I have studied literature on bifurcation of various systems. I observe two types of bifurcation diagrams, one was a curve(which is easy to interpret) but the second type is like a shaded region, which I am unable to interpret. For example look at the attached diagram. Interestingly, while drawing bifurcation for my model (Predator-prey model) I get something different.
Hello Wajid,
the "shaded regions" are the chaos onset display. The most interesting places inside that regions are the "white spaces", where you can find the 3-point attractors, which have important role in the transition to chaos. The "white regions" and their boundaries also show the instability of the initial conditions - another important aspect of the chaos.
The logistic map is a very well studied map. You can find the details in a lot of places, including Wikipedia. Or, look at https://demonstrations.wolfram.com/LogisticMapOnsetOfChaos/ for the "time series view" on the logistic map.
I add to the answer by Michael Zakharov the fact that in the "white spaces", for instance after 3.8 there are orbits of every period, which is a consequence of Sharkoski theorem, which says that the existence of a period-3 orbits implies that there are orbits of every period. They are not seen in the diagram since they are unstable.
At the white space after approximately 3.8 there are subsequent bifurcations which produce periodic orbits of period 6, 12 etc..
Also recall how the bifurcation diagram is generally produced. For a given parameter, one generates a large number of iterates of the function and generally the first iterates are not considered since the long run behaviour is what matter. One then plots these iterates. In the shaded region, for each parameterit appears that these iterates fulfil a whole interval, which shows that pobably there is a dense orbit, a orbit that is not periodic and fill and entire continuum, which is one characteristic of chaotic behaviour.
Naked-eye visual study of bifurcation diagrams can be tricky. One should also consider the impact of numerical simulation tools (see https://www.researchgate.net/publication/305682196_Bifurcation_Diagram_Behavior_for_Discrete_models_of_Rossler_system_depending_on_timestep) and possible initial conditions effect. It is always helpful to apply additional methods of analysis. E.g. plot the lyapunov spectrum beneath the bifurcation diagram to see where the oscillation mode really changed. Another interesting tool is phase shift diagrams, because amplitudal bifurcation diagrams are not that representative for some specific cases of chaos, e.g. Gavrilov-Shilnikov case and spiking systems. A further step to a deep analysis is the dynamical map.
Hi, Ram Singh
I flag it as a possible duplicate (joke, see the stackoverflow site). Anyway, on this site:https://www.researchgate.net/post/How_can_I_draw_a_bifurcation_plot_in_MATLAB
and many similar discussions can be found here.
The darker areas of the chaotic region imply that the system while being in the state of chaos, spends more time there than the lightly shaded regions. Despite chaos being highly sensitive to initial conditions, there is a trend observed in the bifurcation diagrams of a system of it spending more time in a specific region than others.
Have also look at http://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch2/bifurcation.html
https://www.youtube.com/watch?v=kczfyIdGxAw
https://www.youtube.com/watch?v=cC2w2z_i2DA
See this video, which explains what the diagram depicts, for continuous systems https://www.youtube.com/watch?v=OXTS-d9HdB4
If you are interested in discrete time systems, this video explains what the diagram depicts and how to interpret it https://www.youtube.com/watch?v=JgLyk304MYk
there is also a Matlab code available