### Multidimensional manifold continuation for adaptive boundary-value problems

- Harry Dankowicz
- Yuqing Wang
- et al.

- 2020
- J. Comput. Nonlinear Dyn.

I am a member of the Environmental Science and Natural Resources Group at IBM's T.J. Watson Research Center. I have a BS in Engineering Science from The Pennsylvania State University, and a PhD in Applied Mathematics from The California Institute of Technology. My basic research interest is nonlinear systems, especially bifurcation theory and continuation methods.

Continuation methods are about solving a nonlinear problem that depends on parameters $\lambda$:

$F(u,\lambda) = 0$

Continuation methods assume that one solution $F(u_0,\lambda_0)=0$ is known and smoothly vary $u$ and $\lambda$ to extend the solution manifold away from the initial points. If the Jacobian $F_{u,\lambda}$ is full rank at a point, the Implicit Function Theorem applies, and near the point the solution manifold is a smooth $k$-dimensional surface.

If $\lambda$ is a scalar the method is sometimes called path following. My speciality is algorithms which can be used when $\lambda$ is a vector of length $k$.

My multidimensional continuation code 'Multifario' is available under an Open Source license from SourceForge

When the Jacobian isn't full rank, several smooth $k$--dimensional manifolds may cross at the point, and bifurcation theory can be used to determine what happens. If $F$ is the flow field of a dynamical system

$\frac{d u}{d t} = F(u,\lambda)$

the solution manifold is a manifold of fixed points, and bifurcations to other motions can occur.

I'm a member of The Jefferson Project, and environmental monitoring and modeling project on Lake George, in the Adirondack Mountains, and Chautauqua Lake, both in New York. From my standpoint the challenge is the same -- to find and make visible complicated dynamics, natural or mathematical, so that they can be studied and understood.

### Multidimensional manifold continuation for adaptive boundary-value problems

- Harry Dankowicz
- Yuqing Wang
- et al.

- 2020
- J. Comput. Nonlinear Dyn.

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