Limit Cycle Behavior in Three- Or Higher-Dimensional Non-Linear Systems: The Lotka–volterra Example

Periodic Orbits for a Three-Dimensional Biological Differential Systems

“…In this work we are concerned with the proof of existence of periodic orbits (see [17,18] for examples in the classical case), and we will use Hopf bifurcation theory in order to prove existence of periodic orbits. Since the system depends on 7 parameters, the conditions to have Hopf bifurcation are quite complicated and for sake of simplicity we also present a particular case in which all the parameters are fixed but 0 .…”

Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

“…The analysis of bifurcation and limit cycles for such systems is much more difficult. Some theorems and methods such as the generalized Poincaré-Bendixon theorem and the describing function method may be applied for the analysis (see, e.g., [17]), but they are applicable only to certain high-dimensional systems. The methods of center manifold [4,10] and Liapunov-Schmidt reduction [5] allow one to reduce any system of dimension n > 2 to a planar system without losing any significant aspect of the dynamic characters.…”

“…This system has been studied in [26,13] (see also [17]) and it has a unique positive steady state (1, 1, 1).…”