Re: dynamical systems in R^2

spradlig@xxxxxxxx schrieb:
I have a question. Let f be a smooth vector field on R^2 and consider
the dynamical system x' = f(x), x:R->R^2. Suppose w(t) is a bounded
orbit of the dynamical system. I would like to assert that the limit
points of w(t) as t goes to infinity, that is, the omega-set of w, is
a fixed point of f or form a periodic orbit. Is this true, and can
anyone suggest a reference?

Yes, this is (almost) true. That's the Poincaré-Bendixson Theorem. I cite (from Hirsch/Smale: Differential Equations, Dynamical Systems And Linear Akgebra):

"A nonempty compact limit set of a C^1 planar dynamical system, *which contains no equilibrium point*, is a closed orbit."

(A dynamical system is here understood to be a continous flow map phi_t which arises for example from the solution set of planar vectorfields).

Do you see the subtle difference to your statement? There are examples of planar vectorfields where the limit set of a bounded orbit consists of 1 or more equilibrium points *together* with orbits joining them.

But otherwise, you're correct.

--
Thomas Nordhaus
.

Relevant Pages

  • Re: dynamical systems in R^2
    ... Greg Spradlin ... On Oct 24, 5:12 pm, Robert Israel ... orbit of the dynamical system. ... a fixed point of f or form a periodic orbit. ...
    (sci.math)
  • dynamical systems in R^2
    ... Let f be a smooth vector field on R^2 and consider ... orbit of the dynamical system. ... I would like to assert that the limit ... a fixed point of f or form a periodic orbit. ...
    (sci.math)