Four-dimensional Phase Space Research Articles

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

Time-Symmetry Breaking in Hamiltonian Mechanics

Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates {q} satisfying Hamilton's motion equations will likewise satisfy them when played backwards, with the corresponding momenta changing signs : {+p} --> {-p}. Here we adopt Levesque and Verlet's precisely bit-reversible motion algorithm to ensure that the trajectory reversibility is exact, with the forward and backward sets of coordinates identical. Nevertheless, the associated instantaneous instability, or sensitive dependence on initial conditions of (or Lyapunov unstable) bit-reversible coordinate trajectories can still exhibit an exponentially growing time-symmetry-breaking irreversibility. Surprisingly, the positive and negative exponents, as well as the forward and backward spectra, are usually not closely related, and so give four differing topological measures of local chaos. We have demonstrated this symmetry breaking for fluid shockwaves, for free expansions, and for chaotic molecular collisions. Here we illustrate and discuss this time-symmetry breaking for three statistical-mechanical systems, [1] a minimal (but still chaotic) one-body cell model with a four-dimensional phase space; [2] relatively small colliding crystallites, for which the whole spectrum is accessible; [3] a near-continuum inelastic collision of two larger 400-particle balls. In the last two of these pedagogical problems the two colliding bodies coalesce. The particles most prone to instability are dramatically different in the two time directions. Thus this Lyapunov-based symmetry breaking furnishes an interesting Arrow of Time.

Controllable electrostatic surface guide for cold molecules with a single charged wire

We demonstrate a controllable highly efficient electrostatic surface guide for ND${}_{3}$ molecules in the weak-field-seeking states on a ceramic substrate over a distance of 840 mm, and study the dependences of the relative molecule number (or the overall transmission efficiency) of our single-wire guide and the guiding-center positions on the guiding voltages, both experimentally and theoretically. Our study shows that the guiding-center position and the number of the guided molecules can be easily controlled by adjusting the guiding voltages, and find that an overall transmission efficiency of higher than 50$%$ in a single quantum state can be obtained. Our experimental results are consistent with ones of Monte Carlo simulations. Also, we discuss the transverse velocity filtering effect and the acceptance of the guided molecules in four-dimensional phase space. Both the transmission efficiency and the acceptance in two-dimensional position space are higher than that in our previous two-wire guide [Y. Xia, Y. Yin, H. Chen, L. Deng, and J. Yin, Phys. Rev. Lett. 100, 043003 (2008)].

MULTI-PULSE HETEROCLINIC ORBITS AND CHAOTIC MOTIONS IN A PARAMETRICALLY EXCITED VISCOELASTIC MOVING BELT

This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for nonlinear, nonplanar oscillations of the parametrically excited viscoelastic moving belts by using an extended Melnikov method in the resonant case. Applying the method of multiple scales, the Galerkin's approach and the theory of normal form, the explicit normal form is obtained for the case of 1:1 internal resonance and primary parametric resonance. Studies are performed for the heteroclinic bifurcations of the unperturbed system and for the characteristics of the hyperbolic dynamics of the dissipative system, respectively. The extended Melnikov method is used to investigate the Shilnikov type multi-pulse bifurcations and chaotic dynamics of the viscoelastic moving belt. Based on the investigation, the geometric structure of the multi-pulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multi-pulse chaotic motions can occur. Furthermore, numerical simulations lead to the discovery of the new shapes of chaotic motion. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.

DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

ELECTROMAGNETIC EXCITATIONS OF HALL SYSTEMS ON FOUR-DIMENSIONAL SPACE

The noncommutativity of a four-dimensional phase space is introduced from a purely symplectic point of view. We show that there is always a coordinate map to locally eliminate the gauge fluctuations inducing the deformation of the symplectic structure. This uses the Moser's lemma; a refined version of the celebrated Darboux theorem. We discuss the relation between the coordinates change arising from Moser's lemma and the Seiberg–Witten map. As illustration, we consider the quantum Hall systems on CP2. We derive the action describing the electromagnetic interaction of Hall droplets. In particular, we show that the velocities of the edge field, along the droplet boundary, are noncommutativity parameters-dependents.

Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons

In this paper, we study the dynamics of a system of two model neurons interacting via the electrical synapse. Each neuron is described by a two-dimensional discontinuous map. A chaotic relaxational-type attractor, which corresponds to the spiking-bursting chaotic oscillations of neurons is shown to exist in a four-dimensional phase space. It is found that the dynamical mechanism of formation of chaotic bursts is based on a new phenomenon of generation of transient chaotic oscillations. It is demonstrated that transition from the chaotic-burst generation to the state of relative rest occurs with a certain time delay. A new characteristic which estimates the degree of synchronization of the spiking-bursting oscillations is introduced. The dependence of the synchronization degree on the strength of coupling of the ensemble elements is studied.

