Chaos, Complexity and Transport

The following sections are included:

• PREFACE
• CONTENTS

Operation of high performance fusion plasmas relies on self-organised properties to reach appropriate working points that are compatible with both high confinement performance to achieve a burning plasma, and controlled ageing of the confinement device. The latter conditions requires a trade-off between simplicity of the operation point and reaching conditions that can be sustained in steady state. The issue of heat flux control at the plasma edge and onto the plasma facing components is an example of this synergy. We address in this framework the problem of radiative divertor operation. The simplified 1D problem is recast in Hamiltonian formalism, the effective energy being invariant. This property is most efficient to address bifurcations and critical points leading to no-solution regions of the parameter space. Analytical investigation of these solutions indicates that taking into account the radiative front location and constraints on the upstream temperature reduces the operation space. Furthermore, one finds that radiative divertor operation tends to lead to operation at reduced plasma pressure, unless stable conditions and hot upstream plasma temperature can be sustained at vanishing divertor temperature.

Magnetic vortices are topologically stable magnetization configurations at the nanoscale. When subject to a current polarized in spin, exchange interaction with the itinerant electrons, modify the configuration of the fixed spins. The spin-transfer torque effect, as experimentally observed, is able to change the magnetization topology. We investigate the relation between the dynamics of magnetic vortices driven by a spin-polarized current and the changes in the magnetization topology. These transitions are characterized by the appearance of magnetic singularities.

Quantifying the dynamics of bacteria is a sinuous road with multiple challenges and subtleties. In this work, we take advantage of the natural evaporationdriven flow in a microscope sample to gain insight into the dynamics of motile bacteria in a unidirectional flow. We find that bacteria tend to align with or against the background flow. We also present in a didactic way every step of the study, with particular attention to explaining the cell tracking algorithm.

We consider low–dimensional dynamical systems with a mixed phase space and discuss the typical appearance of slow, polynomial decay of correlations: in particular we emphasize how this mixing rate is related to large deviations properties.

This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type trajectories, power-law convergence/divergence of trajectories, period doubling bifurcations with changes in the memory parameter, intersection of trajectories, and overlapping of attractors. In the limit of small time steps these maps converge to nonlinear fractional differential equations.

The distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincaré section, also called first-return time in dynamical systems community) are investigated numerically for 3D stationary chaotic mixers. Above all, we study the large tails of those distributions, and show that two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. Hence long time of flight correspond to particles that are stuck for a while in the vicinity of stable manifolds of fixed points. This property is then applied to the rate of decay of scalar energy in the case of diffusing species with global chaos. More precisely, we suggest that mixing in 3D-flows is governed by regions of slow rate of strain rather than by the Lyapunov exponent. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to -3. The model suggests that long times of flights correspond to particles stuck for a while in the vicinity of solid walls.

The Belousov-Zhabotinsky reaction embedded into water-in-oil reverse microemulsions (BZ-AOT system) is a well-known prototype for studying pattern formation. The diversity of self-organized structures observed in the BZ-AOT system, most of them unobtainable without the confined media, lies in the tunability of the medium. These complex patterns not only depend upon changes in the concentration of the BZ reactants but also modifications in the size and concentration of the droplets. Firstly, in this chapter we will give a general description of the BZ-AOT system. Then, we will present several fascinating self-organized patterns and, finally, we will describe a parameter that characterizes the morphology of the patterns.

We study the transport and diffusion of a scalar in a two-dimensional flow generated by the co-rotation of two cylinders and perturbed periodically by an oscillating cylinder. The flow is both stratified with a linear density gradient using salted water, and viscous in order to prevent Ekman pumping and centrifugal instabilities. The evolution of a blob of scalar injected close to the stagnation point is observed. First, the distance between the stable and the unstable manifolds is measured as the distance between the maximum and the minimum of the dye undulating pattern, and is compared successfully to the Melnikov function. Second, the diffusion of the stretched filament close to the hyperbolic point has been measured and validated theoretically. Thirdly, the fractal structure created by the flow and its homogenization by diffusion has been observed qualitatively at late stages.

The transport of passive tracers inside a two-dimensional differentially heated cavity is investigated numerically in the oscillatory regime, both in the vicinity and far from the corresponding Hopf bifurcation. Differential transport at low Rayleigh number is essentially linked to the exchange of fluid between two symmetric homoclinic tangles. A hierarchy of models is developed for the transient homogenisation process, the third order model reproducing well the observed behaviour. The vertical transport is efficient only at higher values of the Rayleigh number once all invariant tori have resonated. Signatures of non-hyperbolicity are shown by monitoring the variance of the coarse-grained concentration vs. time.

