Showing papers on "Nonlinear system published in 1974"

Linear and Nonlinear Waves

Abstract: Introduction and General Outline. HYPERBOLIC WAVES. Waves and First Order Equations. Specific Problems. Burger's Equation. Hyperbolic Systems. Gas Dynamics. The Wave Equation. Shock Dynamics. The Propagation of Weak Shocks. Wave Hierarchies. DISPERSIVE WAVES. Linear Dispersive Waves. Wave Patterns. Water Waves. Nonlinear Dispersion and the Variational Method. Group Velocities, Instability, and Higher Order Dispersion. Applications of the Nonlinear Theory. Exact Solutions: Interacting Solitary Waves. References. Index.

Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos

Abstract: Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.

Effective Action for Composite Operators

Abstract: An effective action and potential for composite operators is obtained. The formalism is used to analyze, by variational techniques, dynamical symmetry breaking and coherent solutions to field theory. A Rayleigh-Ritz procedure is introduced which replaces arbitrary variations with parametric variations. Previously unsolved nonlinear equations become, in the Rayleigh-Ritz approximation, solvable algebraic equations.

On three-dimensional packets of surface waves

Abstract: In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

The Fitting of Power Series, Meaning Polynomials, Illustrated on Band-Spectroscopic Data

Abstract: The prototype of fitting polynomials to equally-spaced data—in which the equalspacing is theoretically precise and the data is accurate to many decimal places—arises in the analysis of band spectra. A hard look at such examples forces us to reexamine our thinking on such diverse issues as: How to formulate such problems, the use of robust/resistant techniques in polynomial regression, which coordinates to use and why, the basic properties of linear least squares, choices in stopping a fit, and improved ways to describe our answers. Our results and attitudes apply rather directly to other situations where we are fitting a sum of functions of a single variable. When two or more different variables, subject to error, blunder, or omission, underlie the carriers to be considered, regression/fitting problems are likely to need not only the considerations presented here, but others as well. To a varying extent, the same will be true of nonlinear fitting/regression problems.

Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades

Abstract: The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.

Finite element solution of non‐linear heat conduction problems with special reference to phase change

Abstract: The paper presents a generally applicable approach to transient heat conduction problems with non-linear physical properties and boundary conditions. An unconditionally stable central algorithm is used which does not require iteration. Several examples involving phase change (where latent heat effects are incorporated as heat capacity variations) and non-linear radiation boundary conditions are given which show very good accuracy. Simple triangular elements are used throughout but the formulation is generally valid and not restricted to any single type of element.

Model reference adaptive control with an augmented error signal

Abstract: It is shown how globally stable model reference adaptive control systems may be designed when one has access to only the plant's input and output signals Controllers for single input-single output, nonlinear, nonautonomous plants are developed based on Lyapunov's direct method and the Meyer-Kalman-Yacubovich lemma Derivatives of the plant output are not required, but are replaced by filtered derivative signals An augmented error signal replaces the error normally used, which is defined as the difference between the model and plant outputs However, global stability is assured in the sense that the normally used error signal approaches zero asymptotically

Mathematics Applied to Deterministic Problems in the Natural Sciences

Abstract: A. An Overview of the Interaction of Mathematics and Natural Science: 1. What is applied mathematics? 2. Deterministic systems and ordinary differential equations 3. Random processes and ial differential equations 4. Superposition, heat flow, and Fourier analysis 5. Further developments in Fourier analysis B. Some Fundamental Procedures Illustrated on Ordinary Differential Equations: 6. Simplification, dimensional analysis, and scaling 7. Regular perturbation theory 8. Illustration of techniques on a physiological flow problem 9. Introduction to singular perturbation theory 10. Singular perturbation theory applied to a problem in biochemical kinetics 11. Three techniques applied to the simple pendulum C. Introduction to Theories of Continuous Fields: 12. Longitudinal motion of a bar 13. The continuous medium 14. Field equations of continuum mechanics 15. Inviscid fluid flow 16. Potential theory.

A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting

Abstract: In this paper the well-known modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a "nearly singular" Jacobian at some iterate. A special technique also derived herein may be useful, if only bad initial guesses of the solution point are available. Difficulties that arose previously in the numerical solution of nonlinear two-point boundary value problems by multiple shooting techniques can be removed by means of the results presented below.

Analysis of nonlinear systems with multiple inputs

Abstract: Analytical modeling of communication receivers to account for their nonlinear response to multiple input signals is discussed. The method is based on the application of the Wiener-Volterra analysis of nonlinear functionals. The derived analytical relations were embodied in a computer program which provides nonlinear transfer functions of large circuits specified by their parameters. This method was applied to the prediction of behavior of communication receivers in the presence of interference. Examples illustrate the method and demonstrate its validity in the small-signal region.

Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients

Abstract: Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined. © 1974 American Mathematical Society.

