Question

can we have two limit cycles in a 2d state space? if possible how can we draw them?

Answer 1

Yes ,There is a quadratic system with 4 Limit cycles

Example by Shi Songling:

has four cycles when

Example by Chen Lan-sun and Wang Ming-Shu:
The following system

has four cycles when

The method of construction of a quadratic system with 4 limit cycles is based on the following:

They pick a quadratic sytem with 2 hyperbolic limit cycles with disjoint interior. Existence of these 2 limit cycles is a consequence of Poincare-Bendixson theorem. One of thses 2 limit cycles suround a weak focus of order 2. So after perturbation we obtain two additional (smal) limit cycles. So we have 4 limit cycles with (3,1) distribution

It is unknown whether there is a quadratic system with more than 4 limit cycle. It is also unknown that whether there is a (2,2) distribution of limit cycles for quadratic systems.

The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case