Search Results (21 total)

We formulate a fictitious-time-flow equation which drives an initial guess torus to a torus invariant under a given dynamics, provided such a torus exists. The method is general and applies in principle to continuous time flows and discrete time maps in arbitrary dimension and to both Hamiltonian and dissipative systems.

We analyze spatiotemporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt’s method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits (“modulated amplitude waves”) on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.

A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The “Newton descent” partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Hénon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits. The evaluation of perturbative corrections is easier to implement in this framework than in the standard Feynman diagram perturbation theory. The results are perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum-mechanical applications.

We investigate intermittent diffusion using cycle expansions and show that a truncation based on cycle stability achieves reasonable convergence, without using detailed information about the analytic structure of the zeta function.

Proofs that Fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators. We construct such operators both for the Gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism, and introduce a new functional determinant which is expected to be entire for Axiom A flows, and whose zeros coincide with the zeros of the Gutzwiller-Voros zeta function.

We demonstrate the utility of the periodic-orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic 3-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic-orbit estimates and the exact quantum poles is observed.

Nonanalyticities in the generalized dimensions of fractal sets of physical interest are interpreted as phase transitions. We apply the thermodynamical formalism to the fractal set formed by the irrational winding parameter values of critical circle maps and introduce and investigate in detail several distinct fractal measures on this set. The thermodynamic functions associated with different measures are distinct: We discover that, in all cases that we study, they exhibit phase transitions. The numerical estimates of the Hausdorff dimension from various versions of the thermodynamical formalism and a variety of circle maps yield D_H=0.8701±0.0003 and are consistent with the conjectured universality of D_H.

We argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible, but is also a theoretically optimal measurement of the invariant properties of a dynamical system.

We use the set of all periodic points of Hénon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a Hénon-type attractor is conveniently represented by a two-dimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the ‘‘pruning front.’’ The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation ‖b‖<1, it is well approximated by a few steps, or, in the symbolic dynamics language, by a finite grammar. Thus equipped with the complete list of allowed periodic points, we reconstruct (to resolution of order bⁿ) the physical attractor by piecing together the linearized neighborhoods of all periodic points of cycle length n. We use this representation to compute the singularity spectrum f(α). The description in terms of periodic points works very well in the ‘‘hyperbolic phase,’’ for α larger than some α_c, where α_c is the value of α corresponding to the (conjectured) phase transition.

The fractal invariant measure of chaotic strange attractors can be approximated systematically by the set of unstable n-periodic orbits of increasing n. Algorithms for extracting the periodic orbits from a chaotic time series and for calculating their stabilities are presented. With this information alone, important properties like the topological entropy and the Hausdorff dimension can be calculated.

The apparent universality of the fractal dimension of the set of quasiperiodic windings at the onset of chaos in a wide class of circle maps is described by construction of a universal one-parameter family of maps which lies along the unstable manifold of the renormalization group. The manifold generates a universal ‘‘devil’s staircase’’ whose dimension agrees with direct numerical calculations. Applications to experiments are discussed.

We show that the color-charge algebra in the three-quark sector generated by the matrices of the fundamental representation of U(n) does not have the trace properties required in Adler's extension of chromodynamics. We also discuss a diagrammatic representation of algebras generated by quark and antiquark charges in general, and an embedding of the N-quark algebra in the symmetric group S_N+1.

The functional techniques of field theory are adapted to the problem of evaluating sums of combinatoric and group-theoretic weights of Feynman diagrams in φ^N, quantum electrodynamics and non-Abelian theories. Considered are various classes of diagrams such as connected, one-particle-irreducible, and skeleton diagrams. For finite orders exact sums are given by compact recursion formulas. For higher orders estimates are obtained from the exact results or by steepest-descent methods.

Gauge-invariant mass-shell amplitudes for quantum electrodynamics (QED) and Yang-Mills theory are defined by dimensional regularization. Gauge invariance of the mass-shell renormalization constants is maintained through interplay of ultraviolet and infrared divergences. Quark renormalizations obey the same simple Ward identity as do the electron renormalizations in QED, while the gluon contributions to gluon renormalizations are identically zero. The simplest amplitude finite in QED, the magnetic moment, is gauge-invariant but divergent in Yang-Mills theory for both external gluon and external photon.

A simple and systematic method for the calculation of group-theoretic weights associated with Feynman diagrams in non-Abelian gauge theories is presented. Both classical and exceptional groups are discussed.

We have evaluated the contribution of 50 Feynman diagrams of three-photon-exchange type to the electron magnetic moment by two independent methods. The results are mutually consistent and are several times more accurate than previously reported calculations. If we combine the analytic result of Levine and Roskies for 10 diagrams and our numerical result for the remaining 40 diagrams, we obtain the best estimate available at present: (0.922±0.024)×(α / π)³. Including the contribution from the remaining 22 diagrams calculated previously, the complete theoretical prediction for the electron anomaly up to the order α³ is 1 / 2α / π-0.328 48(α / π)²+(1.195±0.026)(α / π)³, in fair agreement with the latest experimental result.

A scheme for systematically separating ultraviolet divergences of Feynman amplitudes in parametric space is developed. It is summarized by an explicit formula which enables us to incorporate readily the ultraviolet-finite remainders thus constructed into the usual renormalization scheme. It is shown further that infrared divergences can be treated in a very similar way. Our method is particularly suitable for numerical integration.

Concise and practical formulas for Feynman-parametric integrals are assembled and presented in the form of Feynman-Dyson rules in parametric space. These rules enable us to write down S-matrix elements directly in terms of some parametric functions. They are particularly useful for construction of ultraviolet- and infrared-divergence-free Feynman-parametric integrals in a form suitable for numerical integration.

Landshoff has recently pointed out that in composite models for hadron-hadron wide-angle scattering, processes where the constituents remain close to their mass shells throughout the scattering can dominate the wide-angle cross sections. We use a Feynman parametric integral to estimate the phase space for such a process and confirm Landshoff's result. The differential cross section for elastic π-π scattering at fixed c.m. angle is found to fall off as s^-5, rather than as s^-6, which is predicted by the dimensional-counting rules.

We have calculated the contribution of fifty Feynman diagrams of the order α³ to the electron magnetic moment. Our result, (1.02±0.04)(α / π)³, agrees with the result of Levine and Wright, and is about 5 times more accurate. Together with the contribution from the rest of the α³ diagrams calculated previously, the complete theoretical prediction for the electron anomaly up to the order α³ is α / 2π-0.328 48(α / π)²+(1.29±0.06)(α / π)³, in good agreement with the latest experimental result.