I know very little about complex (meaning over the field of complex numbers, and not in the sense of complexity) dynamical systems. But by chance I’m invited to give a talk in a seminar on complex dynamics in Paris this week, so I have to find a topic which is close to my work and which would at the same time interest “complex dynamicians”. The topic will be “complexification of dynamical systems
Jacques Hadamard said: “Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.” This is indeed also true for dynamical systems. I will give a series of examples to show how the complexification of real dynamical systems can be useful in various problems such as:
* Local normal forms
* Semi-local and global invariants
* Local symmetry groups
* Obstructions to integrability
Example 1. Smooth vector fields on a circle.
In the case when the vetor field is non-singular, i.e. it does not vanish anywhere on the circle, the only invariant up to smooth isomorphisms is the period, i.e. the time it takes to go one full cycle around the circle.
If denotes the vector field, and denotes the 1-form dual to the vector field (so that the contraction of and is 1), then the period (which is the inverse of the frequency) can be defined as:
Consider now the case when admits nondegenerate singular points on the circe, i.e. singular points at which has a non-trivial linear part. Then it is easy to see that:
* The number of singular points is an even number: half of them are attractive (sinks) and half of them are repulsive (sources), each sink is adjacent to two sources and vice versa. The number of singular points is an invariant of up to isomorphisms.
* The eigenvalues of at the singular points, in a cyclic order and up to an orientation, are also smooth invariants of .
* In order to have a complete set of invariants of up to isomorphisms, one needs to add one more invariant, namely the regularized period, or also called the monodromy of X.
This regularized period can be defined by “jumping over the walls” using local (-preserving) involutions near nondegenerate singular points. This definition works very well, but may look a bit misterious. Another more intuitive definition of the regularized period is to complexify the system (assuming for the moment that it is analytic) near the circle. Then the regularized period is nothing but the period along a circle in the complex plane which is near the original circle but which doesn’t contain any singular point. In fact, all the eiganvalues can also be recovered by the difference of the periods of different (non-homotopic) loops which avoid the singular points: the eigenvalue at a singular point corresponds (up to multiplication by ???) the period of the loop around that point in the complex plane.
The complexification does not give a proof why the period (defined in the complex domain) is a real invariant, and why it is the only missing invariant. Nevertheless, it gives the invariant in a very intuitive way. The formal proof which comes after (or before — actually we found the invariant before realizing that it could be obtained easily by complexification) is not very difficult.
Example 2: Regularized periods & cohomology classes of integrable Hamiltonian systems.
Consider now a slightly more complicated example: a 1-degree-of freedom Hamiltonian function on a 2D surface, with a nondegeerate hyperbolic singular level set. How to classify neighborhoods of such level sets (up to fibration-preserving symplectomorphisms, or up to exact system-preserving symplectomorphisms) ? What about the higher-dimensional case ?
This question had been studied by many people: Dufour / Molino / Toulet / Bolsinov / Kruglikov / Vu Ngoc / Dullin / etc. Their definitions of the invariants didn’t involve any complex analysis (the considered systems are real). But with the help of complexification, the invariants will become more natural and easier to understand.
Example 2′. Similar semi-global invariants, but for non-Hamiltonian systems (Minh & Zung, in terms of Puiseux series)
Example 3. Existence of torus actions near singularities of integrable systsems.
The conditions under which the torus action exists are very mild, but they involve a local complexification of the system.
Example 4. Differential Galois obstructions to integrability, based on partiular complex solutions (even though the original system is real)
Example 5. Local normal forms for real systems, based on intrinsic local torus actions in the complex domain.
The above 5 examples are probably enough for a 1-hour talk ?!