Gyroaverage effects on nontwist Hamiltonians: Separatrix reconnection and chaos suppression

A study of finite Larmor radius (FLR) effects on E × B test particle chaotic transport in non-monotonic zonal flows with drift waves in magnetized plasmas is presented. Due to the non-monotonicity of the zonal flow, the Hamiltonian does not satisfy the twist condition. The electrostatic potential is modeled as a linear superposition of a zonal flow and the regular neutral modes of the Hasegawa-Mima equation. FLR effects are incorporated by gyro-averaging the E × B Hamiltonian. It is shown that there is a critical value of the Larmor radius for which the zonal flow transitions from a profile with one maximum to a profile with two maxima and a minimum. This bifurcation leads to the creation of additional shearless curves and resonances. The gyroaveraged nontwist Hamiltonian exhibits complex patterns of separatrix reconnection. A change in the Larmor radius can lead to heteroclinic-homoclinic bifurcations and dipole formation. For Larmor radii for which the zonal flow has bifurcated, double heteroclinic-heteroclinic, homoclinic-homoclinic and heteroclinic-homoclinic separatrix topologies are observed. It is also shown that chaotic transport is typically reduced as the Larmor radius increases. Poincare sections show that, for large enough Larmor radius, chaos can be practically suppressed. In particular, changes of the Larmor radius can restore the shearless curve.     

Nonlinear reflecting diffusion process,and the propagation of chaos and fluctuations associated

A nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven. Then a gaussian limit for the fluctuation field which is defined in L₀²(B) of a Wiener type space B is obtained. The covariance of the gaussian limit is computed in terms of a Hilbert-Schmidt operator on L₀²(B).     

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process ?? on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of ??. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener?CIt? chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris?CFKG-inequalities for monotone functions of ??.     

Identification of chaos in a regenerative cutting process by the 0-1 test

We examine the regenerative cutting process by using a single degree of freedom non-smooth model with a friction component and a time delay term. Instead of the standard Lyapunov exponent calculations, we propose a statistical 0-1 test for chaos. This approach reveals the nature of the cutting process signaling regular or chaotic dynamics. We are able to show that regular or chaotic motion occur in the investigated model depending on the delay time. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear processes in Hamiltonian reconnection

The generation of small spatial scales and their interplay with large scale coherent structures is one of the outstanding phenomena of plasma physics and fluid mechanics. In high temperature space and laboratory plasmas dissipative effects become important at length scales that are much smaller than those where microscopic dynamical effects, related e.g., to electron inertia, come into play. Here we discuss the role of this dissipationless small scale dynamics on the nonlinear evolution of collisionless magnetic reconnection within the framework of the so called “two-field” and “four-field models”.     

Identification of chaos in a cutting process by the 0–1 test

We have examined the cutting process by using a two degrees of freedom non-smooth model with a friction component. Instead of the standard Lyapunov exponent treatment a statistical ‘0–1’ test based on the asymptotic properties of a non-harmonic Brownian motion chain has been successively applied to reveal the nature of the cutting process. In this test we calculated the control parameter K which is approaching asymptotically to 0 or 1 for regular and chaotic motions, respectively. The presented approach is independent on the integration procedure as we defined a characteristic distance between the points forming the time series used in the test separately.     

On the scenario of reconnection in non-twist cubic maps

In this paper, we study the reconnection process in the dynamics of cubic non-twist maps, introduced in [Howard JE, Humpherys J. Nonmonotonic twist maps. Physica D 1995; 256–76]. In order to describe the route to reconnection of the involved Poincaré–Birkhoff chains we investigate an approximate interpolating Hamiltonian of the map under study. Our study reveals that the scenario of reconnection of cubic non-twist maps is different from that occurring in the dynamics of quadratic non-twist maps.     

