I have elementary questions about the construction of phase portraits and limit cycles using Mathematica.
It should be remarked that I have also searched and read previous discussions from Mathematica Forum. In addition, I have read Mathematica's documentation. However, these did not help me with the questions to be presented here. Owing to the fact that phase portrait and phase spaces are quite interesting topics on nonlinear dynamics, I think that these questions, and their possible answers, may be of great interest to those that subscribe to the Mathematica forum.
With that said, suppose that we have the following nonlinear dynamical system
$\displaystyle\frac{d x}{dt}=x^{2}+xy+y$,
$\displaystyle\frac{d y}{dt}=a_{2}x^{2}+b_{2}xy+c_{2}y^{2}+\alpha_{2}x+\beta_{2}y$,
in which $a_{2}=-10,b_{2}=2.2,c_{2}=0.7,\alpha_{2}=-72.7778,\text{and}\, \beta_{2}=0.0015$.
It is already well-known that the dynamical system has four limit cycles. Based on the above, I ask:
How may I construct the phase portrait of this system without using StreamPlot? (Attached below, you may see my notebook, in which a phase portrait is made by employing the StreamPlot command. However, I cannot visualize the isolated periodic trajectories.)
How may I highlight the isolated periodic trajectories in the phase portrait?
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