When plotting a bifurcation diagram in nonlinear dynamics, the axis x displays a given phase parameter. Are there examples in which the phase parameter stands for time passing (for example, from the value T0 to the value T200 seconds, or months, or years)?
Thanks!
To make an example, I was thinking to something like the one in the Figure below, concerning the phase transitions among liquids, solids and gases: if you leave, e.g,., that the temperature raises of one degree every second, can we say that the axis x displays time (apart temperature values)?
This is a great question with lots of possible paths to following in looking for answers.
A comprehensive view of bifurcation diagrams for dynamical systems is given in
Sergei N. Taraskin and Francisco J. P´erez-Reche
Bifurcations in synergistic epidemics on random regular graphs
https://arxiv.org/pdf/1809.05575.pdf
Four sample BFs plus a view of the time aspects of the BFs are are given in the attached images.
Bifurcation diagrams from a perturbed vortex system dynamics is discussed in
P. E. Ryabov
Bifurcation diagram of one perturbed vortex dynamics problem
https://arxiv.org/pdf/1811.10512.pdf
Dear Jim, Hi!
Thanks for your very skillful answer!
However, the examples you mention have phase parameters (called alpha, beta, p, k) that are not properly the once i require, i.e., the arrow of time.
I would like to know whether a bifurcation diagram does exist where the single line bifurcates in two lines with time passing, and not due to atemporal changes in a phase parameter.
I want a single path in a phase space which bifurcates irreversibly, when the progression of the arrow of time takes place...
Dear Jim, one second ago I provided an example of what I'm saying in the text of my question (see above, under the question).
Dear Jean-Pierre,
Hi!
I started from the old medical technique of Thromboelastography (which was a lot of time ago performed by my father), which describes peculiar curves (SEE THE FIGURE BELOW) based on the viscoelastic properties of the blood during normal coagulation and its diseases.
I world like to extend these peculiar curves in order to describe other dynamical systems, such as, for example, Volterra equations for preys.
Put simply, I'm looking for dynamical systems which trajectories in time can be described in terms of viscoelastic changes occurring either in the (biological of physical) system, or in the medium in which the system is embeddded.
Look at the recapitulative Figure below. I want to find the same type of curves, but in other contexts. To make two examples, remind that in Alzheimer's disease the brain viscoelasticity is increased, and that neurons are stiffer than glial cells. If I put somehow the brain electric waves'paths in the Figure, I may describe brain temporal activity in terms of viscoelastic variations, both normal and pathological... Remind, also, that materials with different viscoelasticity transmit electric waves with different frequency (an explanation for EEG's different waves, such as alpha, gamma and so on?). Is anyone interested in such approach?
Dear Jean-Pierre,
thanks for your interest!
gimme a couple of days, and I will let you know what I have in mind...
My thoughts are in progress, but I'm starting to focus the feasible result... I will need you, if, of course, you will like my idea...
...this is what I had in mind...
Preprint RELATIONSHIPS BETWEEN BRAIN STIFFNESS AND EEG FREQUENCIES SU...
Dear Arturo,
your question is indeed interesting. The conventional bifurcation diagrams are plotted for time-invariant dynamical systems, that is, the dynamics do not change with time, but of course they change with the parameters. Your question implies a time-varying system, which is not the conventional scenario considered in the literature in the context of bifurcations and that is what makes the question interesting. What comes to my mind (for this is, I believe, what would happen in practice) is a system for which the dynamics depend on parameters (nothing new up to here), but the parameters depend on time in a possibly known way. So, suppose the dynamics depends on a parameter alpha, but alpha changes with time in a certain way... then we indicate alpha(t). Now, if you make a bifurcation diagram in terms of alpha (as it would usually be the case), but if you know how alpha behaves with time (here you probably have another dynamical system that determines the change of alpha... which could just be a drift), then it would not be absurd to show a bifurcation diagram in terms of time... Well, just a few thought for you to considier.
Dear Louis Antonio, you exactly catched what I had in mind... In my paper on Alzheimer's disease I took a different path, but your words suggest that my starting idea is worth to be pursued...In touch with your idea, in a published paper of mine, we inserted the parameter time in the context of the Shannon entropy's plot.
Article A timeless biology
Dear Arturo,
great! Glad to know you are in what seems to be a promissing direction.
All the best in your research.
Luis
see my theory
Research Sample 7 Update
and also a question I posted https://www.researchgate.net/post/Thermal_Entropy_and_Ships_Floating3
In which I discuss the possibility of rapid movement given specific situations arise in my theory... You speak of some in your question...
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