Dear Sundarapandian Vaidyanathan

As you mentioned, the bifurcation diagram allows us to analyze the dynamics of the systems, in this case, chaotic systems. The idea in eliminating a certain amount of data in the time series (no matter if are 1000 secs, 1500 points in the time series, or whatever) before constructing the bifurcation diagrams, lies in eliminating the transitory state of the system and focus on what it is known as steady-state dynamics. I think that is what is referred to in the articles you describe, where the amount of time referred to is not so relevant, and it would suffice to say that the dynamics of the steady-state is analyzed, or that they have not been considered the transitory states for the construction of the bifurcation diagrams.

Dear Sundarapandian, My guess is that this is because Bifurcation Diagrams tend to exhibit exponential and chaotic growth, certainly very quickly and in the long-term. I would assume that authors omit the first 1000 seconds precisely because of this chaotic behaviour, which can never be perfectly reproduced (unless, arguably, one knew the exact initial conditions). So, early on, as José Luis Echenausía-Monro suggests, it makes no difference whether you start off with 1000, 1500, 2000, or 3000, time coordinates etc, or anything else of that kind of magnitude, the overall and long-term picture will have the same kind of "form" irrespective of the initial period. Whatever the initial series of time coordinates are, the long-term nature of the bifurcation process is to settle down to a steady-state, even if it is an unstable one, so an exact picture of the first 1000 seconds is immaterial when it comes to understanding the system's long-term behaviour.

A bifurcation diagram shows changes in a system's long-term behaviour when a system parameter is varied. As such, the transient information is not relevant and only the long-term behaviour of the system is taken into account (hence the omission of the initial number of samples).

If transient behaviour is also of interest, then one would use a phase portrait in conjunction with a bifurcation diagram.

Dear Colleagues,

I totally agree with the above. I would like to mention, however, that if we want to have a clear picture of the long-term behavior of a chaotic system, the amount of data to be eliminated is important. I think of the so-called sticky orbits, where the transition from order to chaos or vice versa is very slow (see, for example, Henon - Heyles system or Gumowski - Mira map). If you want to analyze a very large number of orbits, the computation time must be taken into account. A compromise is required between precision and efficiency.

Dear Sundarapandian Vaidyanathan,

In the same line, I will like to strengthen that bifurcation diagram is a qualitative measure which highlights the abrupt changes which occur in a dynamical system when varying certain control parameter. The knowledge of metastable chaotic dynamics helps to discard sufficiently long transient in order to reveal only steady-state oscillations. The choice of this transient regime is somehow tricky and depends on investigated systems. One of our recent work which will be soon published underlined the existence of double transient regimes is dynamical systems. This justifies that scientist should look (if it doesn't exist yet) for straight methods in identifying transients regimes.

The bifurcation diagram shows and confirms the existence of the chaotic behavior, all depends on the parameters of the non-linear dynamic system. In general, the amount of data to be taken to draw the bifurcation diagram, depends on the dynamic system type.

Bifurcation diagram helps us to understand the limiting behaviour of the systems. After any steps dynamical systems approximately approach to the limiting behaviour.

This is done to remove possible transient and to ensure states are close to or on the attractor.

Dear Sundarapandian Vaidyanathan,

A number of the first points of the orbit are discarded, to allow time for the orbit to evolve into its final behavior, whether this is a periodic or chaotic orbit - this is called eliminating the transient. Because the interest is to show the orbit convergence values, therefore the transient values are not used.

Best regards.