Hilbert’s 16th Problem: How Many Cycles?

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

“…However, it is not an easy task to study the quantity of limit cycles that can be generated throughout the bifurcation of a center-focus (Chicone 2006). This problem is related to Hilbert's well-known 16th Problem for polynomial systems (Gaiko 2003), and it is a question that has remained unanswered for the predation model, particularly for the Gause type system described by system (1) (Coleman 1983), suggesting: "Find a predator-prey or other interacting system in nature, or construct one in the laboratory, with at least two ecologically stable cycles".…”

Global Bifurcation of a Perturbed Double-Homoclinic Loop*

“…C. S. Coleman in his survey (cf. [1]) stated that "For n > 2, the maximal number of eyes is not known, nor is it known just which complex patterns of eyes within eyes, or eyes enclosing more than a single critical point can exist." Here so called "eye" means the limit cycle.…”

“…2 appears near L * 2 and outside L (2) 2 ; when 0 < a 3 − K 3 (ε, a 1 , a 4 ) |a 1 − K 1 (ε, a 4 )| a 4 , L * 2 breaks up and a large unstable limit cycle Γ (2) appears near Γ * and inside Γ (1) . Finally, for fixed a 1 , a 3 , a 4 , if 0 < a 2 − K 2 (ε, a 1 , a 3 , a 4 ) |K 3 (ε, a 1 , a 4 ) − a 3 | |a 1 − K 1 (ε, a 4 )| a 4 holds, then the homoclinic loop L * 1 breaks up, and a small stable limit cycle L…”

“…[6][7][8]). Then there is a large stable limit cycle Γ (1) which appears near and outside the double-homoclinic loop Γ * , with two small stable limit cycle L (1) 1 , L (2) 2 appeared near inside L * 1 , L * 2 and outside L…”

Linearizability conditions for Lotka–Volterra planar complex quartic systems having homogeneous nonlinearities

“…If F and G are of degree greater or equal 2, of particular significance in applications is the existence of limit cycles and the number of them that can arise. This type of problem goes back to Coleman [23] who poses the question whether a predator-prey system can have two or more ecologically stable limit cycles in a function of the degree of F and G. A first step to approach the problem of the number of limit cycles and the problem of bifurcation of critical periods for real polynomial systems of the form (1) is to solve the center problem and the linearizability problem for the associated complex system (2). Once we know solutions to the latter problems, we can study the bifurcations of limit cycles from the period annulus and bifurcations of critical periods for the real systems, see for instance [24,2].…”