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

#### Abstract

Aniterative 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.