Lagrangian chaotic saddles and objective vortices in solar plasmas.

We report observational evidence of Lagrangian chaotic saddles in plasmas, given by the intersections of finite-time unstable and stable manifolds, using an ≈22h sequence of spacecraft images of the horizontal velocity field of solar photosphere. A set of 29 persistent objective vortices with lifetimes varying from 28.5 to 298.3 min are detected by computing the Lagrangian averaged vorticity deviation. The unstable manifold of the Lagrangian chaotic saddles computed for ≈11h exhibits twisted folding motions indicative of recurring vortices in a magnetic mixed-polarity region. We show that the persistent objective vortices are formed in the gap regions of Lagrangian chaotic saddles at supergranular junctions.

Objective magnetic vortex detection.

Magnetic coherent vortical structures are ubiquitous in space and astrophysical plasmas and their detection is key to understanding the nature of the intrinsic turbulence in those conducting fluids. A recently developed method to detect magnetic vortices is explored in problems of two- and three-dimensional magnetohydrodynamic simulations. The integrated averaged current deviation, the normed difference of the current density at a point and the mean current density in the domain, integrated along a magnetic field line, is proved to be objective, i.e., invariant under rotations and translations of the observer. The method is shown to detect accurately the boundary of magnetic vortices in two-dimensional simulations, as well as magnetic flux ropes in three dimensions.

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.

Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation.

The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs.

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.

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.

Self-modulation of nonlinear Alfvén waves in a strongly magnetized relativistic electron-positron plasma.

We study the self-modulation of a circularly polarized Alfvén wave in a strongly magnetized relativistic electron-positron plasma with finite temperature. This nonlinear wave corresponds to an exact solution of the equations, with a dispersion relation that has two branches. For a large magnetic field, the Alfvén branch has two different zones, which we call the normal dispersion zone (where dω/dk>0) and the anomalous dispersion zone (where dω/dk<0). A nonlinear Schrödinger equation is derived in the normal dispersion zone of the Alfvén wave, where the wave envelope can evolve as a periodic wave train or as a solitary wave, depending on the initial condition. The maximum growth rate of the modulational instability decreases as the temperature is increased. We also study the Alfvén wave propagation in the anomalous dispersion zone, where a nonlinear wave equation is obtained. However, in this zone the wave envelope can evolve only as a periodic wave train.

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.

Self-modulation of nonlinear waves in a weakly magnetized relativistic electron-positron plasma with temperature.

We develop a nonlinear theory for self-modulation of a circularly polarized electromagnetic wave in a relativistic hot weakly magnetized electron-positron plasma. The case of parallel propagation along an ambient magnetic field is considered. A nonlinear Schrödinger equation is derived for the complex wave amplitude of a self-modulated wave packet. We show that the maximum growth rate of the modulational instability decreases as the temperature of the pair plasma increases. Depending on the initial conditions, the unstable wave envelope can evolve nonlinearly to either periodic wave trains or solitary waves. This theory has application to high-energy astrophysics and high-power laser physics.

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.

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.

Chaotic saddles at the onset of intermittent spatiotemporal chaos.

In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)], it has been shown that nonattracting chaotic sets (chaotic saddles) are responsible for intermittency in the regularized long-wave equation that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity and temporal chaos. In the present paper, it is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation. Prior to the onset of STC, a spatiotemporally chaotic saddle coexists with a spatially regular attractor. After the transition to STC, the chaotic saddle merges with the attractor, generating intermittent bursts of STC that dominate the post-transition dynamics.

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.

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.

Critical dynamic events at the crisis of transition to spatiotemporal chaos.

In the driven/damped drift-wave plasma system a collision of the weak chaotic attractor with a saddle point is demonstrated at the crisis that induces a transition from a spatially coherent state to spatiotemporal chaos (STC). The phenomenon of the collision is consistent with the previous observation of the "pattern resonance" that triggers the crisis. Subsequent to the collision, before the system is ejected to the STC attractor, there is evidence of another critical dynamic event involving state transition of a mode phase. The second event plays a crucial role in the destruction of the spatial coherence.

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

A new type of synchronization, on-off collective imperfect phase synchronization, is found in a turbulent state. In the driver frame the nonlinear wave system can be transformed to a set of coupled oscillators moving in a potential related to the unstable steady wave. In "on" stages the oscillators in different spatial scales adjust themselves to collective imperfect phase synchronization, inducing strong bursts in the wave energy. The interspike intervals display a power-law distribution. In addition to the embedded saddle point, it is emphasized that the delocalization of the master mode also plays an important role in developing the on-off synchronization.