Bifurcation diagrams illustrating the qualitative similarity of the cascading when either increasing or decreasing the bifurcation parameter. (a) Period-1 solutions exist on both extremes of the diagram. The box marks the region magnified on the right. (b) Period-3 solutions exists on both ends of the diagram.
(a) Bifurcation diagram along , indicated by the horizontal line in the right panel. (b) Lyapunov phase diagram discriminating periodicity and chaos in the control space. Colors denote chaos (i.e., positive exponents) while the darker shadings mark periodicity. Chaos prevails in the accumulation region around the letter despite the coloration. The color scale is linear on both sides of zero but not uniform. Note the high spread of exponents, indicating stiffness. The bifurcation diagram has a resolution of , while the phase diagram displays Lyapunov exponents.
Top row: phase diagrams illustrating structuring for three distinct parameter cuts over wide parameter ranges. Chaos prevails around the letter . Middle row: magnifications of the boxes in the corresponding panel above. Note the great similarity of the two rightmost diagrams with that in Fig. 2(b). Bottom row: replot of the panels in the middle row, but using a strict cut for zero exponents, as indicated by the color scales. To enhance contrast, the intensity of red was slightly increased. The granularity exposes intrinsic difficulties of calculating exponents that are close to zero (see text).
Successive magnifications illustrating the alternation of periodic and chaotic solutions in the space. Note the structural resemblance to parameter cuts shown in Figs. 2(b) and 3. Chaos prevails around the letter . Both axis were multiplied by to avoid unnecessarily long sequences of zeros. Thus, the minimum values of and in the upper leftmost panel are and , respectively.
Progressively more highly resolved phase diagrams illustrating the fine structure of the chaotic phases for selected regions of the parameter space. In the center panel of the middle row one sees period-adding cascades and accumulations similar to those observed in lasers, electric circuits and other models (Ref. 15). The bottom row shows that under high resolution the chaotic phase of the BZR is also riddled with shrimps (Refs. 17 and 18).
Numerical values of rate constants and parameters fixed in our simulations, taken from Ref. 20, in the same units.
Article metrics loading...
Full text loading...