Edge of chaos and genesis of turbulence

Homoclinic bifurcation and switching of edge state in plane Couette flow

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

Complexity and transition to chaos in coupled Adler-type oscillators

Coupled nonlinear oscillators are ubiquitous in dynamical studies. A wealth of behaviors have been found mostly for globally coupled systems. From a complexity perspective, less studied have been systems with local coupling, which is the subject of this contribution. The phase approximation is used, as weak coupling is assumed. In particular, the so-called needle region, in parameter space, for Adler-type oscillators with nearest neighbors coupling is carefully characterized. The reason for this emphasis is that, in the border of this region to the surrounding chaotic one, computation enhancement at the edge of chaos has been reported. The present study shows that different behaviors within the needle region can be found and a smooth change of dynamics could be identified. Entropic measures further emphasize the region's heterogeneous nature with interesting features, as seen in the spatiotemporal diagrams. The occurrence of wave-like patterns in the spatiotemporal diagrams points to nontrivial correlations in both dimensions. The wave patterns change as the control parameters change without exiting the needle region. Spatial correlation is only achieved locally at the onset of chaos, with different clusters of oscillators behaving coherently while disordered boundaries appear between them.

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.

The Continuum Between Temperament and Mental Illness as Dynamical Phases and Transitions

The full range of biopsychosocial complexity is mind-boggling, spanning a vast range of spatiotemporal scales with complicated vertical, horizontal, and diagonal feedback interactions between contributing systems. It is unlikely that such complexity can be dealt with by a single model. One approach is to focus on a narrower range of phenomena which involve fewer systems but still cover the range of spatiotemporal scales. The suggestion is to focus on the relationship between temperament in healthy individuals and mental illness, which have been conjectured to lie along a continuum of neurobehavioral regulation involving neurochemical regulatory systems (e.g., monoamine and acetylcholine, opiate receptors, neuropeptides, oxytocin), and cortical regulatory systems (e.g., prefrontal, limbic). Temperament and mental illness are quintessentially dynamical phenomena, and need to be addressed in dynamical terms. A meteorological metaphor suggests similarities between temperament and chronic mental illness and climate, between individual behaviors and weather, and acute mental illness and frontal weather events. The transition from normative temperament to chronic mental illness is analogous to climate change. This leads to the conjecture that temperament and chronic mental illness describe distinct, high level, dynamical phases. This suggests approaching biopsychosocial complexity through the study of dynamical phases, their order and control parameters, and their phase transitions. Unlike transitions in physical systems, these biopsychosocial phase transitions involve information and semiotics. The application of complex adaptive dynamical systems theory has led to a host of markers including geometrical markers (periodicity, intermittency, recurrence, chaos) and analytical markers such as fluctuation spectroscopy, scaling, entropy, recurrence time. Clinically accessible biomarkers, in particular heart rate variability and activity markers have been suggested to distinguish these dynamical phases and to signal the presence of transitional states. A particular formal model of these dynamical phases will be presented based upon the process algebra, which has been used to model information flow in complex systems. In particular it describes the dual influences of energy and information on the dynamics of complex systems. The process algebra model is well-suited for dealing with the particular dynamical features of the continuum, which include transience, contextuality, and emergence. These dynamical phases will be described using the process algebra model and implications for clinical practice will be discussed.

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.

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.