Georgia Institute of Technology College of Sciences

We present a method for the detection of stable and unstable fibers of invariant manifolds of periodic orbits. We show how to propagate the fibers to prove transversal intersections of invariant manifolds. The method can be applied using interval arithmetic to produce rigorous, computer assisted estimates for the manifolds. We apply the method to prove transversal intersections of stable and unstable manifolds of Lyapunov orbits in the restricted three body problem.

Conformally symplectic systems send a symplectic form into a multiple of itself. They appear in mecanical systems with friction proportional to the velocity and as Euler-Lagrange equations of the time discounted actions common in economics. The conformaly symplectic structure provides identities that we use to prove "a-posteriori" theorems that show that if we have an approximate solution which satisfies some non-degeneracy conditions, we can obtain a true solution close to the approximate one. The identities used to prove the theorem, also lead to very efficient algorithms with small storage and operation counts. We will also present implementations of the algorithms.

A common practice in aerospace engineering has been to carry out deterministicanalysis in the design process. However, due to variations in design condition suchas material properties, physical dimensions and operating conditions; uncertainty isubiquitous to any real engineering system. Even though the use of deterministicapproaches greatly simplifies the design process since any uncertain parameter is setto a nominal value, the final design can have degraded performance if the actualparameter values are slightly different from the nominal ones.Uncertainty is important because designers are concerned about performance risk.One of the major challenges in design under uncertainty is computational efficiency,especially for expensive numerical simulations. Design under uncertainty is composedof two major parts. The first one is the propagation of uncertainties, and the otherone is the optimization method. An efficient approach for design under uncertaintyshould consider improvement in both parts.An approach for robust design based on stochastic expansions is investigated. Theresearch consists of two parts : 1) stochastic expansions for uncertainty propagationand 2) adaptive sampling for Pareto front approximation. For the first part, a strategybased on the generalized polynomial chaos (gPC) expansion method is developed. Acommon limitation in previous gPC-based approaches for robust design is the growthof the computational cost with number of uncertain parameters. In this research,the high computational cost is addressed by using sparse grids as a mean to alleviatethe curse of dimensionality. Second, in order to alleviate the computational cost ofapproximating the Pareto front, two strategies based on adaptive sampling for multi-objective problems are presented. The first one is based on the two aforementionedmethods, whereas the second one considers, in addition, two levels of fidelity of theuncertainty propagation method.The proposed approaches were tested successfully in a low Reynolds number airfoilrobust optimization with uncertain operating conditions, and the robust design of atransonic wing. The gPC based method is able to find the actual Pareto front asa Monte Carlo-based strategy, and the bi-level strategy shows further computationalefficiency.

For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.