KAM Islands Research Articles

A mapping tool using anisotropic unstructured meshes to study mixing in periodic flows

The objective of the present work is to describe a new mapping tool using anisotropic unstructured meshes to study mixing within a spatially periodic pipe flow. Instead of tracking the boundaries of elementary cell flow domains as it was done in the original mapping method established by Kruijt et al. (A.I.Ch.E. J. 47 (5) (2001a) 1005; Int. Polym. Process. 16 (2) (2001b) 151) and Galaktionov et al. (Comput. Fluids 30 (3) (2001) 271), the deformation of elementary triangles (only three nodal points) between the inlet and exit pipe cross-sections is followed. It is however necessary to adapt the initial mesh according to criteria which takes into account the spatial stretching and folding of fluid elements. The method developed is applied to the twisted curved pipes (TCP) three-dimensional (3D) flow. We show the evolution of concentration distributions along the TCP mixer for chaotic advection flow regimes. This method allows the emphasis of isolated unmixed regions (KAM islands). The flexibility of the method allows also the possibility of studying multiple stirring protocols, thus contributing to a better comprehension of the physical phenomena involved in chaotic mixing. The method developed is also applicable to 2D temporally periodic flows.

Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos.

We introduce a technique, which we term bailout embedding, that can be used to target orbits having particular properties out of all orbits in a flow or map. We explicitly construct a bailout embedding for Hamiltonian systems so as to target invariant tori. We show how the bailout dynamics are able to lock onto extremely small regular islands in a chaotic sea.

Mushrooms and other billiards with divided phase space.

We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a "chaotic sea" (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary. (c) 2001 American Institute of Physics.

Clustered motion in symplectic coupled map systems

Clustered motion of particles is found in Hamiltonian dynamics of symplectic coupled map systems. Particles assemble and move with strong correlation. The motion is chaotic but is distinguishable from random chaotic motion. Lyapunov analysis distinguishes global instability from local fluctuations. Clustered motions have finite lifetime. They have fractal geometric structure in the phase space, as the orbits are trapped to ruins of KAM tori and islands.

CHAOTIC SCATTERING IN TIME-DEPENDENT HAMILTONIAN SYSTEMS

We consider the classical scattering of particles in a one-degree-of-freedom, time-dependent Hamiltonian system. We demonstrate that chaotic scattering can be induced by periodic oscillations in the position of the potential. We study the invariant sets on a surface of section for different amplitudes of the oscillating potential. It is found that for small amplitudes, the phase space consists of nonescaping KAM islands and an escaping set. The escaping set is made up of a nonhyperbolic set that gives rise to chaotic scattering and remains of KAM islands. For large amplitudes, the phase space contains a Lebesgue measure zero invariant set that gives rise to chaotic scattering. In this regime, we also discuss the physical origin of the Cantor set responsible for the chaotic scattering and calculate its fractal dimension.