Nonlinear Invariants Research Articles

Relating crevassing to non-linear strain in the floating part of Jakobshavn Isbræ, West Greenland

AbstractJakobshavn Isbræ is a major ice stream that drains the west-central Greenland ice sheet and becomes afloat in Jakobshavn Isfjord (69° N, 49° W), where it has maintained the world’s fastest-known sustained velocity and calving rate (7 kma−1) for at least four decades. The floating portion is approximately 12 km long and 6 km wide. Surface elevations and motion vectors were determined photogrammetrically for about 500 crevasses on the floating ice, and adjacent grounded ice, using aerial photographs obtained 2 weeks apart in July l985. Surface strain rates were computed from a mesh of 399 quadrilateral elements having velocity measurements at each corner. It is shown that heavy crevassing of floating ice invalidates the assumptions of linear strain theory that (i) surface strain in the floating ice is homogeneous in both space and time, (ii) the squares and products of strain components are nil, and (iii) first- and second-order rotation components are small compared to strain components. Therefore, strain rates and rotation rates were also computed using non-linear strain theory. The percentage difference between computed linear and non-linear second invariants of strain rate per element were greatest (mostly in the range 40–70%) where crevassing is greatest. Isopleths of strain rate parallel and transverse to flow and elevation isopleths relate crevassing to known and inferred pinning points.

Estimating Lyapunov exponents in biomedical time series.

Among nonlinear dynamical invariants, determination of the largest Lyapunov exponent is well suited to positive identification of chaos in observed time series. When analyzing the dynamics of biomedical series, such as an electro-encephalogram (EEG), model-based methods should be used. Moreover, in the absence of any well founded theoretical model, and because of unexplained variability in the data, candidate models must provide for a stochastic component. Here we use nonlinear autoregressive stochastic modeling to estimate the dominant Lyapunov exponent in an EEG series and compute confidence intervals from surrogate data. The results are found to differ from those of approaches which aim at deleting noise prior to analysis.

A Shallow-Water Model that Prevents Nonlinear Steepening of Gravity Waves

The shallow-water model is used as a testbench for understanding many fundamental dynamical problems (e.g. certain wave–mean interaction problems). One sometimes wants to allow large-amplitude gravity waves to propagate significant distances in such models without forming shocks. This paper presents a simple, and apparently unique, modification of the standard shallow-water model that prevents gravity wave shock formation, but which, at the same time, introduces only minimal changes in other aspects of the behavior. For instance, the presented modification is nondissipative as well as nondispersive, and it preserves the linear structure of the shallow-water equations as well as the nonlinear functional form and material invariance of shallow-water potential vorticity. The modification is derived theoretically and has been tested numerically in several ways in one and two dimensions.

Exactly Conservative Integrators

Traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants. In this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly. We illustrate the method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator-prey model, and the Kepler problem. The ideas are discussed in the context of symplectic (phase--space-conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.

1-17-02 Characterization of brain structures through EEG activity by nonlinear dynamical invariants

1-17-02 Characterization of brain structures through EEG activity by nonlinear dynamical invariants

A remarkable nonlinear invariant for evolution with heterogeneous rates

A model for DNA or protein sequence evolution is proposed where each position belongs to one of two distinct classes. The two classes evolve at different rates. For a phylogeny on four species, we find a cubic function of 4-tuple occurrence frequencies that is nontrivially invariant no matter what the proportion of positions in each rate class. This result refutes the major criticism of nonlinear polynomial invariants.

Harmonic superspace geometry of (4,4) supersymmetric sigma models with torsion.

