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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 …

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 …

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 …

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 …

Model predictive control (MPC, aka receding horizon control) is one type of sub-optimal control method that is extremely well studied and popular. The "sub-optimality" of this type of control methods …

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. " …

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 …

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 …

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 …

Robinson: Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors

I guess "Geometrical Methods in the Theory of Ordinary Differential Equations" by Arnold should be in the list too, although it doesn't satisfy the "purely" topological criteria.

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

I am assuming you are interested in multidimensional case $x\in\mathbb R^n$. Let $\Omega$ be the set whose invariance you are interested in estabilishing. There are two types of problems here: A). …

You can get chaos with a cubic term for $x_1$. In applied dyanmical systems, there has been considerable interest in last decade to study systems of the following form: $\ddot{x_1}=-ax_1^3+\epsilon(x_ …

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 …