2025/10/06 by Gábor Benedek, Benedek, Gábor, Tibor Krisztin +1
Computer Science · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2510.04505
openalex publication_date 2025/10/06 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28
This paper considers a class of delay differential equations with unimodal feedback and describes the structure of certain unstable sets of stationary points and periodic orbits. These unstable sets consist of heteroclinic connections from stationary points and periodic orbits to stable stationary points, stable periodic orbits and some more complicated compact invariant sets. A prototype example is the Mackey--Glass type equation y'(t)=-ay(t)+b (y2(t-1))/(1+yn(t-1)) having three stationary solutions 0, ξ1,n and ξ2,n with 0<ξ1,n<ξ2,n, provided b>a>0, and n is large. The 1-dimensional leading unstable set Wu(ξ1,n) of the stationary point ξ1,n is decomposed into three disjoint orbits, Wu(ξ1,n)=Wu,-(ξ1,n)∪ \ξ1,n\ ∪ Wu,+(ξ1,n).. Here ξ1,n is a constant function in the phase space with value ξ1,n. Wu,-(ξ1,n) is a connecting orbit from ξ1,n to 0. There exists a threshold value b^*=b^*(a)>a such that, in case b∈ (a,b^*), Wu,+(ξ1,n) connects ξ1,n to 0; and in case b>b^*, Wu,+(ξ1,n) connects ξ1,n to a compact invariant set An not containing 0 and ξ1,n. Under additional conditions, there is a stable periodic orbit On with An= On. Analogous results are obtained for the 2-dimensional leading unstable sets Wu(Qn) of periodic orbits Qn close to ξ1,n, establishing connections from Qn to On.