucidated in terms of alternate switching between two chaotic saddles embedded in the merged chaotic attractor.