Some breathers and multi-breathers for FPU-type chains

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are close to (possibly infinite) sums of breather solutions

Torsional instability in suspension bridges: the Tacoma Narrows Bridge case

All attempts of aeroelastic explanations for the torsional instability of suspension bridges have been somehow criticised and none of them is unanimously accepted by the scientific community. We suggest a new nonlinear model for a suspension bridge and we perform numerical experiments with the parameters corresponding to the collapsed Tacoma Narrows Bridge. We show that the thresholds of instability are in line with those observed the day of the collapse. Our analysis enables us to give a new explanation for the torsional instability, only based on the nonlinear behavior of the structure

On a nonlinear nonlocal hyperbolic system modeling suspension bridges

We suggest a new model for the dynamics of a suspension bridge through a system of nonlinear nonlocal hyperbolic differential equations. The equations are of second and fourth order in space and describe the behavior of the main components of the bridge: the deck, the sustaining cables and the connecting hangers. We perform a careful energy balance and we derive the equations from a variational principle. We then prove existence and uniqueness for the resulting problem

On the dynamics of coupled oscillators and its application to the stability of suspension bridges

We describe and provide a computer assisted proof of the bifurcation graph for a system of coupled nonlinear oscillator described in a model of a bridge. We also prove the linear stability/instability of the branches of solutions

Optimization of the Forcing Term for the Solution of Two-Point Boundary Value Problems

We present a new numerical method for the computation of the forcing term of minimal norm such that a two-point boundary value problem admits a solution. The method relies on the following steps. The forcing term is written as a (truncated) Chebyshev series, whose coefficients are free parameters. A technique derived from automatic differentiation is used to solve the initial value problem, so that the final value is obtained as a series of polynomials whose coefficients depend explicitly on (the coefficients of) the forcing term. Then the minimization problem becomes purely algebraic and can be solved by standard methods of constrained optimization, for example, with Lagrange multipliers. We provide an application of this algorithm to the planar restricted three body problem in order to study the planning of low-thrust transfer orbits

Periodic orbits, symbolic dynamics and topological entropy for the restricted 3-body problem

This paper concerns the restricted 3-body problem. By applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits, the existence of symbolic dynamics and we give a rigorous lower estimate for the topological entropy.Comment: 22 pages. Includes Mathematica source file r3b.n

Some symmetric boundary value problems and non-symmetric solutions

We consider the equation −Delta u = wf ′(u) on a symmetric bounded domain in Rn with Dirichlet boundary conditions. Here w is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions u that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric solutions and the non-existence of symmetric solutions. In particular, we construct a solution u for the disk in R2 that has index 2 and whose modulus |u| has only one reflection symmetry. We also provide a corrected proof of [12, Theorem 1]

Validated numerical solutions for a semilinear elliptic equation on some topological annuli in the plane

We consider the equation −Δw=w3 with zero boundary conditions on planar domains that are conformal images of annuli. Starting with an approximate solution, we prove that there exists a true solution nearby. Our approach is computer-assisted. It involves simultaneous and accurate control of the (inverse) Dirichlet Laplacean, nonlinearities, and conformal mappings

A Hopf bifurcation in the planar Navier-Stokes equations

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.Comment: 16 pages, 1 figur