Show that there exists a non-constant periodic trajectory for the system of ODEs I am having a hard time finding information online about how to show that there exists a non-constant periodic trajectory for the system of ODEs.
For example if the system is $$x'=1-4x+x^2y$$ $$y'=3x-x^2y$$ Does anyone know where I can read about this type of problem or what steps to take to find the trajectories? 
Thanks.
 A: The main rigorous theorem for showing existence of a periodic solution is the Poincare-Bendixson theorem. This is specific to 2D. The main assumption that the theorem requires is that the dynamics will remain in a bounded region forever. It then gives a full classification of the $\omega$-limit sets of the system. In particular it tells you that if there are no fixed points then there must be a periodic orbit (under this boundedness assumption). So if you can check that your system remains in a bounded region forever then you can take care of the matter. This is frequently difficult to check (even if it holds). Usually the way to do it is basically to construct a Lyapunov function of some kind.
You can get a quick conjecture about where periodic orbits might be by looking for fixed points and taking a linearization about them. This does not require 2D. For this we have three rules of thumb, which basically amount to "trajectories started close enough to the fixed point behave like the linear system".


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*If the linearization has all eigenvalues with negative real part, then trajectories started near the fixed point will decay to the fixed point (not oscillate). 

*If the linearization has all eigenvalues with nonpositive real part but some with zero real part, then there is frequently a periodic solution. 

*If the linearization has any eigenvalues with positive real part then usually there is no periodic solution reasonably close to that point.


The first rule is always correct, but exceptions to the second and third rules exist. For the second: there are cases where a fixed point with all the eigenvalues of the linearization having nonpositive real part is still (subexponentially) unstable. For the third: there are cases where a fixed point has positive real part eigenvalues that repel you away from the fixed point, but where you go straight out toward a limit cycle from there.
