Selçuk Journal of Applied Mathematics

 Selçuk Journal of
  Applied Mathematics

 Winter-Spring, 2002
Volume  3
 Number 1

Research Center of 
  Applied Mathematics

 SJAM Winter-Spring 2002, Volume 3 - Number 1

Computation of periodic orbits  in a three-dimensional kinetic model of catalytic hydrogen oxidation *

Natalia A. Chumakova1, Lubov’ G. Chumakova2, Anna V. Kiseleva3, Gennadii A. Chumakov4

1 Boreskov Institute of Catalysis, SB RAS, Prospect Lavrentieva 5, 630090 Novosibirsk, Russia;

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;

Received: March 16, 2002


An  iterative method for solving periodical boundary-value problem (BVP) for autonomous ordinary differential equations (ODEs) is applied to calculations of periodic orbits and their stability in a three-dimensional kinetic model of catalytic hydrogen oxidation.

According to the method, the periodic orbit is decomposed into pieces by local cross-sections $\{\pi_i\}$ and between $\pi_i$ and $\pi_{i+1}$ the integration of the system is to be accomplished. Hence we obtain an $\alpha$-pseudo-orbit 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 co-operation with scientists from the New Independent States of the former Soviet Union (INTAS) grant No. 99-01882.

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