When we want to plot a bifurcation diagram for a flow or map, we should consider some important points. Some of them have been mentioned in [1]. For example we should be careful about coexisting attractors, transient behaviors, proper quality of the diagram (resolution, marker size,…), and dealing with more than one bifurcation parameters.
I want to know if there are additional important points that some may encounter during obtaining a bifurcation diagram.
Thanks in advance,
Sajad
[1] J. C. Sprott, “A proposed standard for the publication of new chaotic systems” Int. J. Bifurcation Chaos, 21, 2391 (2011).
I also agree that a bifurcation diagram is not the only tool to analyze evolutionary behaviour of any dynamical system. As real systems are mostly nonlinear, bifurcation diagram may not be sufficient to discuss complete dynamic properties. Along with this, one should compute phase diagram, Lyapunov exponents, power spectrum plot, plot of topological entropy etc to understand complexities in the system. If the system show chaos, then correlation dimensions also help in understanding fractal properties.
Dear Sajad,
If you want ot have a thorough reference of all the challenges involved in bifurcation analysis, I recommend you this book:
http://www.springer.com/mathematics/dynamical+systems/book/978-0-387-21906-6?token=gbgen&wt_mc=GoogleBooks.GoogleBooks.3.EN
The XPPAut by Ermentrout could be also very useful for you:
http://books.google.es/books?id=Qg8ubxrA060C&printsec=frontcover&hl=es&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
With best regards,
David Arroyo
I think the only thing required is the novelty of your results concerning such a system and its bifurcation parameters space. Otherwise anything else can be considered as standard.
If you are talking about bifurcation diagrams obtained by using brute-force methods, as I guess, another important issue is the transient duration, which must be fixed according to two opposite constraints: short enough to avoid too long simulation times and long enough to avoid missclassifications of the asymptotic behaviors. A standard trick is to start the next trajectory from the last point of the previous one, with additional care when you detect a bifurcation.
You can have a look at http://sourceforge.net/projects/biflib/
Hope this helps!
Many of the problems have alreaddy been answered. In my opinion th most cucial is (as Marco mentioned) to find the right time for each simulation in order to get a fast and accurat simulation. Since one normally follows one attractor by taking the previous state in to the next simulation the transients can be significantly reduced. The problem of multiple solutions is somthing to be aware of and the result you get is one branche of the dynamics. A normal problem is also a deviation of phase in the poincare section. Since the simulations are made during long time it is easy to get too big numbers in the time. When a small timestep is added it will result in errors. A good approach here is to use a local time scale and a global time scale. Then the phase can be between 0-2pi.
Another suggestion is to use a simulation with adaptive time step which is much faster.
Finally I like to have a simulation of largest lyapunov exponent together with the bifurcation diagram. It can normally explain a lot of the behaviours and also give hint to wats going on.
While plotting a bifurcation diagram following points necessary to note:
(1) Fixed points of the system with respect to a set of parameters and study their stability criteria.
(2) Drawing bifurcation diagram by varying only one parameter within its allowable range while keeping other parameters. Evolution should start from a point nearby a stable fixed point.
(3) If it is a multidimensional system, bifurcation diagram should be drawn with respect to each variable for thorough analysis
Do not forget that a bifurcation diagram by itself is a nice tool but not the only tool to analyze the dynamic behaviour of a dynamical system. You should combine with other tools, such as computing orbits in phase space or Lyapunov exponents diagrams, etc...
I also agree that a bifurcation diagram is not the only tool to analyze evolutionary behaviour of any dynamical system. As real systems are mostly nonlinear, bifurcation diagram may not be sufficient to discuss complete dynamic properties. Along with this, one should compute phase diagram, Lyapunov exponents, power spectrum plot, plot of topological entropy etc to understand complexities in the system. If the system show chaos, then correlation dimensions also help in understanding fractal properties.
In touch with the last answer, I think that the use of general informational entropies, such as Rénji's, and/or the Jaynes' max entropy principle, may give good clues about changes in phase transitions and the onset of bifurcations.
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