need an example for an ode system with 3 limit cycles

Question

I was trying to find an ode system in predator-pray model with at least 2 limit cycles and different foci but I had trouble finding any, does anybody have an example in mind? thanks in advance

Answer 1

Answer based on previous formulation of question: Consider a planar system (i.e. for $x(t)$ and $y(t)$), and write it in polar coordinates. Suppose that system has the form \begin{align} \dot{r} &= f(r),\\ \dot{\theta} &= 1. \end{align} Then, there exists a limit cycle, taking the form of a circle, for every root of $f(r)$. This way, you can construct a system with an arbitrary number of limit cycles by taking $f(r) = (r-r_1)(r-r_2)(r-r_3)\ldots(r-r_n)$, for example. Reverting the result to Cartesian coordinates $(x,y)$ yields a planar system with $n$ limit cycles.

Answer to current formulation of question: I'm afraid that the assignment 'predator-prey' model is an interpretation of a dynamical system, rather than a categorization. Also, I have the strong impression that you're not interested in more than one predator or prey, which means that the system is two-dimensional.

As you can learn from the relevant Scholarpedia lemma, the most general type of predator-prey model is called the Kolmogorov model, which takes the form \begin{align} \dot{x} &= x\,f(x,y),\\ \dot{y} &= y\,g(x,y), \end{align} where $f$ and $g$ are such that \begin{equation} \frac{\partial f}{\partial y}(x,y)< 0\quad\text{and}\quad \frac{\partial g}{\partial x} > 0. \end{equation} The condition that the system has at least two different equilibria means that there are at least two different points $(x_1,y_1)$ and $(x_2,y_2)$ where both $f$ and $g$ are zero. The limit cycles are a bit more work, I'm afraid. I would suggest to follow the line of approach of the first part of my answer, as follows.

Take as $f(r) = r(1-r)$, for example, yielding a circular limit cycle around the origin. Now shift the entire system to a point $(x_1,y_1)$ (rewritten in polar coordinates). Do the same for a second point $(x_2,y_2)$ which is sufficiently far away from the first point (such that the limit cycles around those two points don't intersect), and add the two systems. Then, you might have to introduce a coordinate shift (rescaling + rotation) to bring the system in the Kolmogorov form.