Finite-dimensional dynamical systems and complex behavior
in the hydrodynamical models with the memory effects
Alexander Makarenko
National Technical University of Ukraine "KPI"
Department of Mathematical Methods of System Analysis (1920)
Kiev, Ukraine
One of the approaches to the investigation of hydrodynamics equations (for example Navier-Stokes equations) is Galerkin method. Using this method it is easy to construct a low-dimensional dynamical system. In particular, for the Navier-Stokes equations one of such system is well known Lorentz system. But as was described previously by us more accurate is generalized hydrodynamics with memory effects. In this paper there are exposed some results on the construction of a low-dimensional analogs for such hydrodynamics and their comparison with common equations obtained under the same initial and boundary conditions.
In 1979 Boldrighini and Franceschini investigated five-dimensional systems on torus. For the sake of comparison we derive the analogous systems for the generalized hydrodynamics.
When 
(no memory effects) this system coinside with the
system from Boldrighini- Franceschini (1979). And in the case 
this
form is the singular perturbation of
Boldrighini- Franceschini system.
Also, in Makarenko (1994)
there was described systems of o.d.e. for generalized hydrodynamics with
memory effects for three-dimensional flows with slip boundary
conditions.
We present some properties of such systems and results of
computer modelling of them. First distinctive feature from the case
without the memory consists in the appearance of neutrally stable
oscillations in such systems. The second is the types of chaotic
behavior. In case typical is attractor of "butterfly" type as in
Lorentz's system. With 
there are complex behavior of new type.
The trajectory fill densely some bounded volume ("container").
Trajectory have broken form in many points (see Makarenko (1994)).
Visually behavior is familiar with these of two- dimensional mappings with
homoclinic tangencies and quasiattractors described by Gonchenko-
Shilnikov - Turaev (1993). Although some additional investigations are needed,
a previous analysis of the bifurcation points of our system (creation the
pairs of
conjugate roots of Jacobian and existence of neutrally stable
oscillations) support the possibility of such mechanism. Also some ideas of
memory effects in turbulence is considered. Remark also that the complex
behavior described may serve as prototype of new possible type of chaos in
the media with memory or in the media with finite speed of propagation.
This work is partially supported by State Science and Technology Committee of Ukraine and by ISF Grant No UAM 000 and Grant No UAM 200.