The navigation satellite constellations in medium-Earth orbit exist in a background of third-body secular resonances stemming from the perturbing gravitational effects of the Moon and the Sun. The resulting chaotic motions, emanating from the overlapping of neighboring resonant harmonics, induce especially strong perturbations on the orbital eccentricity, which can be transported to large values, thereby increasing the collision risk to the constellations and possibly leading to a proliferation of space debris. We show here that this transport is of a diffusive nature and we present representative diffusion maps that are useful in obtaining a global comprehension of the dynamical structure of the navigation satellite orbits.

A new chaos indicator Ultra Fast Lyapunov Indicator (UFLI) is proposed in this paper. UFLI uses second order derivatives and can distinguish chaotic orbits from torus orbits (orbits on a torus) more sensitively than FLI, OFLI and OFLI² in a short iterations for the standard map for δ = 0.9 and the Froeschlé map ε = 0.6 if we chose an initial variational vector properly. For the generalized Boole transformations (Boole transformation) whose analytic formula of Lyapunov exponent is given, the value of UFLI grows more rapidly than FLI except for α = 0.999, which is the weak chaos near to the infinite ergodic point while the difference in growing speed of the indicator values is relatively small compared with the standard map or the Froeschlé map.

We investigate the linear and the nonlinear responses to the external field in the mean-field interaction systems whose dynamics is described by the Vlasov equation in the limit of large population. A nonlinear response formula is derived by the asymptotic-transient decomposition, which is developed for analyzing the nonlinear Landau damping and plasma waves. The formula gives a self-consistent equation for the asymptotic state, which is in general not in thermal equilibrium but in a long-lasting nonequilibrium quasistationary state due to the long-range nature of the interaction. This formula is useful to compute the critical exponent analytically for the nonlinear response at the critical point, denoted by δ. Further advantage of this formula is that it includes the linear response formula obtained from the naively linearized equation, and unifies the nonlinear and the linear responses, where the latter induces another critical exponent γ for the zero-field susceptibility. Interestingly, these critical exponents for the quasistationary states differ from ones in thermal equilibria, but the Widom’s scaling relation holds true even for the non-classical critical exponents. We discuss universality of the critical exponents and the scaling relation for spatially periodic one-dimensional systems.

We study the effect of magnetic impurities in a two-dimensional topological insulator under voltage bias. The quantized conductance of this system is computed, and we study the influence of magnetic impurities coupling with charge carriers on the transition between phase with conductance (in fundamental units e²/h) of G = 2 (integer quantum spin Hall effect) and G = 1 (anomalous quantum Hall effect). We assume a ferromagnetic coupling between impurities and electron-like carrier, and two kind of coupling with hole-like. We show that the phase G = 1 exists for ferromagnetic hole-impurities coupling, in the strong coupling limit, in contrast with the prediction of the mean field approximation. This result is supported by direct numerical computations using Landauer transport formula, and by analytical calculations of the chemical potential and mass renormalization as a function of the disorder strength, in the self-consistent Born approximation. The transition is related to the suppression of one of the spin conduction channels, for strong enough disorder, by selective spin scattering and localization.

The Cahn-Hilliard-Oono equation describes the dynamics of phase separation process of co-polymers alloys and other binary systems. This process is driven by the minimization of the free energy. However the lower energy state in rarely reached and only for particular initial states. Misbah and Politi [1] have proposed a dynamical criterion which leads to final states that don’t correspond to those obtained by minimization of the free energy. In this article we investigate possible transition modes that may describe the dynamic of segregation and coalescence in those systems, in order to understand the difference between these criteria.

To detect and estimate the position and the velocity of moving objects, a passive radar system uses the signal transmitted by “illuminators-of-opportunity”, such as digital television transmitter. A new efficient method, based on the characteristics of digital telecommunication signals, is proposed. Results on simulated and real data are presented. Due to the low cost of the required hardware, experiments in real context could easily be considered for studying human activities like air, land or maritime traffic.

Metal electrodeposition systems exhibit tree-like structures with fractal character. Although such ’dendritic’ deposits are typically obtained by electrolysis, we can grow beautiful ramification patterns via a simple redox reaction. We present here a study of 1. silver (Ag) deposits from the reduction of Ag⁺ in (AgNO₃) solution by metallic copper and 2. copper (Cu) deposits from the reduction of Cu²⁺ by metallic zinc. The experiments are carried out in discotic geometry in a Petri dish. A wide variety of deposited structures and patterns, with different fractalities are obtained. We undertake the task of varying both the concentration of Ag⁺ and Cu²⁺ in the two sets of experiments. Whereas the Ag deposits showed little effect of concentration in the 2D spread, the Cu deposits exhibited a rich and varied spectrum of morphologies and textures.

The following section is included:

• AUTHOR INDEX