Kristian Uldall Kristiansen

Kristian Uldall Kristiansen

Assistant Professor

Matematiktorvet, Building 303 B, Room 155

2800 Kgs. Lyngby

Denmark

Phone: 45253063

Fax: 4588 1399

2008 Cand. Scient. Applied Mathematics DTU

2010 PhD Applied Mathematics Uni Surrey, UK

2011 Visiting fellow, visiting assistant professor City U Hong Kong

2012 HC Ørsted Post doc DTU Mathematics

2013 onwards Assistant Professor DTU Mathematics

Research interests: Dynamical systems and perturbation theory

Recent Results:

With Claudia Wulff I have proved some conjectures of MacKay and have shown that in analytic slow-fast systems there exists an almost invariant slow manifold. Here almost is understood in the sense that the error-field is exponential small in epsilon, a small number measuring the time-scale separation. The almost invariant slow manifold is epsilon-close to a critical manifold. These results complement Fenichel's theorem as we only require that the normal motion is fast, including the normally elliptic case which is relevant for the Hamiltonian case. In this case the slow manifold can be made sympletic and we can define a reduced slow Hamiltonian system.

The method used to obtain these results is constructive and has previously been applied in computations of normally hyperbolic slow manifolds. There are alternatives to this method in the literature. It is unclear how these method compare. However, the CSP method due to Kaper, Zagaris and other co-workers can also approximate fiber directions. With Morten Brøns and Jens Starke I have shown that it is also possible to add a step to the method previously used by Claudia and I, so that fiber directions are also approximated. This is an iterative method, at each step envolving only linear equations, that seek to remove the part of the slow vector-field that is linear in the fast variable. This is even possible in the normally elliptic case. We are currently investigating the possibility of combining this method with collacation in computations of saddle-type slow manifolds.

Some future work:

* Extending results on almost invariant slow manifolds to unbounded vector-fields - currently the method can handle unbounded fast vector-field

* Application of result on almost invariant slow manifolds. Possibly looking at structural mechanical applications using the results in conjunction with modal expansions. This would complement centre manifold approaches that (artificially) adds dissipation in the form u_xxxxt to  push the eigenvalues corresponding to higher order modes far off the imaginary axis.

* Slow-fast Hamiltonian systems and (degenerate) KAM

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