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Results tagged with ds.dynamical-systems
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user 2906
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2
votes
0
answers
89
views
Counting orbits of the standard map
Consider the standard map. Might it happen that for some nonzero parameter value $K$ and for some positive integer $q$ that there exist an infinite number of periodic orbits having period $q $ I woul …
- 8,336
1
vote
Anosov diffeomorphisms and the chaotic hypothesis
Take the standard map with its break down of invariant circles and all,
Cantorii and the like. How would you start to approximate by Anosov?
I am thinking of Aubry/Mather/Bangert results on 2 degr …
- 8,336
2
votes
Accepted
Non-polynomial integrals of motion for polynomial dynamical systems
Here is a tentative Hamiltonian answer having 2 degrees of freedom.
Take $H = (1/2)(1 + a x^2 + bxy + cy^2) (p_x ^2 + p_y ^2)$.
for essentially any parameters $a, b, c$ for which
$ax^2 + bxy + cy …
- 8,336
4
votes
0
answers
48
views
Getting analytic center manifolds
The center manifold of a degenerate zero of an analytic vector field need not be unique nor analytic. But say I want it analytic. Does anyone know of additional conditions to be imposed on the vecto …
- 8,336
3
votes
Accepted
The fibers of the momentum map for the $SO(n+1)$ symmetry of the geodesic flow on $S^n$
You may want to look into the notion of a dual pair in symplectic and Poisson geometry
to place your examples in a more general context.
Your first example is one of the canonical examples of a dual …
0
votes
Persistence of fixed points under perturbation in dynamical systems
In order to develop your intuition, you might want to start
with a flow having no fixed points -- so a constant nonzero vector field, say
in the plane. Perturb it by a nice vector field, eg. polynomi …
- 8,336
3
votes
Accepted
symplectic topology of (perturbed) KAM tori
No. (Answer courtesy of Jacques Féjoz, via email correspondence.) The invariant torus might be translated in the actions,in which case the perturbed torus is not Hamiltonian isotopic to the original. …
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4
votes
4
answers
1k
views
Continuous pointwise ergodic theorem?
Let $\Phi$ be a homeomorphism of a compact metric space $M$
which preserves a regular Borel
probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. )
Under these hypothesis, I have two questio …
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2
votes
Good books on Geometric Theory of Dynamical Systems
Take a look at Bob Easton's book Geometric Methods for Discrete Dynamical Systems'.
He was a student of Conley's, who took a very topological/geometric perspective. Conley's tiny monograph,Isolated In …
3
votes
Is there a singularity theorem in higher-dimensional Newtonian gravity?
Following the lines Willie followed, but allowing for unequal masses $m_i$, set $I(x) = \langle x, x \rangle $ where $\langle v, w \rangle = \Sigma m_i v_i \cdot w_i$ is the so-called mass metric …
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4
votes
Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$
There is a wonderful trick - I think promulgated by Moser - for viewing Hamiltonian flows on the cotangent bundle of the sphere as the reduction of a Hamiltonian flow on the ambient phase space of ( …
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2
votes
Certain flow on space of lattices in $\mathbb C$
Please look at
http://www.josleys.com/articles/ams_article/Lorenz3.htm
section 3 on Modular flows and see if that answers your question.
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5
votes
When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riema...
This is a kinda stupid answer but it is the simplest most useful test I know.
Does your Hamiltonian vector field have a zero? If `yes': sorry, not a geodesic flow!
(Could be the reduction of a geodes …
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2
votes
Stable periodic orbits for three equal masses
The proposer gives as their definition of stability the standard notion
of Lyapunov stability. Unfortunately, there are no known solutions for the planar or spatial three-body problem which are Lyapun …
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7
votes
How can generic closed geodesics on surfaces of negative curvature be constructed?
Are you willing to buy that the set of non-closed geodesics are dense? If so, here is an argument that goes back to Birkhoff and Hadamard. Take the surface's
universal cover -- which is to say the …
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