We are working on some projects about dynamical analysis of two dimensional oscillators described by differential equations. We appreciate if you can introduce us the interesting cases you have dealt with. E.g. we know that Van der Pol oscillator is a famous well-investigated example of such systems. Or Shi-Songling system is an important example related to “Hilbert's 16th problem for quadratic vector fields”. Multi stable oscillators are very interesting for us and we would be glad to know important cases from this category.
You can start with Stuart Landau oscillator. It is simple and people have used it to study steady states and chimera in coupled systems.
You can also try non isochronous limit cycle oscillator. For instance, Stuart Landau oscillators with the presence of non isochronous term, sometimes called as uncoupled CGLE.
See the equation (36) and (37) in the following paper:
http://aip.scitation.org/doi/pdf/10.1063/1.1525170
The model (in the polar coordinate) becomes interesting because the frequency has dependence on amplitude, changing initial conditions in amplitude equation will change the frequency.
You can start with Stuart Landau oscillator. It is simple and people have used it to study steady states and chimera in coupled systems.
You can also try non isochronous limit cycle oscillator. For instance, Stuart Landau oscillators with the presence of non isochronous term, sometimes called as uncoupled CGLE.
See the equation (36) and (37) in the following paper:
http://aip.scitation.org/doi/pdf/10.1063/1.1525170
The model (in the polar coordinate) becomes interesting because the frequency has dependence on amplitude, changing initial conditions in amplitude equation will change the frequency.
Try autonomous Duffing oscillator, then physical pendulum (Josephson's junction, if you like). Then you can go to nonautonomous case, i.e., those oscillators, harmonically forced in a weakly nonlinear limit. You have to account time, if systems are forced, so those cases are actually 3D, however, using any averaging procedure, you would eliminate time variable and switch back to 2D. You will discover as many multistable states as you want (in the case of pendulum).
You can also look at the Rayleigh-Plesset equation for the description of gas bubbles' dynamics in a liquid, for example see https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Plesset_equation
See my activity "Journal of Mathematical Sciences, January 2015, Volume 04, Issue 4, pp 379–530, Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconservative Force Fields"
Dear Sajad,
We focused in our works on the Hénon map because of its simplicity of computing and good chaotic proprieties..
Dear Sajad,
See our article in SIAM "Baer, S.M., Kooi, B.W., Kuznetzov, Yu.A. and Thieme H.R. 2006.
Multidimensional bifurcation analysis of a basic two-stage population
model. SIAM Applied Mathematics, 66(4):1339-1365."
Dear Sajad,
recently, we submitted a paper on a single complex delay differential equation for a class A semiconductor lasers. This model can be regarded as two-dimensional oscillator exhibiting multi-stability of steady states (which is actually harmonic oscillation), mode locking, multi-stability of mode-locked regimes, pulse shape transformation, etc.
Article Class-A mode-locked lasers: Fundamental solutions
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