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 Senior scientific researcher, Dr. Gennadii A. Chumakov Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, prospect Koptyuga, 4, 630090 Novosibirsk, Russia Phone: (+7-383)363-45-37 Fax: (+7-383)333-25-98 E-mail: chumakov@math.nsc.ru

 Personal data Education and degrees Experience in scientific institutes Teaching experience Research interests Main research fields Current research Selected publications

Last name: Chumakov
Middle name: Aleksandrovich
Data of birth: March 1, 1951 (Herson region, Ukraine)
Languages: Russian (native language), English (fluently)

1969-1974 - Department of Mathematics, Novosibirsk State University (Novosibirsk).

1974 - Diploma (M. S.), Novosibirsk State University (Novosibirsk).

1985 - Candidate of Physical and Mathematical Sciences (Ph. D.), Novosibirsk State University (Novosibirsk), Ph. D. Thesis "Analysis of Mathematical Models of Rate Autooscillations in Heterogeneous Catalytic Reactions".

1974-1977 - Junior Scientific Researcher, Boreskov Institute of Catalysis of the Siberian Branch of the USSR Academy of Sciences (Novosibirsk).

1977-1982 - Junior Scientific Researcher, Computing Center of the Siberian Branch of the USSR Academy of Sciences (Novosibirsk).

1982-1985 - Junior Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).

1985-1992 - Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).

1992-present - Senior Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).

1983-1989 - Assistant Professor, Novosibirsk State University (Novosibirsk). ("Ordinary Differential Equations")

1989-1992 - Associate Professor, Novosibirsk State University (Novosibirsk). ("Ordinary Differential Equations")

1992-1993 - Associate Professor, Novosibirsk State University (Novosibirsk). ("Limit Cycles and Chaos in Dynamical Systems")

1993 - Associate Professor, Novosibirsk State University (Novosibirsk). ("Numerical Analysis of Ordinary Differential Equations")

1993-1999 - Associate Professor, Novosibirsk State University (Novosibirsk). ("Advanced Ordinary Differential Equations")

1997-present - Associate Professor, Novosibirsk State University (Novosibirsk). ("Mathematical Methods in Chemistry")

Structured quasi-isometric grids, domain decomposition, ordinary differential equations, bifurcation theory, chaos.

Nonlinear Dynamics and Chaos

A possible mechanism by which turbulent solutions in the autonomous systems of three ordinary differential equations
$$\mu \frac{dx}{dt} = f(x,y,z), \quad \frac{dy}{dt} = g(x,y,z),\quad \frac{dz}{dt} = \varepsilon h(x,y,z)$$
could appear as small parameters $\mu$ and $\varepsilon$ are varied as originally offered in [3, 5] and developed in [14, 15] . Chaotic behavior in numerical solutions of a mathematical model of the chemically reacting system was found in [5, 14].

In recent papers [23-26] we discuss the phenomenon of "weakly stable dynamics" of the global reaction rate in the nonlinear three-dimensional model of catalytic hydrogen oxidation with fast, intermediate and slow variables. Its connection with both a strange attractor in the 3-D system and an influence of the global error in long-term numerical integration of ODE's as a sorce of stochastic effects is also demostrated.

In the papers [12] and [24] a method being effective in parallel computation for finding of a structure-stable periodic orbit for autonomous ODEs is proposed. Interval of integration is divided into several sub-intervals and on each of them the integration of the system is to be accomplished. Then we construct a point mapping that takes into account not only the boundary conditions of periodicity, but also condition of continuity of the solution at boundary points of subintervals. Thus the BVP for ODE system is reduced to a system of nonlinear algebraic equations which after linearization has a band structure. For solving such a system the orthogonal sweep method proposed by S.K.Godunov is extremely effective.