Characterization of an ArF excimer laser beam from measurements of the Wigner distribution function

An ArF excimer laser beam at a wavelength of 193 nm has been characterized by a quantitative determination of the Wigner distribution function. The setup, comprising a spherical lens, a rotating cylindrical lens and a moveable ultraviolet-sensitive CCD detector, enabled the mapping of the entire four-dimensional phase space within less than 20 min. Experiments yielded complete information about second-order moment-based parameters, spatial coherence, wavefront, beam profiles, as well as beam propagation and local distributions of radiant intensities.

A dynamical system study of the inhomogeneous Λ-CDM model

We consider spherically symmetric inhomogeneous dust models with a positive cosmological constant, Λ, given by the Lemaître–Tolman–Bondi metric. These configurations provide a simple but useful generalization of the Λ-CDM model describing cold dark matter (CDM) and a Λ term, which seems to fit current cosmological observations. The dynamics of these models can be fully described by scalar evolution equations that can be given in the form of a proper dynamical system associated with a four-dimensional phase space whose critical points and invariant subspaces are examined and classified. The phase space evolution of various configurations is studied in detail by means of two two-dimensional subspaces: a projection into the invariant homogeneous subspace associated with Λ-CDM solutions with FLRW metric, and a projection into a subspace generated by suitably defined fluctuations that convey the effects of inhomogeneity. We look at cases with perpetual expansion, bouncing and loitering behavior, as well as configurations with ‘mixed’ kinematic patters, such as a collapsing region in an expanding background. In all cases, phase space trajectories emerge from and converge to stable past and future attractors in a qualitatively analogous way as in the case of the FLRW limit. However, we can identify in both projections of the phase space various qualitative features absent in the FLRW limit that can be useful in the construction of toy models of astrophysical and cosmological inhomogeneities.

Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell

In this paper the global dynamics and topological integrity of the basins of attraction of a parametrically excited cylindrical shell are investigated through a two-degree-of-freedom reduced order model. This model, as shown in previous authors’ works, is capable of describing qualitatively the complex nonlinear static and dynamic buckling behavior of the shell. The discretized model is obtained by employing Donnell shallow shell theory and the Galerkin method. The shell is subjected to an axial static pre-loading and then to a harmonic axial load. When the static load is between the buckling load and the minimum post-critical load, a three potential well is obtained. Under these circumstances the shell may exhibit pre- and post-buckling solutions confined to each of the potential wells as well as large cross-well motions. The aim of the paper is to analyze in a systematic way the bifurcation sequences arising from each of the three stable static solutions, obtaining in this way the parametric instability and escape boundaries. The global dynamics of the system is analyzed through the evolution of the various basins of attraction in the four-dimensional phase space. The concepts of safe basin and integrity measures quantifying its magnitude are used to obtain the erosion profile of the various solutions. A detailed parametric analysis shows how the basins of the various solutions interfere with each other and how this influences the integrity measures. Special attention is dedicated to the topological integrity of the various solutions confined to the pre-buckling well. This allows one to evaluate the safety and dynamic integrity of the mechanical system. Two characteristic cases, one associated with a sub-critical parametric bifurcation and another with a super-critical parametric bifurcation, are considered in the analysis.

Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration

We demonstrated very recently [Lenz, New J. Phys. 11, 083035 (2009)] that an ensemble of particles in the driven elliptical billiard shows a surprising crossover from subdiffusion to normal diffusion in momentum space. This crossover is not parameter induced, but rather occurs dynamically in the evolution of the ensemble. In this work, we consider three different driving modes of the elliptical billiard and perform a comprehensive analysis of the corresponding four-dimensional phase space. The composition of this phase space is different in the high-velocity regime compared to the low-velocity regime. We will show, among others, by investigating periodic orbits and probability distributions of laminar phases that the stickiness properties, which eventually determine the diffusion, are intimately connected with this change in the composition of the phase space with respect to velocity. In the course of the evolution, the accelerating ensemble thus explores regions of varying stickiness, leading to the mentioned crossover in the diffusion.

Semiclassical propagation of Wigner functions

We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.

