### Dynamically meaningful latent representations of dynamical systems

- 2024
- Mathematics

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.

### Dynamically meaningful latent representations of dynamical systems

- 2024
- Mathematics

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

- Harry Dankowicz
- Yuqing Wang
- et al.

- 2020
- J. Comput. Nonlinear Dyn.

### Planning Singularity-Free Paths on Closed-Chain Manipulators

- Oriol Bohigas
- Michael E. Henderson
- et al.

- 2013
- IEEE Transactions on Robotics

### A singularity-free path planner for closed-chain manipulators

- Oriol Bohigas
- Michael E. Henderson
- et al.

- 2012
- ICRA 2012

### Flow Box Tiling Methods for Compact Invariant Manifolds

- 2011
- SIADS

### Assisted seismic matching: Joint inversion of seismic, rock physics and basin modeling

- 2007
- SEG 2007

### Computing Invariant Manifolds by Integrating Fat Trajectories

- 2005
- SIADS

### Multiparameter Parallel Search Branch Switching

- 2005
- IJBC

### A survey of methods for computing (un)stable manifolds of vector fields

- Bernd Krauskopf
- Hinke M Osinga
- et al.

- 2005
- IJBC

### Classification of the Spatial Equilibria of the Clamped Elastica: Numerica Continuation of the Solution Set

- Michael E. Henderson
- Sebastien Neukirch

- 2004
- IJBC

- 26 Feb 2024
- US
- 11914043

- 02 May 2022
- US
- 11322033

- 26 Jul 2021
- US
- 11074319

- 25 Jan 2021
- US
- 10902260

- 10 Jun 2019
- US
- 10319229

IN

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