On the gluing and ungluing of strange attractors in the study of the Lorenz system

The ungluing of a strange attractor, gluing of strange attractors, and the coexistence of strange attractors, not reported earlier in the study of the Lorenz system, are discovered numerically.

Singularity structure and chaotic behavior of the homopolar disk dynamo

We study in great detail a system of three first-order ordinary differential equations describing a homopolar disk dynamo (HDD). This system displays a large variety of behaviors, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The nonintegrability aspects of the HDD system are studied by investigating analytically the singularity structure of the system in the complex domain. Coexisting attractors (including period-doubling sequence) and coexisting strange attractors appear in some parametric regimes. The gluing of strange attractors and the ungluing of a strange attractor are also shown to occur. A period of bifurcation leading to chaos, not observed for other chaotic systems, is shown to characterize the chaotic behavior in some parametric ranges. The limiting case of the Lorenz system is also studied and is related to HDD.

Predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach

Perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one. Even though the sensitivity of initial conditions limits the predictability in chaotic systems, an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent. All of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions. Conceptually, global models are ineffective in modeling the highly unstable structure of the chaotic attractor. This paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters, i.e., embedding dimension and delay time to quantify the uncertainty in initial conditions. The ensemble approach is implemented through a local learning wavelet network model with a global feed-forward neural network structure for the phase space prediction of chaotic streamflow series. Quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times. The ensemble approach is proved to be 50% more efficient than the single prediction for both local approximation and wavelet network approaches. The wavelet network approach has proved to be 30%-50% more superior to the local approximation approach. Compared to the traditional local approximation approach with single initial condition, the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times. Localization property of wavelets, utilizing different dilation and translation parameters, helps in capturing most of the statistical properties of the observed data. The need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated.

Chaotic and power law states in the Portevin-Le Chatelier effect

Recent studies on the Portevin-Le Chatelier effect report an intriguing crossover phenomenon from low-dimensional chaotic to an infinite-dimensional scale-invariant power law regime in experiments on CuAl single crystals and AlMg polycrystals, as function of strain rate. We devise fully dynamical model which reproduces these results. At low and medium strain rates, the model is chaotic with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, power law statistics for the magnitudes and durations of the stress drops emerge as in experiments and concomitantly, the largest Lyapunov exponent is zero.

Dynamical behavior in the nonlinear rheology of surfactant solutions

Several surfactant molecules self-assemble in solution to form long, flexible wormlike micelles which get entangled with each other, leading to viscoelastic gel phases. We discuss our recent work on the rheology of such a gel formed in the dilute aqueous solutions of a surfactant CTAT. In the linear rheology regime, the storage modulus G′(ω) and loss modulus G″(ω) have been measured over a wide frequency range. In the nonlinear regime, the shear stress σ shows a plateau as a function of the shear rate math above a certain cutoff shear rate mathc. Under controlled shear rate conditions in the plateau regime, the shear stress and the first normal stress difference show oscillatory time-dependence. The analysis of the measured time series of shear stress and normal stress has been done using several methods incorporating state space reconstruction by embedding of time delay vectors. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent, unambiguously implying that the dynamics of the observed mechanical instability can be described by that of a dynamical system with a strange attractor of dimension varying from 2.4 to 2.9.

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L-2m-norms of vorticity D-m (1 <= m <= infinity). The first in this hierarchy, D-1, is the global enstrophy. Three regimes naturally occur in the D-1-D-m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime 1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

Attractive holographic c-functions

Using the attractor mechanism for extremal solutions in N = 2 gauged supergravity, we construct a c-function that interpolates between the central charges of theories at ultraviolet and infrared conformal fixed points corresponding to anti-de Sitter geometries. The c-function we obtain is couched purely in terms of bulk quantities and connects two different dimensional CFTs at the stable conformal fixed points under the RG flow.