Showing papers on "Nonlinear system published in 1969"
TL;DR: An algorithm for the analysis of multivariate data is presented along with some experimental results that is based upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data "structure" is approximately preserved.
Abstract: An algorithm for the analysis of multivariate data is presented along with some experimental results. The algorithm is based upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data "structure" is approximately preserved.
3,460 citations
TL;DR: In this paper, a set of three-dimensional constitutive equations is proposed which is consistent with nonlinear behavior of some metals and plastics, and which enables all properties to be evaluated from uniaxial creep and recovery data.
Abstract: Starting with specific constitutive equations, methods of evaluating material properties from experimental data are outlined and then illustrated for some polymeric materials; these equations have been derived from thermodynamic principles, and are very similar to the Boltzmann superposition integral form of linear theory. The experimental basis for two equations under uniaxial loading and the influence of environmental factors on the properties are first examined. It is then shown that creep and recovery data can be conveiently used to evaluate properties in one equation, while two-step relaxation data serve the same purpose for the second equation. Methods of reducing data to accomplish this characterization and to determine the accuracy of the theory are illustrated using existing data on nitrocellulose film, fiber-reinforced phenolic resin, and polyisobutylene. Finally, a set of three-dimensional constitutive equations is proposed which is consistent with nonlinear behavior of some metals and plastics, and which enables all properties to be evaluated from uniaxial creep and recovery data.
1,004 citations
01 Jan 1969
536 citations
TL;DR: In this paper, a perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein-Gordon equation.
Abstract: A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein‐Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrodinger equation.
447 citations
TL;DR: In this article, the system of equations for the magneto-acoustic wave propagating along a critical direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one.
Abstract: The hydromagnetic waves with small but finite amplitude in a cold collision-free plasma are investigated by using a nonlinear perturbation method. In the lowest order of perturbation, we can show that the system of equations for the magneto-acoustic wave propagating along a `critical' direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one. An extension of the problem to more general dispersive system is also made. On the other hand, the system of equations for the Alfven wave is reduced to a modified Korteweg-de Vries equation in the sense that the non-linear term f ∂ f /∂ξ in the Korteweg-de Vries equation is replaced by f 2 ∂/∂ξ. In the case of steady propagation this equation can be integrated to give a solution in closed form, which exhibits a solitary wave. Two kinds of solitary wave (both compressive and rarefied) are found to be possible.
363 citations
TL;DR: In this article, a saturable resonator is used to switch out the CO2 laser light from its cavity and for repetitive Q•switching for variable length pulses, infinite length pulse trains, logical operations on two signals, and memory functions.
Abstract: Optical resonators containing saturable absorbers (saturable resonators) have nonlinear characteristics and can exhibit hysteresis. This is demonstrated experimentally at 10.6 μ wavelength. A saturable resonator is used to switch out the CO2 laser light from its cavity and for repetitive Q‐switching. Devices are described to obtain variable length pulses, infinite pulse trains, logical operations on two signals, and memory functions.
345 citations
Book•
31 Dec 1969
TL;DR: This edition retains the emphasis on practical systems, develops transfer-function and state-variable analog models, reviews matrices, develops an analytical design procedure that applies both root-locus design and Bode design, and presents a table of Laplace transforms and z-transforms in the appendices.
Abstract: From the Publisher:
Organized into three principal areas (analog control systems, digital control systems, and nonlinear analog control systems), Feedback Control Systems has been significantly revised to provide more practical examples, more thorough design coverage, and increased emphasis on computer-aided analysis and design using MATLAB. Specifically, the Third Edition features: clearer, more detailed explanations of basic material; a new section (4.5) dealing with time-scaling differential equations designed to help readers better relate the transfer functions of the systems of the examples to those of practical systems; additional practical applications in the form of examples and end-of-chapter problems that better relate mathematical developments to physical systems; and completely new end-of-chapter problems, many using MATLAB programming for solution verification. In addition, this edition retains the emphasis on practical systems, develops transfer-function and state-variable analog models, reviews matrices, develops an analytical design procedure that applies both root-locus design and Bode design, and presents a table of Laplace transforms and z-transforms in the appendices.
