Hopf bifurcation in higher dimensional differential systems via the averaging method
This interesting article treats Hopf bifurcations in higher dimensions. The authors show in their main result that from one singularity with eigenvalues \(\pm bi\) and \(n-2\) zeros, the bifurcation of exactly \(l\) cycles is possible for all \(l\in\{1,\dots, 2^{n-3}\}\). In particular, this shows that the number of limit cycles of a Hopf bifurcation scenario can grow exponentially with the dimension of the system. The result is proved by using first-order averaging theory and Brouwer degree theory. Moreover, the shape and stability of the limit cycles is analyzed in dimension four, and the results are applied to certain fifth-order differential equations. Finally, the authors examine a simplified Marchuk system which models immune response.
scientific article; zbMATH DE number 5541997