Temporal oscillations in Becker–Döring equations with atomization

Abstract

We prove that time-periodic solutions arise via Hopf bifurcation in a finite closed system of coagulation-fragmentation equations. The system we treat is a variant of the Becker-Döring equations, in which clusters grow or shrink by addition or deletion of monomers. To this is added a linear atomization reaction for clusters of maximum size. The structure of the system is motivated by models of gas evolution oscillators in physical chemistry, which exhibit temporal oscillations under certain input/output conditions. All solutions of the system (2)-(4) conserve mass, since ∂ t M =1 n = 0. R. L. Pego and J. J. L. Velázquez 3 1. If β 0 < κ < β k , then any unstable eigenvalue of B is non-real and simple, and satisfies |λ| ≤Ĉ k M −3/2 .

Oscillations in a Becker-Döring model with injection and depletion

“…Recently, oscillations for a Becker-Döring model with atomization were proved to exist by two of the present authors in [27]. The model in [27] is closed and has no external source or removal terms (S = r = 0 in (1.9)-(1.10)).…”

“…Recently, oscillations for a Becker-Döring model with atomization were proved to exist by two of the present authors in [27]. The model in [27] is closed and has no external source or removal terms (S = r = 0 in (1.9)-(1.10)). The atomization of clusters of a maximal size M into M monomers provides a closed feedback mechanism from large clusters to monomers, which could be considered to replace injection and depletion.…”

Hopf bifurcation in addition-shattering kinetics

“…These solutions are born through the Hopf bifurcation mechanism: The steady states exist whenever β ≥ 0, but become unstable for parameters from (23) and give birth to never-ending oscillations via Hopf bifurcation. Oscillatory solutions in other models have been detected recently [28,29,39]. For instance, Hopf bifurcation has been found in a finite Becker-Döring system with constant kinetic coefficients [39].…”

“…In our case, the application of the Dulac criterion shows the absence of limit cycles independently of the rates (see SM). Recent results on Hopf bifurcation in a finite Becker-Döring exchange model also show that the number of ODEs in such finite systems has to be sufficiently large to obtain oscillatory solutions [39]. This perhaps explains why despite years of searching, the oscillatory solutions have not been observed.…”

Stochastic gel-shatter cycles in coalescence-fragmentation models

“…in the absence of cluster influx/outflux [17]. Most recently, it has been shown that deterministic temporal oscillations can arise in a class of coalescence-fragmentation (C-F) processes where clusters grow or shrink by addition or deletion of monomers [18]. All of these findings predicted deterministic oscillations and relied on mean-field descriptions.…”