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The keyword you are looking for is " Microorganism Billiards". Very recent topic, but now seems to be catching up in fluids/bio community.

This topic has long history. Here are some references: Bloch, Anthony M. "Steepest descent, linear programming and Hamiltonian flows." Contemp. Math. AMS 114 (1990): 77-88. Brockett, Roger W. Dynami …

For the Restricted three-body problem, I suggest: Dynamical Systems, the Three-Body Problem and Space Mission Design By Marsden,Koon,Lo and Ross Available free at: www2.esm.vt.edu/~sdross/books This …

Dynamical systems is a huge field, with at least 3 (or more) subdisciplines which often interact with each other, but also have self-contained advances. Ergodic theory, topological dynamical systems, …

Q: Might there be a reason why stable physical processes would tend to have low-dimensional phase spaces. Yes. One reason is physical processes have dissipation. E.g., turbulence is "known" to be cha …

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional …

The problem of 'optimal path' for going to moon has been studied under the topic of "Circular Restricted three-body problem" and "Planar circular three-body problem (PCR3BP)". Poincare' made major con …

Please see: Maruskin, Jared M., Daniel J. Scheeres, and Anthony M. Bloch. "Dynamics of symplectic subvolumes." SIAM Journal on Applied Dynamical Systems 8.1 (2009): 180-201. Scheeres, D. J., et al. " …

Looking for literature / known results on the following class of problems: Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar fie …

It would be quite hard to give a purely analytical proof for continuous systems, since period doubling analysis (which is typically via Lyapunov-Schmidt bifurcation theory) will need to be carried on …

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional …

It cannot happen in a continuous time 2D system, simply due to uniqueness of ODE property. At least (2+1)-D is needed, i.e. this phenomenon can be seen in 2D maps derived from taking time-T sections o …

In dynamical systems, there is a concept of "almost-invariance", which generalizes invariance of a set, under the action of dynamics. The analogy is roughly the following: If you create a markov chai …

IF you are willing to extend into "braid theory and dynamics", there is quite a bit of activity in the field of "topological fluid mechanics" in last decade. Some of this work is directed at determi …

Here's my go to links: PDE flavor notes by Ryzhik: https://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf Probability flavor notes by Lacker: http://www.columbia.edu/~dl313 …