Minimizing the delay at traffic lights

Vehicles holding at traffic lights is a typical queuing problem.At crossings the vehicles experience delay in both directions.Longer periods with green lights in one direction are disadvantageousfor the vehicles coming from the other direction. The totaldelay for getting through the traffic point is what counts.This article presents an expression to calculate the optimaltime periods of red lights and green lights starting from afixed-cycle time. The solution is optimal if it makes the trafficjam delay at the road crossing minimal. As these solutions dependon the number of cars arriving in the different directions,which is not constant during the day, the application can beenlarged to a system where the time periods of red and greenlights change during the day.     

Simulation of turbulent magnetic reconnection in the small-scale solar wind

Some observational examples for the possible occurrence of the turbulent magnetic reconnection in the solar wind are found by analysing Helios spacecraft's high resolution data. The phenomena of turbulent magnetic reconnections in small scale solar wind are simulated by introducing a third order accuracy upwind compact difference scheme to the compressible two_dimensional MHD flow. Numerical results verify that the turbulent magnetic reconnection process could occur in small scale interplanetary solar wind, which is a basic feature characterizing the magnetic reconnection in high_magnetic Reynolds number (RM=2 000-10 000) solar wind. The configurations of the magnetic reconnection could evolve from a single X_line to a multiple X-line reconnection, exhibiting a complex picture of the formation, merging and evolution of magnetic islands, and finally the magnetic reconnection would evolve into a low_energy state. Its life_span of evolution is about one hour order of magnitude. Various magnetic and flow signatures are recorded in the numerical test for different evolution stages and along different crossing paths, which could in principle explain and confirm the observational samples from the Helios spacecraft. These results are helpful for revealing the basic physical processes in the solar wind turbulence.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

Comments on the non-stationary chaos

Non-stationary chaos is a universal phenomenon in non-hyperbolic dynamical systems. Basic problems regarding the non-stationarity are discussed from ergodic-theoretical viewpoints. By use of a simple system, it is shown that “the law of large number” as well as “the law of small number” break down in the non-stationary regime. The non-stationarity in dynamical systems proposes a crucial problem underlying in the transitional region between chance and necessity, where non-observable processes behind reality interplay with observable ones. The incompleteness of statistical ensembles is discussed from the Karamata's theory. Finally, the significance of the stationary/non-stationary interface is emphasized in relation to the universality of 1/f fluctuations.     

On the genericity of chaos

Let M be a compact manifold with dimM?2. We prove that some iteration of the generic homeomorphism on M is semiconjugated to the shift map and has infinite topological entropy (Theorem 1.1).     

Minimizing Fusion Frame Potential

Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the desired number of fusion frame subspaces is large, and if the desired dimension of these subspaces is small compared to the dimension of the underlying space, then a tight fusion frame of those dimensions will necessarily exist, being a minimizer of the fusion frame potential.     

Turbulent chaos

A new definition of a chaotic invariant set for a continuous semi-flow in a metric space is given. This definition generalizes the known definition of Devaney and allows us to take into account a specific feature arising in a noncompact and infinite-dimensional case, the so-called turbulent chaos.     

Minimizing the sum of many rational functions

We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of variables can be relatively large (10 to 100) provided some kind of sparsity is present. We describe a formulation of the rational optimization problem as a generalized moment problem and its hierarchy of convex semidefinite relaxations. Under some conditions we prove that the sequence of optimal values converges to the globally optimal value. We show how public-domain software can be used to model and solve such problems. Finally, we also compare with the epigraph approach and the BARON software.     

Minimizing the Flow-time Variance in Single-machine Systems

In this paper we deal with an n-job, single-machine scheduling problem. All jobs are available from the start, and the objective is to minimize the variance of job flow-times. A heuristic procedure which is based on the complementary pair-exchange principle is proposed. It has been concluded that this heuristic procedure provides improved results (in terms of objective-function value) when compared with other heuristics. Our heuristic procedure has the complexity of O(n log n).     

Minimizing the overhead for some tree-scheduling problems

This paper is devoted to the study of tree-scheduling problems within the execution model described by Anderson, Beame and Ruzzo. We first prove the NP-completeness of the problem of minimizing the overhead for scheduling trees on m processors, and then we propose an algorithm that provides optimal schedules when complete trees are considered.