Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps

Area-preserving nontwist maps, i.e., maps that violate the twist condition, arise in the study of degenerate Hamiltonian systems for which the standard version of the Kolmogorov-Arnold-Moser (KAM) theorem fails to apply. These maps have found applications in several areas including plasma physics, fluid mechanics, and condensed matter physics. Previous work has limited attention to maps in 2-dimensional phase space. Going beyond these studies, in this paper, we study nontwist maps with many-degrees-of-freedom. We propose a model in which the different degrees of freedom are coupled through a mean-field that evolves self-consistently. Based on the linear stability of period-one and period-two orbits of the coupled maps, we construct coherent states in which the degrees of freedom are synchronized and the mean-field stays nearly fixed. Nontwist systems exhibit global bifurcations in phase space known as separatrix reconnection. Here, we show that the mean-field coupling leads to dynamic, self-consistent reconnection in which transport across invariant curves can take place in the absence of chaos due to changes in the topology of the separatrices. In the context of self-consistent chaotic transport, we study two novel problems: suppression of diffusion and breakup of the shearless curve. For both problems, we construct a macroscopic effective diffusion model with time-dependent diffusivity. Self-consistent transport near criticality is also studied, and it is shown that the threshold for global transport as function of time is a fat-fractal Cantor-type set.     

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of $\text{1}\text{δ}\text{=}\text{1}\text{8.721097200}$…, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = ?4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.     

Natural boundaries for area-preserving twist maps

We consider KAM invariant curves for generalizations of the standard map of the form (x, y)=(x+y, y+f(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation +. In the complex plane, natural boundaries of different shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as tends to the critical value.     

Correlations of periodic,area-preserving maps

A simple analytical decay law for correlation functions of periodic, area-preserving maps is obtained. This law is compared with numerical experiments on the standard map. The agreement between experiment and theory is good when islands are absent, but poor when islands are present. When islands are present, the correlations have a long, slowly decaying tail.     

A semiclassical quantization of area-preserving maps

A semiclassical path integral formalism is developed for a class of area-preserving maps. The quasi-energy spectrum of these systems is represented as a sum over closed orbits in the manner of Gutzwiller. The method is illustrated with some numerical tests of the “standard map”.     

On universality for area-preserving maps of the plane

We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form

$\text{Λ}{}^{\mathrm{?1}}\text{°Φ°Φ°Λ = Φ}$

and

$\text{det}\text{DΦ = 1}$

, Φ:$\text{R}$²→$\text{R}$². In particular we argue that the scaling transformation Λ:$\text{R}$²→$\text{R}$² is conjugate to the transformation Λ₀:(x, y)→(λx, μy), with λ² ≠ μ, and in fact λ² >μ. We present some numerical evidence that

$\text{δ = 8.721}$

…,

$\text{?}\text{1}\text{λ}\text{= 4.018}$

…,

$\text{1}\text{μ}\text{= 16.36}$

…, where δ is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2^k described above.     

Invariant curves for area-preserving twist maps far from integrable

One-parameter families of area-preserving twist maps of the formF (x, y)=(x +y +f(x),y +f(x)) are considered. Various invariant curves, for the maps corresponding tof(x)=sin andf(x)=sinx+(1/50) sin(5x), are rigorously constructed forlarge values of the nonlinearity parameter . For larger values of , close to critical, some numerical experiments are briefly discussed.     

Trajectory scaling function for bifurcations in area-preserving maps on the plane

The trajectory scaling function for area-preserving maps on the plane is found using a calculation of the unstable manifold for the renormalization group operator R·T=Λ·T²·Λ^-1 with $\text{Λ=}\text{α 0}\text{0 β}$. Internal self-similarities of high order cycles and of power spectra are deduced.     

Broad resonances in transport theory

We discuss the extension of the transport theoretical framework to include states with a broad mass distribution. We focus on the proper life-time and cross sections for a state with an arbitrarily given invariant mass.     

Detecting resonances in conservative maps using evolutionary algorithms

A numerical method is proposed for detecting resonances of conservative maps which reduces this task to an optimization problem. We then solve this problem using evolutionary algorithms, which are methods for global optimization inspired by biological evolution. The proposed methodology is simple and can be easily applied to maps of arbitrary dimensions. In this Letter we apply it to several examples of 2- and 4-dimensional conservative maps, with quite promising results concerning integrability, the location of resonances and the presence of chaotic regions surrounding the island chains that correspond to these resonances.     

Frobenius-perron resonances for maps with a mixed phase space

Resonances of the time evolution (Frobenius-Perron) operator P for phase space densities have recently been shown to play a key role for the interrelations of classical, semiclassical, and quantum dynamics. Efficient methods to determine resonances are thus in demand, in particular, for Hamiltonian systems displaying a mix of chaotic and regular behavior. We present a powerful method based on truncating P to a finite matrix which not only allows us to identify resonances but also the associated phase space structures. It is demonstrated to work well for a prototypical dynamical system.     

Universality of k·3n and k·4n bifurcations in area-preserving maps

We study period-trebling and period-quadrupling bifurcations in two-dimensional reversible area-preserving maps. Our numerical results show that there are unique universal limiting behaviors in each of the period-trebling and period-quadrupling sequences.     

Imaging transport resonances in the quantum Hall effect

We use a scanning capacitance probe to image transport in the quantum Hall system. Applying a dc bias voltage to the tip induces a ring-shaped incompressible strip (IS) in the 2D electron system (2DES) that moves with the tip. At certain tip positions, short-range disorder in the 2DES creates a quantum dot island in the IS. These islands enable resonant tunneling across the IS, enhancing its conductance by more than 4 orders of magnitude. The images provide a quantitative measure of disorder and suggest resonant tunneling as the primary mechanism for transport across ISs.     

Correlation-induced resonances in transport through coupled quantum dots

We investigate the effect of local electron correlations on transport through parallel quantum dots. The linear conductance as a function of gate voltage is strongly affected by the interplay of the interaction U and quantum interference. We find a pair of novel correlation-induced resonances separated by an energy scale that depends exponentially on U. The effect is robust against a small detuning of the dot energy levels and occurs for arbitrary generic tunnel couplings. It should be observable in experiments on the basis of presently existing double-dot setups.     

Effect of continuum resonances on electronic transport in quantum wells

We investigate the influence of continuum resonances on the release of electrons by a quantum well. We find that the inclusion of resonance effects leads to a decrease in the rate of scattering by non-polar phonons from bound confined states to unbound continuum states.