Phase Space Of Solutions Research Articles

A geophysically constrained large ensemble analysis of the deglacial history of the North American ice-sheet complex

Past reconstructions of the deglaciation history of the North American (NA) ice-sheet complex have relied either on largely unconstrained and limited explorations of the phase space of solutions produced by glaciological models or upon geophysical inversions of relative sea-level (RSL) data which suffer from incomplete geographic coverage of the glaciated regions, load history amplitude/timing ambiguities, and a lack of a priori glaciological self-consistency. As a first step in the development of a much more highly constrained deglaciation history, we present a synthesis of these two previously disjoint methodologies based on a large ensemble of glacial cycle simulations using a three-dimensional thermo-mechanically coupled ice-sheet model. Twenty glacial system model parameters, chosen so as to best cover the true deglacial phase space, were varied across the ensemble. Furthermore, a new high-resolution digitized ice margin chronology was imposed on the model in order to significantly limit the uncertainties associated with deglacial climate forcing. The model is simultaneously constrained by a large set of high-quality RSL histories, a space geodetic observation of the present-day rate of vertical motion of the crust from Yellowknife and a traverse of absolute gravity measurements from the west coast of Hudson Bay southward into Iowa. The general form of the Last Glacial Maximum (LGM) ice topography that ensues when model results are subject to geophysical constraints is an ice sheet dominated by a large (3.3– 4.3 km maximum ice thickness) Keewatin dome to the west of Hudson Bay connected to a major ice ridge running southeast to the Great Lakes, together with a Hudson Bay region that has relatively thin ice and an Arctic region heavily incised by open water and/or ice shelves. Geographically restricted fast flows due to sub-glacial till deformation are shown to be critical to obtaining such a multi-domed late glacial Laurentide Ice Sheet structure, one that has been previously inferred on the basis of geomorphological data and that is required to fit the geophysical constraints. Our results further suggest that the NA contribution to LGM eustatic sea-level drop is likely to be in the range of 60– 75 m .

Fast-phase space computation of multiple arrivals.

We present a fast, general computational technique for computing the phase-space solution of static Hamilton-Jacobi equations. Starting with the Liouville formulation of the characteristic equations, we derive "Escape Equations" which are static, time-independent Eulerian PDEs. They represent all arrivals to the given boundary from all possible starting configurations. The solution is numerically constructed through a "one-pass" formulation, building on ideas from semi-Lagrangian methods, Dijkstra-like methods for the Eikonal equation, and Ordered Upwind Methods. To compute all possible trajectories corresponding to all possible boundary conditions, the technique is of computational order O(N log N), where N is the total number of points in the computational phase-space domain; any particular set of boundary conditions then is extracted through rapid post-processing. Suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance. As an application, we apply the technique to the problem of computing first, multiple, and most energetic arrivals to the Eikonal equation.

Aspects of the Quasiclassical t-Matrix Boundary Condition

We apply the t-matrix boundary conditions of Rainer to study the critical current of superfluid 3He confined to a thin channel between realistic rough surfaces using quasiclassical techniques. Rainer's conditions lead directly to a simple, stable and comprehensive algorithm for solving the quasiclassical equations which we present in detail. The conditions yielded physical results in all cases studied and their ease of use permitted an exhaustive search through the phase space of solutions. A robust metastable state appears for narrow channel widths suggesting the possibility of two different critical currents.

Tracking unstable steady states by large-amplitude low-frequency periodic modulation of a control parameter: Phase-space analysis

Inhibition of chaos in a dissipative nonlinear system that is slowly (nonresonantly) modulated across an instability domain of a fixed-point solution is investigated in detail, considerably extending previous analyses. Comparison is made between the evolution of the modulated system and the evolution of the steady-state solution in phase space as a function of the modulation parameter. It is shown that tracking of the steady-state solution across the instability domain (which can be achieved for a wide domain of modulation frequencies) occurs in a nonintuitive way, as a result of the combination of two factors, which can be present in many nonlinear systems.

Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations

A method is presented for the exact and complete resolution of Hamiltonian systems only using the corresponding Hamilton-Jacobi equation (we show that it possible to get the ODE solution from the corresponding PDE).

Quantum manifestations of classical chaos in a Fermi accelerating disk

We study the classical and quantum mechanics of a free particle that collides elastically with the walls of a circular disk with the radius varying periodically in time. The quasi-energy spectral properties of the model are obtained from evaluation of finite-dimensional approximations to the time evolution operator. As the scaled hbar is changed from large to small, the statistics of the Quasienergy Eigenvalues (QEE) change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the Quasienergy Eigenfunctions (QEF) is also studied using the chi-squared test with nu degrees of freedom, which goes over to the Porter-Thomas distribution for nu=1. We find that the integrable regime is associated with exponentially localized QEF whereas the eigenfunctions are extended in the chaotic, COE regime. We change the representation of the model to one in which the boundary is fixed and the Hamiltonian acquires a quadratic forcing term. We then carry out a successful comparison between specific classical phase space solutions and their corresponding QEFs in the Husimi representation.

Soluitions of nonlinear partial differential equations in phase space

In this paper we consider the problem of constructing solutions of several well known nonlinear partial differential equations (p.d.e.s) in phase space (i.e,. the Fourier transform domain). We seek solutions representing travelling focussed pulses. As such, based on a technique used to construct such solutions (so called Localized Wave solutions) of linear p.d.e.s, we look for phase space solutions constisting of a generalized funtion whose support is a particular line or surface, together with a suitable weighting function. The support of the phase space solutions must be such that it regenerates itself after the appropriate nonlinear operation. In one spatial dimension we construct the usual well known soliton solutions of several equations. For the case of higher spatial dimensions we construct a travelling “slab” pulse solution of the nonlinear Schrödinger equation. We also discuss some issues involved with the extra freedom one has for the phase space support, leading perhaps to more exotic spacetime domain solutions.

Behaviors of Variational and Nudging Assimilation Techniques with a Chaotic Low-Order Model

Abstract Variational and nudging data-assimilation schemes are examined within the framework of a model initialization problem using the Lorenz three-component model of Rayleigh–Benard convection. Since the intent of this study is to explore what factors influence the abilities of the two assimilation techniques to produce accurate initial conditions, identical twin experiments are conducted for various lengths of the data-assimilation window when the flow is both study state and chaotic. These experiments illustrate that the location of the model solution in phase space is an important consideration when applying either data-assimilation scheme. Decision points, when the model “chooses” which stationary point to orbit, are found to affect the ability of the assimilation techniques to find an accurate initial condition. For chaotic flow, variational assimilation produces a better initial condition when the assimilation window is short, while nudging produces a better initial condition when the assimilatio...

Finite-amplitude baroclinic instability with periodic forcing

A low-order model is used to study the behavior of large-scale planetary waves arising from the baroclinic instability of an east-west zonal wind that includes a weak modulation in time. The finite-amplitude equilibration model is equivalent to the modulated Lorenz equations, containing an externally forced time-periodic Rayleigh parameter. Without modulation the baroclinic instabilities equilibrate to steady waves for all values of the external parameters reasonably near the linear neutral curve. However, a small modulation with a period of the same order as the internal dynamical timescale can completely destabilize the non-linear system, leading to chaotic solutions even for small supercriticality. Low-frequency seasonal modulations give rise to tori in the solution phase space. As the amplitude of the periodic forcing increases to about 10% of the average zonal wind shear, the tori break up and chaotic orbits are again found. This latter behavior is interpreted by studying the effect of seasonal forcing on the adiabatic invariants of the system.

