Return map analysis of the highly nonlinear Bray–Liebhafsky reaction model

Abstract

By numerically simulated Bray–Liebhafsky (BL) reaction under a continuously fed well stirred tank reactor (CSTR) conditions, we discussed the attractors and Poincaré 1D maps with respect to flow rate as the control parameter. The new technique of the return maps from transient trajectories over the slow manifold is developed and applied in order to explore its multilayered structure related to dynamical states (periodic and aperiodic -chaotic oscillating modes) of the system. Kinetic relations underlying the slow manifold structure are briefly discussed.

Appendix: Standard numerical methods

In the return map method, the attractor is first cut by the selected surface in the phase space. Furthermore, only those points are identified where trajectory passes through the selected surface from one chosen direction. Hence, each cycle is counted only once. Thus, quasi-continuous trajectory (line) is completely discretized—transformed to a set of points in the surface. However, the connection between the points is not obvious any more leaving us without the information about their sequence in time. Since we have this information from the original data series, we can use it to construct two-dimensional map, equivalent to the Poincare section, by plotting one chosen variable of the precursor point, against the same variable for the successor point. The Poincaré map, or return map is created this way [22]. Period doubling is indicated by duplicated number of corresponding cross points in the map in comparison to their number in previous dynamical state of the period doubling sequence. Thus, in the case of periodic motions, cross points have finite number and are distinctively localized in the map. In the case of chaotic dynamics, in contrast to periodicity, cross points continuously fill in a part of the map.