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Results tagged with ds.dynamical-systems
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user 5484
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
15
votes
Why do dynamicists worry so much about differentiability hypotheses in smooth dynamics?
There are some fascinating phenomena in dynamical systems and related fields whose existence depends on the degree of differentiability.
Among all of these, my favorite is the fact, proved by Haeflig …
- 2,858
4
votes
Accepted
Noninteger iterates of functions: How to get ODE from flow at a given time?
If g is a real-analytic function defined near x0 with g(x0) = x0 and 0 < λ ≠ 1 where λ := g'(x0), then Koenigs proved that there exists a real-analytic homeomorphism h defined near x0 such that hgh-1( …
- 2,858
-2
votes
Vector field on 3-sphere
Suppose a vector field V on S3 has K > 0 source singularities and L sink singularities, and no others. Then by the Poincaré-Hopf index theorem, K - L = 0, so K = L.
Now for each singularity, remove a …
- 2,858
3
votes
2
answers
457
views
Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
{(a z2 + b …
- 2,858
2
votes
Analytic ODE with complex time
One thing to be careful about is that even for an analytic ODE given on ℂ via
dz/dt = f(z)
where f is an entire function, the solutions Φ(z,t) always exist for all (z,t) in some open neighborhood of …
- 2,858
2
votes
Complementary integrable vector fields
Wasn't it André Haefliger who proved that there exists no real-analytic foliation of the 3-sphere by surfaces?
- 2,858