richard rand - Profile on Academia.edu (original) (raw)

Papers by richard rand

Computer algebra and the elliptic restricted problem of three bodies

Perturbation solutions of differential equations typically involve power series expansions in a s... more Perturbation solutions of differential equations typically involve power series expansions in a small parameter. Computer algebra has proven to be a useful tool for the treatment of perturbation problems due to the large quantities of algebra involved. As an example, a problem which was previously handled both by numerical integration and by hand-derived perturbation analysis is treated by computer algebra. The elliptic restricted problem of three bodies describing the planar motion of a particle (the third body) under the gravitational attraction of two masses (the primaries) which move about their common center of mass in elliptical orbits is examined. Specifically, the stability of the L(sub 4) equilibrium, located at the vertex of an equilateral triangle which is positioned so that the primaries lie at the other two vertices is determined. It is shown that the use of computer algebra yields a vast increase in the number of terms obtained in the perturbation expansion, and raises...

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equa... more We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomena of coexistence. The differential equation studied governs the stability era mode of vibration in an unforced conservative two degree of freedom system used to model the free vibrations of a thin elastica. Using perturbation methods, we show that at parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable, even though linear stability analysis predicts the origin to be stable. We also investigate the bifurcations associated with this instability.

Frequency-locking and other phenomena emerging from nonlinear interactions between mechanical osc... more Frequency-locking and other phenomena emerging from nonlinear interactions between mechanical oscillators are of scientific and technological importance. However, existing schemes to observe such behaviour are not scalable over distance. We demonstrate a scheme to couple two independent mechanical oscillators, separated in frequency by 80kHz and situated far from each other (3.2km), via light. Using light as the coupling medium enables this scheme to have low loss and be extended over long distances. This scheme is reversible and can be generalised for arbitrary network configurations.

Design Engineering Technical Conferences

Plant, Cell and Environment, 1985

Coupled oscillators as a model for nonlinear parametric excitation

Mechanics Research Communications, 1981

ABSTRACT Typescript (photocopy). Thesis (M.S.)--Cornell University, May, 1981. Bibliography: leav... more ABSTRACT Typescript (photocopy). Thesis (M.S.)--Cornell University, May, 1981. Bibliography: leaves 70-71.

Slow Passage through Resonance in Mathieu's Equation

Journal of Vibration and Control, 2003

ABSTRACT We investigate slow passage through the 2:1 resonance tongue in Mathieu's equati... more ABSTRACT We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The Wentzel-Kramers-Brillouin (WKB) method is then applied to the slow flow equations to obtain an analytic approximation.

A hydrodynamical model of bordered pits in conifer tracheids

Journal of Theoretical Biology, 1977

ABSTRACT Bordered pits are small structures in the cell walls of tracheid xylem cells in plants. ... more ABSTRACT Bordered pits are small structures in the cell walls of tracheid xylem cells in plants. Coniferous bordered pits are typified by a closing membrane possessing a torus and margo structure. This paper presents a hydrodynamical model which supports the conjecture that coniferous bordered pits act like valves to fluid flowing in the xylem pathway.By means of an approximate solution and a corresponding stability analysis, the model is shown to permit only flows with flow rates smaller than a certain critical flow rate Q∗. Flow rates larger than Q∗ are shown to be associated with an unstable equilibrium configuration. As a result of this instability, the pit “snaps” shut and stops the flow.

Journal of Mathematical Biology, 1985

A mathematical examination of retinal photochemistry leads to a hypothesis for Mach band phenomen... more A mathematical examination of retinal photochemistry leads to a hypothesis for Mach band phenomena based on eye movements. This retinal model suggests why minimally distinct borders fade under eye fixation and agrees qualitatively with subjective measures of border contrast as a function of overall field luminance.

Communications in Nonlinear Science and Numerical Simulation, 2002

In a previous paper [Chaos Solitons Fract. 14 , the authors investigated the dynamics of the equa... more In a previous paper [Chaos Solitons Fract. 14 , the authors investigated the dynamics of the equation:

Bifurcations in a Mathieu equation with cubic nonlinearities

Chaos, Solitons & Fractals, 2002

We investigate the nonlinear dynamics of the classical Mathieu equation to which is added a nonli... more We investigate the nonlinear dynamics of the classical Mathieu equation to which is added a nonlinearity which is a general cubic in x, ẋ. We use a perturbation method (averaging) which is valid in the neighborhood of 2:1 resonance, and in the limit of small forcing and small nonlinearity. By comparing the predictions of first-order averaging with the results of

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

Journal of Applied Nonlinear Dynamics, 2012

Symbolic computation and perturbation methods using elliptic functions

Dynamics of Coupled Microbubbles with Large Fluid-Compressibility Delays

The theory of differential delay equations is applied to analyze a system of two bubbles related ... more The theory of differential delay equations is applied to analyze a system of two bubbles related via a delayed coupling term. The dynamics of the system are described by perturbation analyses and bifurcations are detected via continuation using DDE-BIFTOOL. It is shown that for these delays, synchronized oscillation and quasiperiodic motions exist for the coupled system.

Dynamic Games and Applications, 2015

We investigate the dynamics of three-strategy (Rock-Paper-Scissors) replicator equations in which... more We investigate the dynamics of three-strategy (Rock-Paper-Scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval T . Taking T as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems, and present an analysis of the resulting limit cycles using Lindstedt's method.

Dynamics of a delay limit cycle oscillator with self-feedback

Nonlinear Dynamics, 2015

MACSYMA Program to Implement Averaging Using Elliptic Functions

The IMA Volumes in Mathematics and Its Applications, 1991

... Springer-Verlag, 1991 MACSYMA PROGRAM TO IMPLEMENT AVERAGING USING ELLIPTIC FUNCTIONS VINCENT... more ... Springer-Verlag, 1991 MACSYMA PROGRAM TO IMPLEMENT AVERAGING USING ELLIPTIC FUNCTIONS VINCENT T. COPPOLA AND RICHARD H. RAND ... Garcia-Margallo and Bejarano [9] find limit cycles in a generalized van der Pol oscillator using generalized harmonic ...

Journal of Applied Nonlinear Dynamics, 2015

Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biologi... more Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The game-theoretic aspect of the model lies in the choice of fitness function, which is determined by a payoff matrix.

Dynamics of a Quasiperiodically-Forced Mathieu Oscillator

Solid Mechanics and its Applications, 1999

... Appl. Math, vol. 28, pp. 205-217. 2. Bender, CM and Orszag, SA, 1978, Advanced Mathematical M... more ... Appl. Math, vol. 28, pp. 205-217. 2. Bender, CM and Orszag, SA, 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York. 3. Davis, SH and Rosenblat, S., 1980," A quasiperiodic Mathieu-Hill equation", SIAM J. Appi. Math. vol. 38, pp. ...

International Journal of Non-Linear Mechanics, 2008

The dynamics of an autonomous conservative three degree of freedom system which exhibits autopara... more The dynamics of an autonomous conservative three degree of freedom system which exhibits autoparametric quasiperiodic excitation is investigated. The system is a generalization of a classical system known as the "particle in the plane". The system exhibits a motion, the z = 0 mode, whose stability is governed by a linear second order ODE with quasiperiodic coefficients. The behavior of the latter ODE is studied by using three different methods: numerical integration, harmonic balance and perturbation methods. ᭧