Recovery of particle distribution function in four-dimensional phase space from measurements with pepper-pot method

The possibility of recovering the particle distribution function at any point of the four-dimensional transverse phase volume “x, x′, y, y′” using the results of measurements with the pepper-pot method is demonstrated. The proposed method for processing experimental data is based on the assumption that the one-to-one correspondence between holes in the mask and their images on the screen is established. Such a preliminary calibration should be experimentally performed on the measured ion beam. The method of moments is used for the recovery of the distribution function. Using Legendre polynomials for processing experimental results reduces the problem to a sequence of recurrence relations. A numerical simulation of the measurement process demonstrated that the recovery accuracy of the particle distribution function is about ±10%.

Development of a pepper-pot device to determine the emittance of an ion beam generated by electron cyclotron resonance ion sources

This paper describes the recent development and commissioning of a pepper-pot emittance meter at the Lawrence Berkeley National Laboratory (LBNL). It is based on a potassium bromide (KBr) scintillator screen in combination with a charged coupled device camera. Pepper-pot scanners record the full four-dimensional transverse phase space emittances which are particularly interesting for electron cyclotron resonance ion sources. The strengths and limitations of evaluating emittances using optical pepper-pot scanners are described and systematic errors induced by the optical data acquisition system will be presented. Light yield tests of KBr exposed to different ion species and first emittance measurement data using ion beams extracted from the 6.4 GHz LBNL electron cyclotron resonance ion source are presented and discussed.

Study of a virus–bacteria interaction model in a chemostat: application of geometrical singular perturbation theory

This paper provides a mathematical analysis of a virus-marine bacteria interaction model. The model is a simplified case of the model published and used by Middelboe (Middelboe, M. 2000 Microb. Ecol. 40, 114-124). It takes account of the virus, the susceptible bacteria, the infected bacteria and the substrate in a chemostat. We show that the numerical values of the parameters given by Middelboe allow two different time scales to be considered. We then use the geometrical singular perturbation theory to study the model. We show that there are two invariant submanifolds of dimension two in the four-dimensional phase space and that these manifolds cross themselves on the boundary of the domain of biological relevance. We then perform a rescaling to understand the dynamics in the vicinity of the intersection of the manifolds. Our results are discussed in the marine ecological context.

Families of isospectral matrix Hamiltonians by deformation of the Clifford algebra on a phase space

By using a recently developed method, we report five different families of isospectral 2 × 2 matrix Hamiltonians defined on a four-dimensional (4D) phase space. The employed method is based on a realization of the supersymmetry idea on the phase space whose complexified Clifford algebra structure is deformed with the Moyal star-product. Each reported family comprises many physically relevant special models. 2D Pauli Hamiltonians, systems involving spin–orbit interactions such as Aharonov–Casher-type systems, a supermembrane toy model and models describing motion in noncentral electromagnetic fields as well as Rashba- and Dresselhaus-type systems from semiconductor physics are obtained, together with their super-partners, as special cases. A large family of isospectral systems characterized by the whole set of analytic functions is also presented.

Deformed Clifford algebra and supersymmetric quantum mechanics on a phase space with applications in quantum optics

In order to realize supersymmetric quantum mechanics methods on a four-dimensional classical phase space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase-space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal–Clifford algebra. Two illustrative examples corresponding to Jaynes–Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigenspinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization.

Evolutionary phase space in driven elliptical billiards

We perform the first long-time exploration of the classical dynamics of a driven billiard with a four-dimensional phase space. With increasing velocity of the ensemble, we observe an evolution from a large chaotic sea with stickiness due to regular islands to thin chaotic channels with diffusive motion leading to Fermi acceleration. As a surprising consequence, we encounter a crossover, which is not parameter induced but rather occurs dynamically, from amplitude-dependent tunable subdiffusion to universal normal diffusion in momentum space.

NESTED INVARIANT 3-TORI EMBEDDED IN A SEA OF CHAOS IN A QUASIPERIODIC FLUID FLOW

Nested invariant 3-tori surrounding a torus braid of elliptic type are found to exist in a model of a fluid flow with quasiperiodic forcing. The Hamiltonian describing the system is given by the superposition of two steady stream functions, one with an elliptic fixed point and the other with a coincident hyperbolic fixed point. The superposition, modulated by two incommensurate frequencies, yields an elliptic torus braid at the location of the fixed point. The system is suspended in a four-dimensional phase space (two space and two phase directions). To analyze this system we define two three-dimensional, global, Poincaré sections of the flow. The coherent structures (cross-sections of nested 2 tori) are found each to have a fractal dimensional of two, in each Poincaré cross-section. This framework has applications to tidal and other mixing problems of geophysical interest.