318 citations
01 Dec 1969
TL;DR: In this paper, a review of tunable optical parametric oscillators is presented, focusing on parametric amplification of Gaussian beams: threshold, tuning techniques, spectral output, and stability.
Abstract: This paper reviews progress on tunable optical parametric oscillators. Topics considered include: parametric amplification of Gaussian beams: threshold; tuning techniques, spectral output, and stability; saturation and power output; spontaneous parametric emission; nonlinear materials; and far infrared generation.
287 citations
TL;DR: In this article, a variational principle is used in conjunction with the finite element method to solve the initial boundary value problem of flow in a saturated porous elastic medium, which results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations.
Abstract: A variational principle is used in conjunction with the finite-element method to solve the initial boundary value problem of flow in a saturated porous elastic medium. This results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations. Direct application is to problems of consolidation and drainage of saturated soils under load. Linear theory of the coupled fields is treated but extension to nonlinear problems is possible through use of incremental procedures.
TL;DR: Topologies introduced on state space for differential equations to obtain dynamical systems were introduced in this article, where the state space was used to obtain the dynamical system topology of dynamical networks.
TL;DR: In this article, a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory, and it is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.
Abstract: Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.
TL;DR: Light pulses propagation in nonlinear laser medium, obtaining equations of motion for density matrix was studied in this paper, where it was shown that light pulses propagate in a nonlinear nonlinear medium.
Abstract: Light pulses propagation in nonlinear laser medium, obtaining equations of motion for density matrix
TL;DR: The method has been extended to the linear integrodifferential equations and ordinary differential equations and it can also be applied to non-linear problems.
Abstract: This paper describes a new method for the numerical solution of linear integral equations of Fredholm type and of Volterra type. The method has been extended to the linear integrodifferential equations and ordinary differential equations. It can also be applied to non-linear problems. In each case numerical examples are treated and the method compares quite favourably with other known methods. (Received March 1969)
TL;DR: In this paper, the nonlinear interactions of electrostatic modes in a plasma are studied within the framework of the weak-turbulence approximation, and it is shown that the multi-wave interaction (resonant mode-mode coupling) can be explosively unstable when, for Σj=1n kj = 0 and Σk= 1nωkj = 0, we also have S1 = S2 = … = Sn, where S j = ( ∂ϵ ∂ω k j ) /| ∂ ϵ ∆ k j |
TL;DR: In this paper, it was shown that if for each t, the frozen system is stable, then the time-varying system should also be stable, provided A(t) is small enough.
Abstract: A limiting case of great importance in engineering is that of slowly varying parameters. For systems described by \dot{x} = A(t)x , one would intuitively expect that if, for each t , the frozen system is stable, then the time-varying system should also be stable. Provided A(t) is small enough, Rosenbrock has shown that this is the case [1]. Rosenbrock used a continuity argument [1, p. 75]. In this correspondence explicit bounds and slightly sharper results are obtained. Finally, it is pointed out that these results are useful in the study of the exact behavior of non-linear lumped systems with slowly varying operating points.
TL;DR: The mathematical results are global limit and oscillation theorems for a class of nonlinear functional-differential systems and pattern completion on recall trials can occur without destroying perfect memory even if the signal thresholds sufficiently large.
Abstract: Learning of patterns by neural networks obeying general rules of sensory transduction and of converting membrane potentials to spiking frequencies is considered. Any finite number of cellsA can sample a pattern playing on any finite number of cells ∇ without causing irrevocable sampling bias ifA = ℬ orA ∩ ℬ =
. Total energy transfer from inputs ofA to outputs of ℬ depends on the entropy of the input distribution. Pattern completion on recall trials can occur without destroying perfect memory even ifA = ℬ by choosing the signal thresholds sufficiently large. The mathematical results are global limit and oscillation theorems for a class of nonlinear functional-differential systems.
TL;DR: In this paper, the theory of nonlinear collisionless acceleration is used to explain the experimentally observed properties of the fast part of plasmas produced by lasers from isolated single small aluminum balls and thick solid targets.
