richard rand | Cornell University (original) (raw)
Papers by richard rand
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...
Plant, Cell and Environment, 1985
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.
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.
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
Communications in Nonlinear Science and Numerical Simulation, 2002
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
Journal of Applied Nonlinear Dynamics, 2012
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
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
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
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...
Plant, Cell and Environment, 1985
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.
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.
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
Communications in Nonlinear Science and Numerical Simulation, 2002
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
Journal of Applied Nonlinear Dynamics, 2012
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
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
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