Grant 96-01-01791

*Mathematical Theory
of Fluid Free Convection *

*(Asymptotics, Transitions,
Turbulence) *

professor Yudovich Victor I.

- assoc. prof. Zenkovskaya Svetlana M.
- assoc. prof. Tsybulin Vyacheslav G.
- assoc. prof. Kurakin Leonid G.
- assoc. prof. Kolesov Vadim V.
- assoc. prof. Revina Svetlana V.
- assoc. prof. Petrovskaya Nataly V.
- ass. prof. Govorukhin Vasiliy N.
- ass. prof. Morgulis Andrew B.
- asp. Shleikel' Alexey L.

Considerably full theory of Poincare -- Andronov -- Hopf bifurcation was developed for dynamical systems with one or several cosymmetries. New effects connected with existence of continuous families of equilibria or stationary regimes (that is typical for the systems with cosymmetries) were found: delay in respect to parameter of branching off of self-oscillating regime, branching off of unstable self-oscillating regimes to a supercritical domain, branching off of quasiperiodic regimes from equilibrium family.

We studied the new regimes of motion (isolated stationary regimes and self-oscillating regimes) arising as a result of cosymmetry-breaking perturbations of a dynamical system.

Analogs of the implicit function theorem were proved for dynamical systems with continuous or discrete time if there exist one or several invariant Pfaff forms. We found the conditions for existence of continuous equilibrium families. The applications to ordinary differential equations, and also to nonlinear vector wave equations are given.

The influence of polutions on vibrational gravitational convection is studied via computer experiment and applying of asymptotical methods. The possibility for control of convection via changing of vibration intensity and concentrations of ingredients under both gravitation and weightlessness are investigated.

We investigate bifurcations near the point of intersection of bifurcations in the systems with cylindrical symmetry, especially in the case of Couette-Taylor flow. Numerical calculation of the coefficients of the amplitude systems are conducted with application of asymptotical and analytical methods. Limit cycles are calculated, and computer experiments on bifurcations leading to chaos are carried out. Some results on the infinite doubling bifurcation of cycles, homoclinic bifurcations, crises of attractors are obtained.

Publications

- Yudovich V. Cosymmetry and dynamical systems. Book of Abstracts. 3d Int. Congress of ICIAM Hamburg, 3-7 July, 1995. ZAMM, 1996. Pp. 556-559.
- Tsybulin V., Yudovich V. Attractors of Quadratic Mapping of Plane: Computer Experiment and Analytical Treatment. Book of Abstracts. 3d Int. Congress of ICIAM Hamburg, 3-7 July, 1995. ZAMM, vol.4, 1996. Pp. 552-555.
- Kolesov V., Ovchinnikova S., Petrovskaya N., Yudovich V. Onset of Chaos through Intersections of Bifurcations in Couette-Taylor Flow. Book of Abstracts. 3d Int. Congress ofICIAM Hamburg, 3-7 July, 1995. ZAMM, 1996. Pp. 548-551.
- Govorukhin V. Computer Experiments with Cosymmetric Models // Proc. ICIAM-95, ZAMM, vol.4, 1996, 556-559.
- Öèáóëèí Â.Ã. Ðåàëèçàöèÿ ðàçíîñòíîé ñõåìû ñóâåðåííûõ ñêîðîñòåé äëÿ ðàñ÷åòà òå÷åíèé âÿçêîé æèäêîñòè // Òðóäû êîíô. "Ñîâðåì. ïðîáëåìû ìåõàíèêè ñïëîøíîé ñðåäû", ò. 1, Ðîñòîâ-íà-Äîíó,