For a system of two differential equations, the eigenvalues determine the stability of the system. When the eigenvalues are positive, the system is unstable. In case of complex eigenvalues, the system is unstable when the real parts of the eigenvalues are positive. In the latter case, how do we mathematically show that the system enters a stable limit cycle when the fixed point is unstable?
Considering the following non-linear system, Bier et al.'s model of yeast glycolysis
\begin{align} \frac{dA}{dt} &= 2k_1GA – \frac{k_pA}{A+K_m}\\\\ \frac{dG}{dt} &= V_{in}-k_1GA\end{align}
where $G$ refers to glucose and $A$ refers to ATP. The values of the parameters are $V_{in}=0.36$, $k_1=0.02$, and $k_p=6$. When $K_m=13$, the following behavior is observed
ref.
From the Jacobian of the matrix, the eigenvalues at the fixed point are $0.0040 + 0.1132i$ and $0.0040 – 0.1132i$. Here, the real parts of both eigenvalues are positive and we observe that the phase portrait shows a limit cycle. I would like to understand how complex eigenvalues with positive real parts are mathematically related to limit cycles.
Any help would be much appreciated
Best Answer
The pattern for a limit cycle in a 2D dynamical system is provided by the Poincaré-Bendixon theorem. What you need is a region that contains only sources as stationary points (or no stationary points at all) while the vector field points inward on the boundary of the region.
Just a source is not sufficient as the solutions originating at the source can move towards infinity.