Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

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 Euler--Lagrange 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 two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE

``Critical'' phonons of the supercritical Frenkel-Kontorova model: renormalization bifurcation diagrams

The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have nontrivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these ``degeneracy points'' the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, nonexponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.Comment: 10 pages, RevTeX, 9 Postscript figure

On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular

The main part of this contribution to the special issue of EJM-B/Fluids dedicated to Patrick Huerre outlines the problem of the subcritical transition to turbulence in wall-bounded flows in its historical perspective with emphasis on plane Couette flow, the flow generated between counter-translating parallel planes. Subcritical here means discontinuous and direct, with strong hysteresis. This is due to the existence of nontrivial flow regimes between the global stability threshold Re_g, the upper bound for unconditional return to the base flow, and the linear instability threshold Re_c characterized by unconditional departure from the base flow. The transitional range around Re_g is first discussed from an empirical viewpoint ({\S}1). The recent determination of Re_g for pipe flow by Avila et al. (2011) is recalled. Plane Couette flow is next examined. In laboratory conditions, its transitional range displays an oblique pattern made of alternately laminar and turbulent bands, up to a third threshold Re_t beyond which turbulence is uniform. Our current theoretical understanding of the problem is next reviewed ({\S}2): linear theory and non-normal amplification of perturbations; nonlinear approaches and dynamical systems, basin boundaries and chaotic transients in minimal flow units; spatiotemporal chaos in extended systems and the use of concepts from statistical physics, spatiotemporal intermittency and directed percolation, large deviations and extreme values. Two appendices present some recent personal results obtained in plane Couette flow about patterning from numerical simulations and modeling attempts.Comment: 35 pages, 7 figures, to appear in Eur. J. Mech B/Fluid

Equilibria in the secular, non-coplanar two-planet problem

We investigate the secular dynamics of a planetary system composed of the parent star and two massive planets in mutually inclined orbits. The dynamics are investigated in wide ranges of semi-major axes ratios (0.1-0.667), and planetary masses ratios (0.25-2) as well as in the whole permitted ranges of the energy and total angular momentum. The secular model is constructed by semi-analytic averaging of the three-body system. We focus on equilibria of the secular Hamiltonian (periodic solutions of the full system), and we analyze their stability. We attempt to classify families of these solutions in terms of the angular momentum integral. We identified new equilibria, yet unknown in the literature. Our results are general and may be applied to a wide class of three-body systems, including configurations with a star and brown dwarfs and sub-stellar objects. We also describe some technical aspects of the semi-numerical averaging. The HD 12661 planetary system is investigated as an example configuration.Comment: 18 pages, 17 figures, accepted to Monthly Notices of the Royal Astronomical Societ

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page