Structured 2-D Quasi-Isometric Grids

A new methodology for the quasi-isometric grid generation was developed in recent papers [8, 11, 16, 17, 22]. The generation of 2-D quasi-isometric grids is connected with the construction of a $\sigma$-quasi-isometric mapping
$$x = x(\xi, \eta), \quad y = y(\xi, \eta) \eqno(1)$$
between points $(\xi, \eta)$ of the unit square ${\cal R} = \left\{(\xi,\eta): 0\leq\xi\leq1, 0\leq\eta\leq1 \right\}$ and the points $(x,y)$ of a physical region ${\cal D}$. By $\sigma$-quasi-isometric mapping an infinitesimal square will approximately go over into a parallelogram with lengths of sides $\Delta S_\xi$ and $\Delta S_\eta$ for which the following estimations hold:
$${\Delta \xi} / {\sigma} \le \Delta S_\xi \le \sigma \Delta \xi, \qquad {\Delta \eta} / {\sigma} \le \Delta S_\eta \le \sigma \Delta \eta.$$
The main advantage of the method proposed in [17] is that under certain condition the mentioned quasi-isometric mapping of ${\cal R}$ onto ${\cal D}$ is proven to exist uniquely and the conformally equivalent metrics of the mapping (1) are available analytically. Moreover the method provides a certain flexibility in the sense that two types of behavior of the boundary points are admitted: they might be chosen fixed or may move along the boundary; the method allows more direct control of the grid cells size and quality as the grids is refined what is important for finite-difference numerical methods.

Domain Decomposition

A class of quasi-conformal mappings for numerical grid generation in complex regions and a methodology for semi-automated block construction has been suggested in [6, 9]. The physical field is segmented into the sub-regions, bounded by four generally curved sides, and in each sub-region an individual coordinate system is generated. Note that the problem of finding the sub-regions sides and coordinate systems formulated as a variational problem, minimizing some quadratic functional. In such sort of methods it is efficient to use parallel computations.

Computational Tools Development

Recently we have proposed the new technique for finding the unknown parameters of the sought $\sigma$-quasi-isometric mapping and have been working on a routine FORTRAN package SiGMA [19, 20] for numerical generation of quasi-isometric grids based on the method mentioned above. The robustness of the algorithm is increased by using the Euler interior boundaries described in [6], by means of which the physical domain ${\cal D}$ can be segmented into sub-regions.

Thus we have got semi-automated tools for domain decomposition, that is an interactive, graphics-oriented technology for 2-D quasi-isometric multi-block grid generation about almost-arbitrary complex geometries of practical interest.

A Kinetic Model with a Strange Attractor

We represent numerical evidence for the existence of the strange attractor $\Lambda$ which involve the identification of a hyperbolic invariant set and provide numerical analysis of the chaotic flow within the attractor $\Lambda$ in terms of a Poincare map. Due to quite different time scales in the model, we replace the actual reversible 3D flow near the attracting set $\Lambda$ by a semiflow $S$ in which solutions are only defined forward in time.

Afterwards we consider a sequence of period doubling bifurcations [27]. As we see, a direct subharmonic cascade and an inverse period-doubling cascade can and do occur in the 3-D system if we let a control parameter vary. At the begining of the direct cascade we find a stable limit cycle $p$. At the first flip bifurcation $p$ becomes unstable and it has stable and unstable manifolds which we denote as $W^s(p)$ and $W^u(p)$. Moreover, the two-dimensional unstable manifold $W^u(p)$ is twisted around the periodic orbit $p$ like a Möbius band around its center line. Numerical observations indicate that the direct and inverse cascades have an accumulation point. Therefore there exists an attractor with an infinite number of unstable periodic orbits. Moreover, we show there is a transversal homoclinic orbit on $\Lambda$, that is, there exists a point $q \neq p$ of transversal intersection between $W^s(p)$ and $W^u(p)$.

In this case the Smale-Birkhoff homoclinic theorem allows us to deduce that the attractor $\Lambda$ has a hyperbolic invariant set: a Smale horseshoe. The sensitive dependence on initial conditions which it engenders in the flow of the 3-D system is of great interest.

As we know, now "homoclinic orbit" and corresponding "Smale's horseshoe" have turned out to be the trademark of chaos and strange attractor. Thus, the results imply that $\Lambda$ is a strange attractor and the typical trajectory on $\Lambda$ will be asymptotically chaotic.

