The goal of this work is to highlight the advantages of using NonStandard Finite Difference (NSFD) numerical schemes for the solution of ordinary differential equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution, such as positivity, are a priori known. The main reference considered is Mickens’ work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena and, therefore, widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous.

We study the impact of delayed feedbacks in the collective synchronization of ensembles of identical and autonomous micro-oscillators. To this aim, we consider linear arrays of Belousov–Zhabotinsky (BZ) oscillators confined in micro-compartmentalised systems, where the delayed feedback mimics natural lags that can arise due to the confinement properties and mechanisms driving the inter-oscillator communication. The micro-oscillator array is modeled as a set of Oregonator-like kinetics coupled via mass exchange of the chemical messengers. Changes in the synchronization patterns are explored by varying the delayed feedback introduced in the messenger species Br2. A direct transition from anti-phase to in-phase synchronization and back to the initial anti-phase scheme is observed by progressively increasing the time delay from zero to the value T₀, which is the oscillation period characterising the system without any delayed coupling. The route from anti- to in-phase oscillations (and back) consists of regimes where windows of in-phase oscillations are periodically broken by anti-phase beats. Similarities between these phase transition dynamics and synchronization scenarios characterising the coordination of oscillatory limb movements are finally discussed.