Energetics, skeletal dynamics, and long-term predictions on Kolmogorov-Lorenz systems

We study a particular return map for a class of low-dimensional chaotic models called Kolmogorov-Lorenz systems, which received an elegant general Hamiltonian description and also includes the famous Lorenz-63 case, from the viewpoint of energy and Casimir balance. In particular, a subclass of these models is considered in detail, precisely those obtained from the Lorenz-63 by a small perturbation on the standard parameters, which includes, for example, the forced Palmer-Lorenz case. The paper is divided into two parts. In the first part the extremes of the mentioned state functions are considered, which define an invariant manifold, used to construct an appropriate Poincaré surface for our return map. From the "experimental" observation of the simple orbital motion around the two unstable fixed points, together with the circumstance that these orbits are classified by their energy or Casimir maximum, we construct a conceptually simple skeletal dynamics valid within our subclass, reproducing quite well the Lorenz cusp map for the Casimir maximum. This energetic approach sheds some light on the "physical" mechanism underlying the regime transitions. The second part of the paper is devoted to an investigation of a type of maximum energy-based long-term predictions, by which knowledge of a particular maximum energy "shell" amounts to knowledge of the future (qualitative) behavior of the system. It is shown, in this respect, that a local analysis of predictability is not appropriate for a complete characterization of this behavior. A perspective on the possible extensions of this type of predictability analysis to more realistic cases in (geo)fluid dynamics is discussed at the end of the paper.