On estimation of the global error of numerical solution on canard-cycles

Modeling the complex dynamics of heterogeneous catalytic reactions with fast, intermediate, and slow variables

“…We have further developed the scenario leading to the chaotic dynamics appearance suggested by Chumakov and Slinko [7]. In addition, we have developed methods for calculation of the periodic solutions with large periods and estimation of the global error of numerical integration (see [12]). …”

“…In [12,13] we showed that a highly sensitive dependence on initial conditions appears due to the existence of a tunnel-type bundle of trajectories which is formed by stable and unstable canards. Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast as time increases.…”

“…The unstable periodic orbits of S u evolve from harmonic to relaxation oscillations as z increases. This happens via canard explosion over a very small range of parameter z since z 0 À z 2 H % 3:8 Á 10 À4 [12]. The unstable periodic orbits of S u emerge from the Andronov-Hopf bifurcation at z ¼ z 2 H , grow explosively in amplitude from the harmonic oscillations of small amplitude (cycle 5 0 ) via the canard-cycles without head (cycles 4 0 , .…”

“…6 presents the chaotic behavior in the phase space of concentrations x, y, and z (a) and the time dependence of z-coordinate (b). To be sure in the computational results, we used some special procedures of higher accuracy with a posteriori estimation of the global error of numerical integration [12].…”

“…The maximal family of periodic orbits of system (9) for k 2 = 8 with the orbits corresponding to the following values of parameter: z = 010), 0.3116 (11), 0.3025(12), and 0.2954(13). The Andronov-Hopf bifurcation also occurs at z 2 H ¼ 0:381826.…”