Selçuk Journal
of
Applied Mathematics
WinterSpring,
2002
Volume 3
Number 1
Research Center
of
Applied Mathematics

SJAM
WinterSpring 2002, Volume 3  Number 1

Computation of periodic orbits in a threedimensional kinetic model
of catalytic hydrogen oxidation * 
Natalia A. Chumakova^{1},
Lubov G. Chumakova^{2}, Anna V. Kiseleva^{3},
Gennadii A. Chumakov^{4} 
^{1 }
Boreskov Institute of Catalysis, SB RAS, Prospect
Lavrentieva 5, 630090 Novosibirsk, Russia;
email:
chum@catalysis.nsc.ru
^{2 }Department of Mathematics, University of Wisconsin
Madison, Madison, WI 53706, USA;
^{3 }Department of Mathematics, Novosibirsk State
University, Pirogova 2, 630090 Novosibirsk, Russia;
^{4 }Sobolev Institute of Mathematics, SB RAS, Prospect
Koptyuga 4, 630090 Novosibirsk, Russia;
email:
chumakov@math.nsc.ru
Received: March 16, 2002^{
} 
Summary
An iterative method for solving
periodical boundaryvalue problem (BVP) for autonomous ordinary
differential equations (ODEs) is applied to calculations of periodic
orbits and their stability in a threedimensional kinetic model of
catalytic hydrogen oxidation.
According to the method, the periodic orbit is
decomposed into pieces by local crosssections $\{\pi_i\}$ and
between $\pi_i$ and $\pi_{i+1}$ the integration of the system is to
be accomplished. Hence we obtain an $\alpha$pseudoorbit and then
construct the generalized Poincare map. Thus the BVP for ODEs is
reduced to a system of nonlinear algebraic equations that takes into
account both the boundary conditions of periodicity and condition of
the solution continuity at boundary points of pieces. Being
linearized, the algebraic system has a band structure and for
solving such a system the orthogonal sweep method is extremely
effective.
In the
model considered we find numerically periodic orbits of rather
complex structure, give an example of weakly stable dynamics, and
show the role of successive period doubling bifurcations in the
creation of weakly stable dynamics.


Key
words
periodic orbit, numerical solution of
ordinary differential equations, chemical kinetic model, period
doubling bifurcation, weakly stable dynamics

Mathematics Subject
Classification (1991): 65L10, 65P30,
65P20, 37M05
^{
*}
This research was supported in part by the International Association
for the promotion of cooperation with scientists from the New
Independent States of the former Soviet Union (INTAS) grant No.
9901882.

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