Showing papers on "Nonlinear system published in 1972"

Stability of time-delay systems

Abstract: A direct method is presented for determining both local and regional stability of systems described by nonlinear differential-difference equations. Prediction of stability is with respect to a general class of initial curves. The practical as well as the conservative nature of the procedure is demonstrated by a numerical example.

Steady-state analysis of nonlinear circuits with periodic inputs

Abstract: In the computer-aided analysis of nonlinear circuits with periodic inputs and a stable periodic response the steady-state periodic response is found for a given initial state by simply integrating the system equations until the response becomes periodic. In lightly damped systems this integration could extend over many periods making the computation costly. In this paper a Newton algorithm is defined which converges to the steady-state response rapidly. The algorithm is applied to several nonlinear circuits. The results show a considerable reduction in the amount of time necessary to compute the steady-state response. In addition, the initial iterates give information on the transient response of the system.

Stochastic differential equations for the non linear filtering problem

Abstract: The general nonlinear filtering or estimation problem may be described as follows. xty (0

Widely convergent method for finding multiple solutions of simultaneous nonlinear equations

Abstract: A new method has been developed for solving a system of nonlinear equations g(x) = 0. This method is based on solving the related system of differential equations dg/dt±g(x)= 0 where in the sign is changed whenever the corresponding trajectory x(t) encounters a change in sign of the Jacobian determinant or arrives ata solution point of g(x)= 0. This procedure endows the method with much wider region of convergence than other methods (occasionally, even global convergence) and enableist to find multiple solutions of g(x)= 0 one after the other. The principal limitations of the method relate to the extraneouss ingularities of the differential equation. The role of these singularities is illustrated by several examples. In addition, the extension of the method to the problem of finding multiple extrema of a function of N variables is explained and some examples are given.

Development of non-linear transformations for improving convergence of sequences

Abstract: A method of generating non-linear transformations for increasing the rate and expanding the domain of convergence of sequences is presented. These transformations represent in a certain sense a generalization of the well-known transformations due to Shanks, and in many cases are more efficient. The transformations would seem to have important application in computing results from formal solutions to problems in applied mathematics when these solutions are obtained in the form of series or sequences having poor convergence. An indication is also given of application to the“evaluation”of divergent formal solutions.

Nonlinear dynamic analysis of complex structures

Abstract: A general step-by-step solution technique is presented for the evaluation of the dynamic response of structural systems with physical and geometrical nonlinearities. The algorithm is stable for all time increments and in the analysis of linear systems introduces a predictable amount of error for a specified time step. Guidelines are given for the selection of the time step size for different types of dynamic loadings. The method can be applied to the static and dynamic analysis of both discrete structural systems and continuous solids idealized as an assemblage of finite elements. Results of several nonlinear analyses are presented and compared with results obtained by other methods and from experiments.

Nonlinear Development of Electromagnetic Instabilities in Anisotropic Plasmas

Abstract: Theory and simulation experiment are presented for a wide variety of transverse electromagnetic instabilities in plasmas with different sources and degrees of anisotropy. In each of the electron bi‐Maxwellian, electron‐pinch, and ion‐pinch experiments, the bulk response of the system during the initial stages of instability is in good agreement with the predictions of quasilinear theory. Furthermore, the two independent energy constants which derive from the fully nonlinear Vlasov‐Maxwell equations are found to remain constant to very good accuracy, even when the magnetic field energy reaches a substantial fraction of the total system energy. In each simulation experiment it is found that the magnetic energy saturates once the magnetic bounce frequency has increased to a value comparable to the linear growth rate prior to saturation, i.e., when ω¯B∼γ¯k. It is concluded that amplitude limitation for Weibel instabilities is a result of magnetic trapping for a broad range of system parameters. In many experi...

On the thermodynamic foundations of non-linear solid mechanics

Abstract: Non-equilibrium thermodynamics with internal variables governed by rate equations is explored as a foundation for non-linear solid mechanics. Rate equations are studied as to type, yielding descriptions of viscoelastic, viscoplastic, and plastic behavior. The special case of uncoupled instantaneous elasticity is considered, as well as materials exhibiting combined behavior.