Starting from the dual action of (4,4) two-dimensional (2D) twisted multiplets in the harmonic superspace with two independent sets of SU(2) harmonic variables, we present its generalization which hopefully provides an off-shell description of general (4,4) supersymmetric \ensuremath{\sigma} models with torsion. Like the action of the torsionless (4,4) hyper-K\"ahler \ensuremath{\sigma} models in the standard harmonic superspace, it is characterized by a number of superfield potentials. They depend on n copies of a triple of analytic harmonic (4,4) superfields. As distinct from the hyper-K\"ahler case, the potentials prove to be severely constrained by the self-consistency condition which stems from the commutativity of the left and right harmonic derivatives. We show that for n=1 these constraints reduce the general action to that of the (4,4) twisted multiplet, while for n\ensuremath{\ge}2 there exists a wide class of new actions which cannot be written only via twisted multiplets. Their most strikng feature is the non-Abelian and in general nonlinear gauge invariance which substitutes the Abelian gauge symmetry of the dual action of twisted multiplets and ensures the correct number of physical degrees of freedom. We show, on a simple example, that these actions describe \ensuremath{\sigma} models with noncommuting left and right complex structures on the bosonic target. \textcopyright{} 1996 The American Physical Society.

Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.

Off-shell (4,4) supersymmetric sigma models with torsion as gauge theories in harmonic superspace

We present a manifestly supersymmetric off-shell formulation of a wide class of (4,4) 2D sigma models with torsion and noncommuting left and right complex structures in the harmonic superspace with a double set of SU(2) harmonic variables. The distinguishing features of the relevant superfield action are: (i) a nonabelian and in general nonlinear gauge invariance ensuring a correct number of physical degrees of freedom; (ii) an infinite tower of auxiliary fields. For a particular class of such models we explicitly demonstrate the noncommutativity of complex stuctures on the bosonic target.

A tri-Hamiltonian formulation of the full Kostant-Toda lattice

We define Poisson structures which lead to a tri-Hamiltonian formulation for the full Kostant-Toda lattice. In addition, a hierarchy of vector fields called master symmetries are constructed and they are used to generate the nonlinear Poisson brackets and other invariants. Various deformation relations are investigated. The results are analogous to results for the finite nonperiodic Toda lattice.

Group invariance of generalized solutions obtained through the algebraic method

Three results are presented as a first in the literature: (1) large classes of nonlinear Lie groups of transformations are defined on algebras of generalized functions containing the L. Schwartz distributions; the definitions do not involve infinite dimensional manifolds, (2) the projectable Lie groups of symmetry for classical solutions of nonlinear PDEs can be extended to symmetry groups for global generalized solutions, (3) applications are given of the nonlinear group invariance for delta wave solutions of semilinear hyperbolic equations and for Riemann type solvers of the nonlinear shock wave equation.

INFINITE-PARAMETER FAMILIES OF SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

A Bäcklund transformation of a nonlinear equation and additional invariance of it under a contact transformation permit the construction of a family of solutions depending on an arbitrary number of parameters. Boundary value problems are solved by choosing the parameters and the initial solution. Bibliography: 5 titles.

Don't bleach chaotic data.

A common first step in time series signal analysis involves digitally filtering the data to remove linear correlations. The residual data is spectrally white (it is "bleached"), but in principle retains the nonlinear structure of the original time series. It is well known that simple linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as fractal dimension and Lyapunov exponents. In theory, bleached data avoids these pitfalls. But in practice, bleaching obscures the underlying deterministic structure of a low-dimensional chaotic process. This appears to be a property of the chaos itself, since nonchaotic data are not similarly affected. The adverse effects of bleaching are demonstrated in a series of numerical experiments on known chaotic data. Some theoretical aspects are also discussed.

