2023/09/18 by Alberto Pérez-Cervera, Pérez-Cervera, Alberto, Antonio E. Teruel +1
Environmental Science · Physics and Astronomy · #Chaos control and synchronization #Dynamical Systems (math.DS) #Ecosystem dynamics and resilience #FOS: Mathematics #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2309.10119
openalex publication_date 2023/09/18 · openalex created_date 2023/09/21 · openalex updated_date 2026/08/01
In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay zd(λ,ε) as a function of the bifurcation parameter λ and the singular parameter ε. We quantify the maximal delay by constructing a piecewise linear (PWL) transcritical minimal model and studying the dynamics near the slow-manifolds. Our findings encompass all potential maximum delay behaviours within the range of parameters, allowing us to identify: i) the trivial scenario where the maximal delay tends to zero with the singular parameter; ii) the singular scenario where zd(λ, ε) is not bounded, and also iii) the transitional scenario where the maximal delay tends to a positive finite value as the singular parameter goes to zero. Moreover, building upon the concepts by A. Vidal and J.P. Françoise (Int. J. Bifurc. Chaos Appl. 2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter λ changes, the system presents a non-bounded canard explosion leading to an enhanced delay phenomenon at the critical value. Our understanding of the maximal delay zd(λ, ε) of a single normal form, allows us to determine both, the amplitude of the canard cycles and, in the enhanced delay case, the increase of the amplitude for each passage.