On biological assays involving quantal responses

Abstract: Quantal biological assays are traditionally performed on grouped data. However, grouping disguises the fundamental all-or-none nature of the responses, is not necessary and may not be possible. The statistical method relevant to quantal assays is reconsidered from the viewpoint of the individual responses and with a view to getting the apparatus for their explicit analysis. Specifically, the logistic function is used as a model for the distribution of the probability of responding as a function of dose or concentration. The parameters of the logistic function are estimated by the method of maximum likelihood. Since the function is nonlinear in both the scale and location parameters, solution of the normal equations is achieved by an iterative technique base on a Taylor series expansion. Observations are not grouped so the method is applicable to cases in which several observations are not available at the same dose level as well as to grouped data. Three variants on the basic model are considered: the classical model in which the parameters estimated are the intercept and slope of the associated linear regression and two models in which the parameter of prime pharmacological interest—the ED50—is estimated directly. In one of the latter the ED5O is considered to be symmetrically distributed on a logarithmic scale, in the other on a linear scale. Similarly, when two curves are compared, potency ratios or their logarithms are estimated directly rather than indirectly as the difference or ratio of two variables. One advantage of such direct estimation is that the error valiance can then be obtained directly from the covariance matric obtained during the solution of the normal equations. Numerical examples are given and illustrative computer programs are given in an appendix.

Computer program for calculation of complex chemical equilibrium compositions

Abstract: Computer program is described for numerical solution of chemical equilibria in complex systems by using nonlinear algebraic equations. Free-energy minimization technique is used.

A nonlinear analysis of pulsatile flow in arteries.

Abstract: An approximate numerical method for calculating flow profiles in arteries is developed. The theory takes into account the nonlinear terms of the Navier-Stokes equations as well as the nonlinear behaviour and large deformations of the arterial wall. Through the locally measured values of the pressure, pressure gradient, and pressure-radius function, the velocity distribution and wall shear at a given location along the artery can be determined. The computed results agree well with the corresponding experimental data.

Controllability in Linear Autonomous Systems with Positive Controllers

Abstract: This paper presents results on the controllability of the autonomous linear control system in $R^n $, $\dot x = Ax + Bu$, where $u \in \Omega \subset R^m $ without the assumption that the origin in $R^m $ is interior to $\Omega $. Necessary and sufficient conditions are given for null-controllability (controllability of each point in some neighborhood of the origin to the origin) and global null-controllability with uniformly bounded controllers. This paper extends some results of Saperstone and Yorke who considered the problem of the controllability of the above system with $m = 1$ and $\Omega = [0,1]$ and obtained necessary and sufficient conditions for controllability for this system. Corollaries to the main result include existence of time-optimal controllers and controllability of nonlinear systems. An example of control of an economic system is presented.

A computer algorithm to determine the steady-state response of nonlinear oscillators

Abstract: In the computer-aided analysis of nonlinear autonomous oscillators, the steady-state periodic response is usually found by integrating the system equations from some initial state until the transient response appears to be negligible. In lightly damped systems, convergence to the steady-state response is very slow, and this integration could extend over many periods making the computation costly. Also, one is never sure if a stable orbit exists or if the response might eventually decay to a singular point. If a stable orbit does exist, sometimes it is difficult to determine the period T of the orbit. In this paper, a Newton algorithm is defined which in the neighborhood of an orbit converges to it rapidly and gives a precise value for the period T of this oscillation. This algorithm represents a substantial step forward in the analysis of nonlinear systems. In addition, the algorithm meshes easily with most computer-aided circuit-analysis programs, and the initial iterates give information on the transient behavior of the circuit.

On the cauchy problem for composite systems of nonlinear differential equations

Abstract: In this paper the solvability of the Cauchy problem in a space of smooth functions is demonstrated for hyperbolic-parabolic composite systems of nonlinear equations which include a broad class of equations of mathematical physics, in particular, symmetric systems of first order and parabolic systems of second order. Cauchy problems for the equations of the dynamics of a viscous compressible fluid and for the equations of gas dynamics are solved as examples. Bibliography: 6 items.

Stability of Large-Scale Systems under Structural Perturbations

Abstract: A large-scale system is considered as a system constituted of subsystems which may be connected or disconnected from each other during operation. A new concept of connective stability is introduced by which a large-scale system is regarded as stable if it remains stable (in the sense of Lyapunov) under structural perturbations produced by the on-off participation of the subsystems. Algebraic conditions are developed that guarantee exponential connective stability of large-scale systems which may be composed of linear and nonlinear time-varying subsystems coupled by linear or nonlinear connections.

