
Personal data  
Education and degrees  
Experience in scientific institutes  
Teaching experience  
Research interests  
Main research fields  
Current research  
Selected publications 
Personal data: 
Last name: Chumakov
First name: Gennadii
Middle name: Aleksandrovich
Data of birth: March 1, 1951 (Herson region, Ukraine)
Languages: Russian (native language), English (fluently)
Education and degrees: 
19691974  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".
Experience in scientific institutes: 
19741977  Junior Scientific Researcher, Boreskov Institute of Catalysis of the Siberian Branch of the USSR Academy of Sciences (Novosibirsk).
19771982  Junior Scientific Researcher, Computing Center of the Siberian Branch of the USSR Academy of Sciences (Novosibirsk).
19821985  Junior Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).
19851992  Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).
1992present  Senior Scientific Researcher, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk).
Teaching experience: 
19831989  Assistant Professor, Novosibirsk State University (Novosibirsk). ("Ordinary Differential Equations")
19891992  Associate Professor, Novosibirsk State University (Novosibirsk). ("Ordinary Differential Equations")
19921993  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")
19931999  Associate Professor, Novosibirsk State University (Novosibirsk). ("Advanced Ordinary Differential Equations")
1997present  Associate Professor, Novosibirsk State University (Novosibirsk). ("Mathematical Methods in Chemistry")
Research interests: 
Structured quasiisometric grids, domain decomposition, ordinary differential equations, bifurcation theory, chaos.
Main research fields: 
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 [2326] we discuss the phenomenon of "weakly stable dynamics" of the global
reaction rate in the nonlinear threedimensional model of catalytic hydrogen oxidation with fast, intermediate and slow variables. Its
connection with both a strange attractor in the 3D system and an influence of the global error in longterm 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
structurestable periodic orbit for autonomous ODEs is proposed. Interval of integration is divided into several
subintervals 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 2D QuasiIsometric Grids
A new methodology for the
quasiisometric grid generation was developed in recent papers
[8, 11, 16, 17, 22]. The generation of 2D quasiisometric grids is
connected with the construction of a $\sigma$quasiisometric
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
$\sigma$quasiisometric 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 quasiisometric 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
finitedifference numerical methods.
Domain Decomposition
A class of quasiconformal mappings for numerical grid generation in complex regions and a methodology for semiautomated block construction has been suggested in [6, 9]. The physical field is segmented into the subregions, bounded by four generally curved sides, and in each subregion an individual coordinate system is generated. Note that the problem of finding the subregions 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 $\sigma$quasiisometric mapping and have been working on a routine FORTRAN package SiGMA [19, 20] for numerical generation of quasiisometric 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 subregions.
Thus we have got semiautomated tools for domain decomposition, that is an interactive, graphicsoriented technology for 2D quasiisometric multiblock grid generation about almostarbitrary complex geometries of practical interest.
Current research: 
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 perioddoubling cascade can and do occur in the 3D 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 twodimensional 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 SmaleBirkhoff 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 3D 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.
QuasiIsometric Grids that are Orthogonal far from Corners
A special class of the quasiisometric mappings for the generation of quasiisometric 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 subregions 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 quasiisometric grid is generated. Orthogonality of coordinate lines holds in the fifth
block which is a conformal image of a nonconvex polygon composed of several rectangles on the plane.
Selected publications: 
1. Chumakov G.A. and Slinko M.G. (1978) Identification of the parameters of the autooscillatory heterogeneous catalytic reaction model. Dokl. Akad. Nauk SSSR, Vol. 243, no. 4, pp. 977980. (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 fourstage reactions. Dokl. Akad. Nauk SSSR, Vol. 253, no. 2, pp. 418421. (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. 653658. (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. 795803.
5. Chumakov G.A. and Slinko M.G. (1982) Kinetic turbulence (chaos) of the hydrogenoxygen interaction rate on metal catalysts. Dokl. Akad. Nauk SSSR, Vol. 266, no. 5, pp. 11941198. (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. 7583. (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. 133151. (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. 172180.
9. Chumakov G.A. and Godunov S.K. (1993) A method for the generation of twodimensional orthogonal far from the corners quasiisometric 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) Quasiisometric parameterization of curvilinear quadrangle and a metric of constant curvature. Siberian Advances in Mathematics, Vol. 5, no. 2, pp. 4867
11. Godunov S.K., Gordienko V.M. and Chumakov G.A. (1995) Variational principle for 2D regular quasiisometric grid generation. Int. J. Comp. Fluid Dyn., Vol. 5, pp. 99118.
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 2426, 1996. Eds.: J. Warnatz and F. Behrendt.
13. Chumakov S.G. and Chumakov G.A. (1997) A method for the generation 2D quasiisometric grids. Abstracts of the Third Mississippi State Conference on Differential Equations & Computational Simulations, Mississippi State University, May 1617, 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. 420421
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 "INPRIM98". 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 2D quasiisometric 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. 5579. http://www.emis.de/journals/EJDE/confproc/1997miss/chumakov
17. Chumakov G.A. and Chumakov S.G. (1998) A method for the 2D quasiisometric regular grid generation. J. Comput. Phys., Vol. 143, pp. 128.
18. Chumakov G.A. and Chumakov S.G. (1999) 2D regular quasiisometric 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) 2D quasiisometric grid around an airfoil, CRC Handbook of Grid Generation, p. B6, Joe F. Thompson, Bharat K. Soni and Nigel P. Weatherill (Eds.) CRC Press.
20. Chumakov G.A. and Chumakov S.G. (1999) SiGMA  a 2D quasiisometric grid generator, CRC Handbook of Grid Generation, p. A53, Joe F. Thompson, Bharat K. Soni and Nigel P. Weatherill (Eds.) CRC Press.
21. Chumakov G.A. and Chumakov S.G. (2000) 2D regular quasiisometric grids and Riemannian metric of harmonic parametrization of geodesic quadrangles on surfaces of constant curvature. In: International Conference "Geometry and Applications" (March 1316, 2000. Novosibirsk, Russia) Abstracts. Novosibirsk: IM SB RAS, 2000, pp. 2628.
22. Chumakov G.A. and Chumakov S.G. (2000) 2D regular quasiisometric 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 2528, 2000. B.K. Soni, J. Haeuser, J.F. Thompson, and P. Eiseman (Eds.) International Society of Grid Generation. September 2000, pp.10071016.
23. Chumakov G.A. and Chumakova N.A. (2001) On a global error estimate in longterm numerical integration of ordinary differential equations. Selcuk J. Appl. Math., Vol. 2, no. 1, pp. 2746.
24. Chumakova N.A., Chumakova L.G., Kiseleva A.V. and Chumakov G.A. (2002) Computation of periodic orbits in a threedimensional kinetic model of catalytic hydrogen oxidation. Selcuk J. Appl. Math., Vol. 3, No. 1, pp. 320.
25. Chumakov G.A. and Chumakova N.A. (2002) Different ways to weakly stable dynamics in a threedimensional kinetic model of catalytic hydrogen oxidation. Book of Abstracts, RussianDutch Workshop "Catalysis for sustainable development", pp. 238244, 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. 23, pp. 151158.
27. Chumakov G.A. and Chumakova N.A. (2003) Weakly stable dynamics in a threedimensional kinetic model of catalytic hydrogen oxidation. Chemistry for Sustainable Development, Vol. 11, pp. 6366.
28. Chumakov G.A. (2003) On 2D quasiisometric regular grids that are orthogonal far from corners. Applied Numerical Mathematics, Vol. 46, No. 34 , pp. 279294.
Last modified 10.01.09