Approximate Solutions to d$^{2}$x/dt$^{2}$ + [1+( dx/dt)$^{2}$]x = 0 Using a Polar Representation

ORAL

Abstract

It can be shown that the following nonlinear differential equation \begin{center} d$^{2}$x/dt$^{2}$ + [1+( dx/dt)$^{2}$]x = 0 \end{center} has only periodic solutions. The application of standard perturbation methods, harmonic balance, and other approximation techniques all reach the conclusion that the angular frequency has a singularity for a finite value of the initial amplitude A, where the initial conditions are x(0) = A and dx(0)/dt = 0. Since a phase-space analysis demonstrates that such a singularity does not exist, we must seek other methods to give the required valid behavior for the angular frequency as a function or the initial amplitude. This presentation reports our work using a method based on a polar representation for the periodic solutions. We compare these results with a priori calculations and give an explanation as to why the earlier calculations were ``interpreted'' as being incorrect.

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Authors

  • 'Kale Oyedeji

    Morehouse College

  • Philip Adams

    Department of Physics and Astronomy, Louisiana State University, ORNL, UT, ORNL/UT, UK, LSU, Louisiana State University, Zhejiang Normal University, Center for Computation and Technology, Louisiana State University, University of Tuebingen, Germany, Aristotle University of Thessaloniki, Greece, Max Planck Institute for Astrophysics, Max Plank Institute for Astrophysics, Laboratoire d'Astrophysique de Marseille, Albert Einstein Institute, California Institute of Technology, Dept. of Physics \& Astronomy, Louisiana State University, Baton Rouge, LA 70803, Dept. of Physics \& Astronomy, Louisiana State Univ., Baton Rouge, LA 70803, NASA Marshall Space Flight Center, Dept. of Physics \& Astronomy, Louisiana State University, Ohio State University, Wright State University, Department of Physics, North Carolina A\&T State University, Clark Atlanta University, Department of Electrical and Computer Engineering, Pennsylvania State University, University Park, P, Princeton University, University of New Orleans, Alabama A\&M University, Vanderbilt Univ., Konstanz Univ., Isik Univ., Department of Physics, Yale University