Abstract: The nonlinear interaction of an intense light wave with an inhomogeneous plasma produces a macroscopic motion. A rigorous treatment of this interaction based on the ponderomotive force description leads to a general equation of motion. In a plasma with collisions and high electron density, the resulting force density can be interpreted as an expression of the radiation pressure. Below special densities there results a nonlinear collisionless force whose direction is toward decreasing density, and which produces a deconfinement. The magnitude of this force has a polarization dependence only in the third‐order terms of the spatial dependence of the density. The total deconfining recoil momentum transferred to the inhomogeneous transition layer is evaluated. The theory of the nonlinear collisionless acceleration is used to explain the experimentally observed properties of the fast part of plasmas produced by lasers from isolated single small aluminum balls and thick solid targets.
TL;DR: In this article, a simplified derivation of the Fokker-Planck equation is given and the uniqueness of the steady-state solution for certain classes of system is discussed.
Abstract: Nonlinear systems disturbed by Gaussian white noises (or by signals obtained from Gaussian white noises) can sometimes be analysed by setting up and solving the Fokker–Planck equation for the probability density in state space. In the present paper a simplified derivation of the Fokker–Planck equation is given. The uniqueness of the steady-state solution is discussed. Steady-state solutions are obtained for certain classes of system. These solutions correspond to or slightly generalize the Maxwell–Boltzmann distribution which is well known in classical statistical mechanics
TL;DR: In this paper, the authors considered the traction boundary value problem in the general case where O(U,, ti) may be nonlinear in both u,, zi,.
01 Dec 1969
TL;DR: In this paper, the authors presented a scheme for obtaining a linear feedback law for a linear system as a result of, minimising a quadratic performance index; the resulting closed-loop system has the property that all its poles lie in a halfplane Re (s) 0 may be chosen by the designer.
Abstract: The paper presents a scheme for obtaining a linear-feedback law for a linear system as a result of, minimising a quadratic-performance index; the resulting closed-loop system has the property that all its poles lie in a halfplane Re (s) 0 may be chosen by the designer. The advantages of this arrangement over conventional optimal design are considered. In particular, it is shown that the reduction of trajectory sensitivity to plant-parameter variations as a result of any closed-loop control is greater for α > 0 than for α = 0, that there is inherently a greater margin for tolerance of time delay in the closed loop when α > 0, that there is greater tolerance of nonlinearity when α > 0, and that asymptotically stable bang-bang control may be achieved simply by inserting a relay in the closed loop when α > 0. The disadvantage of the scheme appears to be that, with α > 0, more severe requirements are put on the power level at which input transducers should operate than for α = 0.
TL;DR: In this article, it was shown that if the short-circuit admittance matrix G of the linear portion of the dc model of a transistor network satisfies a certain dominance condition, then the network cannot be bistable.
Abstract: Several results are presented concerning the equation F(x) + Ax = B (with F(·) a “diagonal” nonlinear mapping of real Euclidean n-space Em into itself, and A a real n × n matrix) which plays a central role in the dc analysis of transistor networks. In particular, we give necessary and sufficient conditions on A such that the equation possesses a unique solution x for each real n-vector B and each strictly monotone increasing F(·) that maps Em onto itself. There are several direct circuit-theoretic implications of the results. For example, we show that if the short-circuit admittance matrix G of the linear portion of the dc model of a transistor network satisfies a certain dominance condition, then the network cannot be bistable. Therefore, a fundamental restriction on the G matrix of an interesting class of switching circuits is that it must violate the dominance condition.
TL;DR: In this article, a compilation of lectures originally presented at the Associated Western Universities Differential Equations Symposium, Baulder, Colorado in the summer of 1967 is given motivating examples and ideas which have influenced the author in his studies of oscillation properties of solutions of linear ordinary differential equations.
TL;DR: In this article, the ground states of infinite quantum lattice systems were investigated and it was shown that a positive energy operator is associated with these ground states, and that the positive energy operators are associated with the ground state.
Abstract: We investigate the ground states of infinite quantum lattice systems. It is shown in particular that a positive energy operator is associated with these states.