Kathleen A. Hoffman


The Forced van der Pol Equation I: The Slow Flow and its Bifurcations J. Guckenheimer, K. Hoffman, W. Weckesser Abstract: We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the EulerLagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values $\sigma \le 1$ at which a second ODE (the Jacobi equation) has a solution vanishing at $0$ and $\sigma$. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter $\lambda$, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of $\lambda$ of interest. Here we present an alternative approach that takes advantage of the presence of a parameter $\lambda$. Rather than search for conjugate points $\sigma \le 1$ at a fixed value of $\lambda$, we search for a set of special parameter values $\lambda_m$ (with corresponding Jacobi solution $\bfzeta^m$) for which $\sigma=1$ is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any $\lambda$ equals the number of these $\bfzeta^m$ for which $\langle \bfzeta^m, \Op \bfzeta^m \rangle < 0$, where $\Op$ is the Jacobi differential operator at $\lambda$. This computation is particularly simple when $\lambda$ appears linearly in $\Op$. We apply this approach to the elastic strut, in which the force appears linearly in $\Op$, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute twodimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of onedimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE. 
