This is a tutorial on some of the main ideas in KAM theory. The goal is to present the
background and to explain and compare somewhat informally some of the main methods of …

A variational principle is proposed and implemented for determining unstable periodic orbits
of flows as well as unstable spatiotemporally periodic solutions of extended systems. An …

In two previous papers [J. Differential Equations, 228 (2006), pp. 530-579; Discrete Contin.
Dyn. Syst. Ser. B, 6 (2006), pp. 1261-1300] we have developed fast algorithms for the …

Rotational invariant circles of area-preserving maps are an important and well-studied
example of KAM tori. John Greene conjectured that the locally most robust rotational circles …

We consider perturbations of integrable, area preserving nontwist maps of the annulus
(those are maps in which the twist condition changes sign). These maps appear in a variety …

In this paper, we develop numerical methods based on the weighted Birkhoff average for
studying two-dimensional invariant tori for volume-preserving maps. The methods do not …

We study numerically the disappearance of normally hyperbolic invariant tori in
quasiperiodic systems and identify a scenario for their breakdown. In this scenario, the …

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by
implementing three different methods. First, we analyze the domains of analyticity of a torus …

Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving
maps. Codimension-one tori are particularly important as they form barriers to transport …

The existence of an invariant surface in high-dimensional systems greatly influences the.
behavior in a neighborhood of the invariant surface. We prove theorems that predict the …