Renormalising dynamical systems

In a general sense, dynamics has two goals: to describe the typical behaviour of trajectories for many systems (of any kind: physical, biological, economic...) as time tends to infinity, and to understand how this behaviour changes with perturbations. The natural question that arises then is to know to what extent a system is stable. That is, we want to find whether it is possible to deform a perturbed system in a way such that we recover the original one. In that case we call the systems equivalent or conjugated, so that the qualitative behaviour (sometimes also quantitative) of the trajectories of each system can be studied on the other one. This task is not easily done and understood for most of the non-linear systems, even the simplest ones. The order of complexity (the so-called chaos) of the solutions can sometimes be too hard to tackle, turning all of the known scientific techniques obsolete.
The idea of "re-normalising" came from physics, inspired by the scaling laws that one observes in phase transitions and their explanation from statistical mechanics. In the late 1970's, Feigenbaum and, independently, Coullet and Tresser performed numerical experiments with one-parameter families of unimodal maps of the interval. They found a remarkable property: the existence of a universal scaling law realised in a constant that appears in the transition between regular and chaotic dynamics. This discovery called for a reflection on the established idea of absence of order in mathematical chaos. The value of the ``Feigenbaum constant'' seems to be common for all such systems, and a renormalisation explanation was proposed for its existence. More specifically, by defining a special scaling transformation in an appropriate space of dynamical systems, the existence of a hyperbolic fixed point with a one-dimensional unstable manifold corresponding to the universal behaviour was conjectured. This is an important result recently proved by the continuous efforts of Sullivan, Lyubich and 1998's Fields medallist McMullen, but extensions for larger dimensions and more "real" systems remain undone. MacKay also found this kind of behaviour numerically for the break up of the invariant tori in two-dimensional area-preserving maps. He constructed a renormalisation operator acting in the space of pairs of commuting maps and developed the same ideas for volume-preserving flows. Others like Stark, Khanin and Sinai have also contributed in these directions.
The renormalisation method can be used to study vector fields on the multidimensional torus close to the linear flow. Thus to prove the local analytic equivalence of flows to the linear one with generic frequency. In a particular case one obtains Arnol'd's famous solution to the problem of local analytic circle diffeomorphism conjugacy to a rotation. Roughly, the renormalisation group technique analyses systems on longer time scales and smaller spatial scales, generating a new system. By iterating this we may get convergence to a limiting behaviour, preferably a fixed point. In such a case, we get self-similarity (which can be non-trivial depending on the problem). The application of this procedure for more complicated systems like Hamiltonian vector fields is also possible.
The well-known work of Kolmogorov, Arnol'd and Moser (KAM theory) showed that invariant tori, typical of integrable Hamiltonian systems of more than one degrees of freedom, persist under small perturbations. This achievement was very important as it finally proved the long-standing "Ergodic Hypothesis" wrong. The main idea of the earlier proofs of the KAM Theorem is to find sequences of canonical coordinate changes which, in the limit, reduce the original Hamiltonian function to an integrable one. That is not always possible because of the divergence associated with the existence of small denominators in the generating functions. The improvement of this theory is the construction of an operator (the renormalisation) that asymptotically separates two distinct classes of Hamiltonians: those with invariant tori with a given frequency in phase space, and those without. The border between these classes will be the set of Hamiltonians with a critical torus on the verge of breaking up. This idea was mainly developed in the works by Koch, Abad, Jauslin and Chandre in recent years. The generalisation of their definitions and results of the renormalisation operator to a larger set of frequencies and higher dimensions remains undone.
There are some applications that may arise from this study. As any system that evolves in time (or is an iterative process) is a dynamical system, the sort of questions above can be of fundamental importance in disciplines as different as genetics, engineering, finance, geophysics, thermonuclear fusion or celestial mechanics. In fact, much work has already been done in those areas based on the dynamical systems theory, despite being a pure mathematical discipline. The essential characteristic of those systems is the nonlinearity of the ruling equations, which makes them unsolvable and chaotic.

non-twist map
strong persistence of the twistless torus

João Lopes Dias

31st January 2001

~ Cambridge ~ England ~

Zooming in the Koch curve.
cf. rescaling and self-similarity