Initial behavior of solutions to the lorenz equations

The early behavior of the solutions to the Lorenz equations is examined for initial conditions far from equilibrium and for large values of the parameter r. The equations are first transformed so that the equilibrium points become independent of r. The initial trajectory of the solution in phase space is seen to be quite different from and essentially independent of the long time behavior. It can be divided into two stages. For short times an approximate non-linear conservative set of equations with exact integrals is obtained which gives an accurate frequency representation of the exact solution. Damping can also be added to this system. For intermediate times, the equations can be approximated by a Bessel equation of high order. The present work is motivated by the one-dimensional modeling of a toroidal natural circulation thermosyphon loop and a physical interpretation of the results in this context is also discussed.

Phase-space solution of the linear mode-conversion problem

The problem of linear mode conversion in a weakly inhomogeneous medium is posed and solved by phase-space methods. The PDEs for the two coupled modes are transformed to a simple first-order ordinary differential equation by a canonical transformation, wherein the two dispersion functions become essentially a locally conjugate pair of coordinates.

Quantisation of limit cycles in a P representation of a dissipative driven anharmonic oscillator

The non-linear polarisability model of dispersive optical bistability with periodic driving field is shown to have non-trivial limit cycle solutions in phase space. The model is quantised rigorously in the positive P representation. For a particular (but typical) limit cycle, the associated steady state probability distribution is shown to be compatible with a bivariate Gaussian. The inverse widths of the Gaussian are calculated as functions of position in both phase space and time.

The stochastic quantum mechanics approach to the unification of relativity and quantum theory

The stochastic phase-space solution of the particle localizability problem in relativistic quantum mechanics is reviewed. It leads to relativistically covariant probability measures that give rise to covariant and conserved probability currents. The resulting particle propagators are used in the formulation of stochastic geometries underlying a concept of quantum spacetime that is operationally based on stochastically extended quantum test particles. The epistemological implications of the intrinsic stochasticity of such quantum spacetime frameworks for microcausality, the EPR paradox, etc., are discussed.

Existence of Travelling Wave Solutions of Predator–Prey Systems via the Connection Index

We consider a reaction–diffusion system which arises as a model of predator–prey interactions in mathematical ecology. The system admits four constant equilibria, two of which are stable. The main result is an existence theorem for travelling wave solutions which connect the two stable equilibria. Most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit. The new aspect of the present discussion lies in devising some novel methods for “locating” the connecting solution in phase space which do not require that the equations be singularly perturbed. In certain cases, the main result holds for an arbitrary choice of diffusion coefficients.

Researches on the restricted three-body problem

This work considers periodic solutions and arc-solutions (solutions with consecutive collisions) of the plane circular restricted problem of three bodies for μ=0. To study the mutual arrangement of these solutions in phase space, we introduce a global section Γ of the phase space. Each solution is represented in Γ by some points (at least one), and families of solutions are represented by curves (the characteristics). We give a concrete description of the arrangement of the characteristics in the three-dimensional space Γ. The solutions studied here play a fundamental role in generating periodic solutions of the restricted problem for small μ≠0.

Phase Space Solution of Buoyant Jets

Abstract The solution of buoyant jets in a calm stratified atmosphere is considered. The Morton model yields a set of differential equations that can be solved in a closed form in a phase space having for coordinates the integral with height of the mass flux, the mass flux, and the derivative with height of the mass flux. By using an approximate expression that relates the mass flux integral with height to a given height, we are able to transfer knowledge of the phase space solution to physical space.

Forced oscillations in a class of self-biased multimode oscillators

The behavior of a class of multimode oscillators, having two nonlinear elements and operating under the influence of an external signal, is analyzed, using approximation procedures of Krylov, Bogoliubov, and Mitropolsky. Although features of the basic circuit are chosen to conform to the general characteristics of common self-biased oscillators, the procedures employed are applicable to a large class of problems involving two nonlinear elements. Through the approximating procedure, the system of nonlinear differential equations, describing the basic circuit, is replaced by a new, more tractable system of three first-order nonlinear differential equations. Phase-space solutions of the approximating system are then used to predict and analyze several features of forced oscillator operation, including frequency entrainment, pulling, and super-regenerative detection. The results of the theoretical analysis are in very close agreement with the actual behavior of an experimental oscillator.