Quasi-Isometric Grids that are Orthogonal far from Corners

A special class of the quasi-isometric mappings for the generation of quasi-isometric regular coordinate systems orthogonal far from corners is discussed [28]. The base computational strategy of our approach is that the physical field is decomposed into five nonoverlapping sub-regions which are automatically generated by solving a variational problem. Four of these blocks containing four corners on the boundary of the physical region are conformal images of geodesic quadrangles on surfaces of constant curvature. Within each of these blocks a quasi-isometric grid is generated. Orthogonality of coordinate lines holds in the fifth block which is a conformal image of a non-convex polygon composed of several rectangles on the plane.

1. Chumakov G.A. and Slinko M.G. (1978) Identification of the parameters of the auto-oscillatory heterogeneous catalytic reaction model. Dokl. Akad. Nauk SSSR, Vol. 243, no. 4, pp. 977-980. (Russian).

2. Chumakov G.A., Beliayev V.D., Plikhta R., Timoshenko V.I. and Slinko M.G. (1980) Number and stability of the steady states of four-stage reactions. Dokl. Akad. Nauk SSSR, Vol. 253, no. 2, pp. 418-421. (Russian).

3. Chumakov G.A., Slinko M.G. and Beliayev V.D. (1980) Complex variations of heterogeneous catalytic reaction rate. Dokl. Akad. Nauk SSSR, Vol. 253, no. 3, pp. 653-658. (Russian).

4. Ivanov E.A., Chumakov G.A., Slinko M.G. e.a. (1980) Isothermal sustained oscillations due to the influence of adsorbed species on the catalytic reaction rate. Chem. Eng. Science, Vol. 35, pp. 795-803.

5. Chumakov G.A. and Slinko M.G. (1982) Kinetic turbulence (chaos) of the hydrogen-oxygen interaction rate on metal catalysts. Dokl. Akad. Nauk SSSR, Vol. 266, no. 5, pp. 1194-1198. (Russian).

6. Godunov S.K., Romenskii E.I. and Chumakov G.A. (1990) Construction of difference grids by the quasiconformal transformation. Proc. of Inst. of Mathematics SB RAS, Vol. 18, pp. 75-83. (Russian).

7. Chumakov G.A. (1992) Riemmanian metric of the harmonic parameterization of geodesic quadrangles onto the surfaces of constant curvature. Proc. of Inst. of Mathematics SB RAS, Vol. 22, pp. 133-151. (Russian).

8. Chumakov G.A. (1993) Conformal parameterization of the curvilinear quadrangles by geodesic quadrangles onto surfaces positive constant curvature. Siberian Math. J., Vol. 34, no. 1, pp. 172-180.

9. Chumakov G.A. and Godunov S.K. (1993) A method for the generation of two-dimensional orthogonal far from the corners quasi-isometric grids. Abstracts of the 7th International Conference on Domain Decomposition Methods. Penn. St. Univ.

10. Godunov S.K., Gordienko V.M. and Chumakov G.A. (1995) Quasi-isometric parameterization of curvilinear quadrangle and a metric of constant curvature. Siberian Advances in Mathematics, Vol. 5, no. 2, pp. 48-67

11. Godunov S.K., Gordienko V.M. and Chumakov G.A. (1995) Variational principle for 2-D regular quasi-isometric grid generation. Int. J. Comp. Fluid Dyn., Vol. 5, pp. 99-118.

12. Chumakov G.A. and Chumakova N.A. (1996) Method of computation of structurally stable periodic solutions of autonomous systems of ODE. Modeling of Chemical Reaction Systems. Proceedings of an International Workshop, Heidelberg, Germany, July 24-26, 1996. Eds.: J. Warnatz and F. Behrendt.

13. Chumakov S.G. and Chumakov G.A. (1997) A method for the generation 2-D quasi-isometric grids. Abstracts of the Third Mississippi State Conference on Differential Equations & Computational Simulations, Mississippi State University, May 16-17, 1997.