Seasonal Adjustment and Relations Between Variables

Abstract: This article studies the effect of official seasonal adjustment procedures on the relations between variables. By considering time-invariant linear filters, and in particular a linear approximation to the Census Method II adjustment program, the effect of adjusting one or both of the variables in a distributed lag relation is examined, and the distortions which can arise are described. Applying the actual (nonlinear) adjustment procedure to artificial data indicates that at least for the particular x-series used, the results of the linear filter analysis provide a good guide to the behavior of estimates obtained from data adjusted by the official method.

On Kolmogorov's inertial-range theories

Abstract: Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety of a priori admissible alternatives.If the Kolmogorov law is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour.

Nonlinear optimization using the generalized reduced gradient method

Abstract: : Generalized Reduced Gradient (GRG) methods are algorithms for solving nonlinear programs of general structure. This paper discusses the basic principles of GRG, and constructs a specific GRG algorithm. The logic of a computer program implementing this algorithm is presented by means of flow charts and discussion. A numerical example is given to illustrate the functioning of this program.

The Generation of Long Nonlinear Internal Waves in a Weakly Stratified Shear Flow

Abstract: A theoretical and experimental analysis is made to understand the generation of finite amplitude internal waves in fluid systems like the oceans and atmosphere in which the Richardson number is generally much greater than ¼. The initial disturbance is assumed to be of finite amplitude with the characteristic horizontal length scale much greater than the vertical length scale. An internal Korteweg and deVries (KdV) type equation for the stream function is derived for a weakly density stratified shear flow by using a three-parameter expansion method where the three small parameters correspond to nonlinear, dispersive, and non-Boussinesq effects. The influences of basic stratification and shear to the nonlinear, dispersive, and non-Boussinesq effects are found. Numerical solutions to this KdV-type equation for a variety of different conditions are then presented to demonstrate the relative importance of nonlinearity and dispersion on the generation of large amplitude internal waves in the breakdown of internal fronts. The numerical results are also in reasonable agreement with laboratory experiments in which a two-dimensional submarine ridge is moved to create transient internal disturbances. Additional numerical calculations show that a nonlinear model accurately describes the principal features of tidal-generated large amplitude internal waves observed in Massachusetts Bay.

Finite amplitude side-band stability of a viscous film

Abstract: The nonlinear stability of a viscous film flowing steadily down an inclined plane is investigated by the method of multiple scales. It is shown that the super-critically stable, finite amplitude, long, monochromatic wave obtained by Lin (1969, 1970, 1971) is stable to side-band disturbances under modal interaction if the bandwidth is less in magnitude than to the ratio of the amplitude to the film thickness. Near the upper branch of the linear neutral-stability curve where the amplification rate ci is O(e2), the nonlinear evolution of initially infinitesimal waves of a finite bandwidth is shown to obey the Landau-Stuart equation, Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived for describing this strong nonlinear development of relatively long waves. In practice, disturbance of this type clusters in the form of a hump which cannot be constructed only by the first few harmonics.

Complete integrability and stochastization of discrete dynamical systems

Abstract: We use the inverse scattering method to study a system of particles with exponential interaction (the Toda chain) and a set of equations describing induced scattering of plasma oscillations by ions. We show that a Toda chain with an arbitrary number of particles is completely integrable. We develop a scheme to integrate these systems and study the interaction between solitons. We indicate a class of completely integrable discrete systems, that is, systems which can not be stochastized.

Computation of set-up, longshore currents, run-up and overtopping due to wind-generated waves

Abstract: The main problem dealt with in this thesis is the calculation of certain effects caused by random waves breaking on a slope. The solution to this problem is greatly complicated by the fact that wave breaking is a highly nonlinear process. The flow field is further complicated by far stronger in homogeneities than those occurring outside the breaker zone, by air entrainment and by generation of turbulence. No realistic deductive treatment of it has been developed so far. Even for the simpler case of periodic waves, empirical knowledge of certain macroscopic properties of the breakers is still an integral part of calculations relating to the surf zone. An attempt has been made in this thesis to apply this knowledge in a formulation incorporating the stochastic nature of wind-generated waves. The computations are of two distinct categories, those relating to comparatively gentle slopes and those relating to comparatively steep slopes. A summary of the results will be given in the following. The energy variation is calculated in chapter 5 by clipping a fictitious wave height distribution, which theoretically would be present if breaking did not occur, at an upper limit which is determined from an adapted breaking criterion for periodic waves. The computed results are in fair agreement with measurements carried out on a plane slope. Knowledge of the energy variation permits the radiation stresses to be evaluated, which in turn are necessary for the calculation of the set-up and the longshore current velocity profiles. A comparison of the calculated set-up profiles with empirical data has not given conclusive results. Good agreement has been found with field data, but not with laboratory data, which locally showed a systematically smaller rise towards the shore than would be expected on the basis of the measured or calculated wave height variations. However, there is some uncertainty with respect to the system used for measuring the set-up in the laboratory, so that is not known to which extent the differences are real or apparent.