Closed orbits of vector fields under perturbation

Question

Consider a vector field $V$ on an annulus $U$, say. Also, assume that the vector field $V$ has a closed orbit. I am looking for a reference that gives stability results of the following type:

If the vector field $V$ satisfies properties (collectively called) A, then under small perturbations of type (collectively called) B of the vector field, the perturbed vector field would still have a closed orbit which is "close" in sense C to the original closed orbit.

The best situation would be to find a body of results for different A, B and C.

My background in dynamical systems is very rudimentary, and the prospect of reading 5 volumes to work a solution out for myself is daunting. Any help would be highly appreciated.

Edit (after Robert Israel's answer): The situation I am looking at does not necessarily have a closed orbit, and now I have checked that the vector fields can indeed be tangent to the boundaries. I was wondering what we can say in such situations (I don't necessarily need existence of a closed orbit. I am more interested in the stability of such an orbit in case it exists). Also, please include a reference if you can, I really need to read the theory up myself.

Answer 1

If you consider for condition B $C^1$ small perturbations of the vector field and view the periodic orbit as an invariant manifold, then a necessary and sufficient condition for it to persist under small perturbations is that it is normally hyperbolic, for original references see Mañé, Transactions AMS 246 for necessity and Fenichel, Indiana Univ. Math. J. 21, and Hirsch, Pugh & Shub, Lecture Notes in Math. 583 for sufficiency.

In the particular case of a periodic orbit, this means that its Poincaré return map has to have its eigenvalues off the complex unit circle. The result is that the periodic orbit persists and is $C^1$ close to the original one (in a sense that can be made more precise by considering the perturbed orbit as a graph over the normal bundle of the original orbit). You may want to for this and "limit cycles" in an advanced ODE textbook. I don't have a precise reference, but possibly "ODEs and applications" by Chicone or "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields" by Guckenheimer and Holmes may be useful.

However, this theory doesn't work (straightforwardly) when the orbit touches the boundary of your domain; it might be modifiable to work there, though.

Answer 2

Suppose there are two positively oriented simple closed curves $C_1$ and $C_2$ with $C_1$ inside $C_2$, such that there are no fixed points of $V$ in the region $R$ between $C_1$ and $C_2$, and at every point of $C_1 \cup C_2$ the vector field "points into" $R$. Thus if $T$ is a tangent vector to $C_1$ in the forward direction at a point $p$ on $C_1$, $V_1 T_2 - V_2 T_1 > 0$, while if $T$ is a tangent vector to $C_2$ in the forward direction at a point $p$ on $C_2$, $V_1 T_2 - V_2 T_1 < 0$. Then there is a closed orbit in the region $R$. The same is true if we change "points into" to "points out of", reversing both inequalities. This condition is stable under small perturbations of $V$.

EDIT: Without some condition similar to this, closed orbits can be unstable. For example, consider a system described in polar coordinates by

$$ \eqalign{\dfrac{dr}{dt} &= f(r)\cr \dfrac{d\theta}{dt} &= 1\cr}$$

where $f(1) = 0$ and $f(r) \ge 0$ in some neighbourhood of $1$. This has a closed orbit $r = 1$. But if you add any $\varepsilon > 0$ to $f(r)$, the closed orbit disappears.