$\dot X = Q(X) $ on $\mathbb{R}^3$, where $Q$ is quadratic. Is there a condition on the coefficients which guarantees a homoclinic point (a complicated attractor)? E.g. like for Reynolds numbers?

# Applications

Among degree $n$ polynomial maps of $[0,1]$ to itself, are Axiom A open and dense. Do bad ones form a stratified set? ...

codon frequencies via equilibrium states for ``some potential''?

Invariant or approximatively invariant finite dimensional subspaces for Navier-Stokes equation

Calculate $h$ for O.D.E. systems on $\mathbb{R}^n$, e.g. linear equations first

Electric circuits

- Analogue computer for finding Axiom A examples
- Is noise sometimes due to hyperbolicity in the dynamics?

Correspondence principle of quantum mechanics. Investigate for some simple mechanical systems. Is h-expansiveness related to quantum ....?

Equilibrium states for 1-dimensional quantum lattice systems without finite range

Is $h(\varphi_t | E)$ differentiable in $E$ for Hamiltonian flows? Any relation to classical or quantum statistical mechanics?

Find Axiom A infinite attractor in some O.D.E. on $\mathbb{R}^3$ (quadratic).

Entropy in Hamiltonian case; for P.D.E.'s? Relation to O.D.E.'s?