14. Chumakov G.A. and Chumakova N.A. (1997) Deterministic approach to kinetic chaos modeling in hydrogen catalytic oxidation. Abstracts of the memorial Boreskov Conference "Catalysis on the eve of the XXI century. Science and engineering". pp. 420-421

15. Chumakov G.A. and Chumakova N.A. (1998) Weakly stable dynamics of heterogeneous catalytic hydrogen oxidation. 3rd International Congress on Industrial and Applied Mathematics "INPRIM-98". Book of abstracts. Part IV. - Novosibirsk, IM SB RAS. P. 81.

16. Chumakov G.A. and Chumakov S.G. (1998) (to appear) Harmonic parameterization of geodesic quadrangles on surfaces of constant curvature and 2-D quasi-isometric grids. Differential Equations and Computational Simulations III, J.Graef, R.Shivaji, B.Soni & J.Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 55-79. http://www.emis.de/journals/EJDE/conf-proc/1997-miss/chumakov

17. Chumakov G.A. and Chumakov S.G. (1998) A method for the 2-D quasi-isometric regular grid generation. J. Comput. Phys., Vol. 143, pp. 1-28.

18. Chumakov G.A. and Chumakov S.G. (1999) 2-D regular quasi-isometric grids. Mathematics in Applications. International Conference honoring academician Sergei K. Godunov. Abstracts, Novosibirsk, Russia. p. 37

19. Chumakov G.A. and Chumakov S.G. (1999) 2-D quasi-isometric grid around an airfoil, CRC Handbook of Grid Generation, p. B-6, Joe F. Thompson, Bharat K. Soni and Nigel P. Weatherill (Eds.) CRC Press.

20. Chumakov G.A. and Chumakov S.G. (1999) SiGMA - a 2-D quasi-isometric grid generator, CRC Handbook of Grid Generation, p. A-53, Joe F. Thompson, Bharat K. Soni and Nigel P. Weatherill (Eds.) CRC Press.

21. Chumakov G.A. and Chumakov S.G. (2000) 2-D regular quasi-isometric grids and Riemannian metric of harmonic parametrization of geodesic quadrangles on surfaces of constant curvature. In: International Conference "Geometry and Applications" (March 13-16, 2000. Novosibirsk, Russia) Abstracts. Novosibirsk: IM SB RAS, 2000, pp. 26-28.

22. Chumakov G.A. and Chumakov S.G. (2000) 2-D regular quasi-isometric grids and Riemannian metric of harmonic parameterization of geodesic quadrangles on surfaces of constant curvature.  In: Proceedings of the 7th International Conference on  Numerical Grid Generation in Computational Field Simulations, September 25-28, 2000. B.K. Soni, J. Haeuser, J.F. Thompson, and P. Eiseman (Eds.) International Society of Grid Generation. September 2000, pp.1007-1016.

23. Chumakov G.A. and Chumakova N.A. (2001) On a global error estimate in long-term numerical integration of ordinary differential equations.  Selcuk J. Appl. Math., Vol. 2, no. 1, pp. 27-46.

24. Chumakova N.A., Chumakova L.G., Kiseleva A.V. and Chumakov G.A. (2002) Computation of periodic orbits in a three-dimensional kinetic model of catalytic hydrogen oxidation. Selcuk J. Appl. Math., Vol. 3, No. 1, pp. 3-20.

25. Chumakov G.A. and Chumakova N.A. (2002) Different ways to weakly stable dynamics in a three-dimensional kinetic model of catalytic hydrogen oxidation. Book of Abstracts, Russian-Dutch Workshop "Catalysis for sustainable development", pp. 238-244, Novosibirsk.

26. Chumakov G.A. and Chumakova N.A. (2003) Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards. Chem. Eng. J., Vol. 91, No. 2-3, pp. 151-158.

27. Chumakov G.A. and Chumakova N.A. (2003) Weakly stable dynamics in a three-dimensional kinetic model of catalytic hydrogen oxidation. Chemistry for Sustainable Development, Vol. 11, pp. 63-66.

28. Chumakov G.A. (2003) On 2-D quasi-isometric regular grids that are orthogonal far from corners. Applied Numerical Mathematics, Vol. 46, No. 3-4 , pp. 279-294.