Nonlinear models, rescaling and test invariance

Several asymptotic tests proposed in the literature are shown not to be invariant to changes in measurement units or, more generally, to various transformations that leave both the model and the null hypothesis invariant. The tests involved include the Wald test (and various generalizations of it), a variant of the Lagrange multiplier test, Neyman's C(α) test, Durbin's (1970) procedure, Hausman-type tests and a number of tests suggested by White (1982). For all these procedures, simply changing measurement units in a way that leaves both the form of the model and the null hypothesis invariant can lead to very different answers. We observe, in particular, that various consistent estimators of the information matrix lead to test procedures with different invariance properties. We then establish general sufficient conditions ensuring that Neyman's C(α) test is invariant to transformations that leave invariant the form of the model. In many practical cases where Wald-type tests lack invariance, we find that a modification of the C(α) test is invariant and hardly more costly to compute than Wald tests. This computational simplicity stands in contrast with other invariant tests such as the likelihood ratio test. The method suggested is applied to regression models with Box-Cox transformations on the variables.

Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion

A Hamiltonian formulation using a noncanonical Poisson bracket is presented for a recently proposed model (Dokl. Akad. Nauk SSSR 293, 818 (1985) [Sov. Phys. Dokl. 32, 262 (1987)]) of two-dimensional shallow-water hydrodynamics with nonlinear dispersion. Nonlinear integral invariants for this model are found to be in the kernel of the noncanonical Poisson bracket. A generalized Kelvin theorem is also given for the model.

Application of nonlinear dynamical invariants in a single electromagnetic wave to the study of the Alfvén-ion-cyclotron instability

Dynamical invariants are derived for particles moving in a single, circularly polarized electromagnetic wave of arbitrary time dependence propagating parallel to a uniform background magnetic field. The invariant associated with helical symmetry is shown to restrict the particle motion to a very narrow region of velocity space. Features of the slow time-scale motion of fixed points associated with the existence of a fourth adiabatic invariant are described for the case of a slowly varying wave. Characteristics of the particle motion thus derived are applied to the analysis of 1d–3v simulations of the saturation of the Alfvén-ion-cyclotron (AIC) instability for a single wave. In particular, an explanation is offered for the appearance of a sharp edge in the velocity distribution function observed in the simulation.

Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations

A Hamiltonian formulation using a noncanonical Poisson bracket is presented for the nonlinear dynamics of baroclinic quasigeostrophic fluid flow. Nonlinear integral invariants for the system are found to be in the kernel of the noncanonical Poisson bracket. A novel feature of this formulation is that dynamical boundary conditions are generated from the constrained energy Hamiltonian, via the noncanonical Poisson bracket.

Electromagnetic solitary waves in magnetized plasmas

A Hamiltonian formulation, in terms of a non-canonical Poisson bracket, is presented for a nonlinear fluid system that includes reduced magnetohydro-dynamics and the Hasegawa–Mima equation as limiting cases. The single-helicity and axisymmetric versions possess three nonlinear Casimir invariants, from which a generalized potential can be constructed. Variation of the generalized potential yields a description of exact nonlinear stationary states. The new equilibria, allowing for plasma flow as well as partial electron adiabaticity, are distinct from those found in conventional magnetohydrodynamic theory. They differ from electrostatic stationary states in containing plasma current and magnetic field excitation. One class of steady-state solutions is shown to provide a simple electromagnetic generalization of drift-solitary waves.

Hamiltonian structure for Alfvén wave turbulence equations

A hamiltonian formulation using a noncanonical Poisson bracket is presented for a nonlinear fluid system that includes reduced magnetohydrodynamics and the Hasegawa-Mima equation as limiting cases. Nonlinear integral invariants for the system are found to be in the kernel of the noncanonical Poisson bracket. This Poisson bracket is given a Lie algebraic interpretation.

Fermion loop contribution to skyrmion stability

We calculate the complete one-fermion loop contribution to the O(∂φ) 4 term in the effective lagrangian of the linear SU(2) × SU(2) σ-model, from which we obtain the corresponding contributions to each of the three possible non-linear chiral invariants of this dimension. Two of these terms are present in the Skyrme phenomenological lagrangian: our calculated terms have the correct signs for soliton stability, and the numerical coefficients are predicted in terms of the symmetry breaking parameter ƒ. The effect of the third term is not yet clear.