Nonlinear vibration of cylindrical shells

Abstract: The large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations. A perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear partial differential equations. The simply-supported boundary condition and the circumferential periodicity condition are satisfied. The resulting solution indicates that in addition to the fundamental modes, the response contains asymmetric modes as well as axisymmetric modes with the frequency twice that of the fundamental modes. In the previous investigations in which the Galerkins procedure was applied, only the additional axisymrnetric modes were assumed. Vibrations involving a single driven mode response are investigated. The results indicate that the nonlinearity is either softening or hardening depending on the mode. The vibrations involving both a driven mode and a companion mode are also investigated. The region where the companion mode participates in the vibration is obtained and the effects due to the participation of the companion mode are studied. An experimental investigation is also conducted. The results are generally in agreement with the theory. "Non-stationary4 response is detected at some frequencies for large amplitude response where the amplitude drifts from one value to another. Various nonlinear phenomena are observed and quantitative comparisons with the theoretical results are made.

Piecewise-Linear Theory of Nonlinear Networks

Abstract: This paper deals with nonlinear networks which can be characterized by the equation ${\bf f}( {\bf x} ) = {\bf y}$, where $f( \cdot )$ is a continuous piecewise-linear mapping of $R^n $ into itself. ${\bf x}$ is a point in $R^n $ and represents a set of chosen network variables, and ${\bf y}$ is an arbitrary point in $R^n $ and represents the input to the network. The Lipschitz condition and global homeomorphism are studied in detail. Two theorems on sufficient conditions for the existence of a unique solution of the equation for all ${\bf y} \in R^n $ in terms of the constant Jacobian matrices are derived. The theorems turn out to be pertinent in the numerical computation of general nonlinear resistive networks based on the piecewise-linear analysis. A comprehensive study of the Katzenelson’s algorithm applied to general networks is carried out, and conditions under which the method converges are obtained. Special attention is given to the problem of boundary crossing of a solution curve.

Analysis and Compensation of Bandpass Nonlinearities for Communications

Abstract: There are many instances in communication systems where bandpass signals are passed through nonlinear devices, such as traveling wave tubes, which exhibit both amplitude and phase nonlinearities. When the input signal is narrow band, the device may be characterized by measurements of its single-carrier amplitude and phase transfer functions. A sufficient model for such a device is a quadrature structure that includes two nonlinearities each of which, acting on its own, would exhibit only amplitude distortion. The outputs of the two halves of this model are linearly independent for arbitrary narrow-band input signals so that their power spectra add. Consequently, almost all previously published results for amplitude nonlinearities can be readily applied to the analysis of the general device. Emphasis is laid on practical procedures for analysis based directly on measured device characteristics rather than analytic approximations and accuracy is checked by comparison of certain intermodulation results with previous results and with measurements. A new result is the performance of an Intelsat IV tube for a large number of independent equal-power-density signals. A heuristically optimal saturating nonlinearity is introduced and analyzed and two methods of compensating an arbitrary saturating device to obtain this optimal characteristic are presented. Two methods of inverting the Chebyshev transform are used in this paper and the choice of basis functions for obtaining series representations of the measured device characteristics is discussed.

Decoupling in a Class of Nonlinear Systems by State Variable Feedback

Abstract: For a class of nonlinear systems we derive a necessary and sufficient condition for the existence of a state variable feedback control law which accomplishes decoupling, as well as some conditions which characterize the class of decoupling control laws. Several examples are presented to illustrate the application of these results. For a special subclass which includes the so-called bilinear systems, we give two equivalent forms of the necessary and sufficient condition.

An Efficient, One-Level, Primitive-Equation Spectral Model

Abstract: A one-level, global, spectral model using the primitive equations is formulated in terms of a concise form of the prognostic equations for vorticity and divergence. The model integration incorporates a grid transform technique to evaluate nonlinear terms; the computational efficiency of the model is found to be far superior to that of an equivalent model based on the traditional interaction coefficients. The transform model, in integrations of 116 days, satisfies principles of conservation of energy, angular momentum, and square potential vorticity to a high degree.

Parametric Excitation of Electromagnetic Waves

Abstract: Electromagnetic waves propagating along a dc magnetic field are shown to excite parametrically decay, purely growing, modulational and beat wave instabilities. Particular attention is given to whistler parametric instabilities, including a discussion of their nonlinear development and saturation.

Localized Spatial Structures and Nonlinear Chemical Waves in Dissipative Systems

Abstract: The problem of chemical instabilities and the subsequent emergence of dissipative structures is studied in the case of systems which are maintained spatially nonuniform. A simple model is analyzed numerically and shown to exhibit, for different ranges of values of the parameters, two types of solution of the kinetic equations: a localized steady state dissipative structure and a time‐dependent solution describing a non‐linear propagating concentration wave. The biological implications of such solutions are briefly discussed.