Homoclinic tangles-classification and applications

@article{RomKedar1994HomoclinicTA,
  title={Homoclinic tangles-classification and applications},
  author={Vered Rom-Kedar},
  journal={Nonlinearity},
  year={1994},
  volume={7},
  pages={441-473}
}
Here we develop the topological approximation method which gives a new description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian flows. It is based upon the structure of the homoclinic tangle, and supplies a detailed solution to a transport problem for this class of systems, the characteristics of which are typical to chaotic, yet not ergodic dynamical systems. These characteristics suggest some new criteria for quantifying transport and mixing-hence… 

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References

SHOWING 1-10 OF 60 REFERENCES

Analysis of chaotic mixing in two model systems

We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible

An analytical study of transport, mixing and chaos in an unsteady vortical flow

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field

Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map

Scattering from a classically chaotic repellor

We report a study of the classical scattering of a point particle from three hard circular discs in a plane, which we propose as a model of an idealized unimolecular fragmentation. The system

Symplectic maps, variational principles, and transport

Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond

Chaotic advection in a Rayleigh-Bénard flow.

  • CamassaWiggins
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1991
TLDR
By modeling the flow with a stream function, it is shown how to construct and identify invariant structures in the flow that act as a ‘‘template’’ for the motion of fluid particles, in the absence of molecular diffusivity.

Trellises formed by stable and unstable manifolds in the plane

A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diSeomorphism of a 2-manifold. This paper describes and classifies some trellises. The set of

Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations

Both upper and lower estimates are establishedfor the separatrix splitting of rapidly forced systems with a homoclinic orbit. The general theory is applied to the equation φ + sin φ =S sin(^t-_Є)

Introduction to Applied Nonlinear Dynamical Systems and Chaos

Equilibrium Solutions, Stability, and Linearized Stability * Liapunov Functions * Invariant Manifolds: Linear and Nonlinear Systems * Periodic Orbits * Vector Fields Possessing an Integral * Index
...