
Professor Richard Rand
Cornell University
Keynote Title: The Nonlinear Dynamics of the Moon-Rand Equations
Abstract: The so-called Moon-Rand equations were developed in 1985 [1] to describe the stiffness control of flexible space structures. The equations take the form of a linear oscillator in x,y coordinates with a control parameter z:
(1) dx/dt = y | (2) dy/dt = -x - z x
The control parameter z satisfies the equation:
(3) dz/dt = - k z + f(x,y)
where f(x,y) is modeled as a truncated power series in x,y:
(4) f(x,y) = G_20 x^2 + G_11 xy + G_02 y^2
The question is to determine the nonlinear dynamical behavior in the neighborhood of the origin. This includes stability of the equilibrium point at the origin, Hopf bifurcation of limit cycles and so on. See [2] Exercise 5, section 5.5, [3],[4],[5],[6],[7].
List of recent publications which refer to the Moon-Rand equations:
[1] F.C. Moon, R.H. Rand, Parametric stiffness control of flexible structures, in: Jet Propulsion Laboratory Publication 85-29, vol. II, California Institute of Technology, 1985, pp. 329–342.
[2] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, third ed., Springer-Verlag, New York, 2004.
[3] A. Mahdi, V.G. Romanovski, D.S. Shafer, Stability and periodic oscillations in the Moon–Rand systems, Nonlinear Anal.: Real World Appl. 14 (2013) 294–313.
[4] L. Barreira, C. Valls, J. Llibre, Integrability and limit cycles of the Moon-Rand system, Int. J. Nonlin. Mech. 69 (2015) 129–136.
[5] J. Llibre, C. Valls, Hopf bifurcation of a generalized Moon–Rand system, Communications in Nonlinear Science and Numerical Simulation, 20 (2015) 1070-1077.
[6] B. Sang, Q. Wang, B.Fercec, Four Limit Cycles in a Generalized Moon-Rand System with Fifth-Order Perturbation, J. Nonlinear Funct. Anal. (2016), 1-12.
[7] J. Giné, C. Valls The generalized polynomial Moon–Rand system, Nonlinear Analysis: Real World Applications 39 (2018) 411–417
Biography: Rand joined the Cornell faculty in 1967 after receiving his doctorate from Columbia University. He was a visiting professor at the University of California at Berkeley in 1981 and at the University of California at Los Angeles in 1989. Rand received teaching awards from the Engineering College at Cornell in 1986, 1993, 1995, 2005 and 2008, and from the Mathematics Department in 2013. In 2017, Rand received the Thomas K. Caughey Dynamics Award from the ASME Applied Mechanics Division.