Using the blow-up technique for a modified Lindemann mechanism
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Using the blow‑up technique for a modified Lindemann mechanism L. Bayón1 · P. Fortuny Ayuso1 · V. M. García Fernández1,2 · C. Tasis1 · M. M. Ruiz1 · P. M. Suarez1 Received: 14 July 2020 / Accepted: 19 September 2020 © Springer Nature Switzerland AG 2020
Abstract We study a modified Lindemann mechanism, in which the second part of the reaction is 2B → P instead of B → P . This model gives rise to a degenerate singularity in the associated system of ordinary differential equations. Using the blow-up technique, we describe this singularity qualitatively, showing that there is an invariant line, and that in the first quadrant (where the reaction takes place), it is stable. As a consequence, numerical methods for integrating differential equations can be used with confidence. Several examples are included. Keywords Lindemann mechanism · Mass action · Ordinary differential equation · Blow-up · Stability Mathematics Subject Classification 92E20 · 34E10 · 34M35
1 Introduction The classical Lindemann Mechanism (LM) states that if a reactant A decays into a product P by colliding with itself, the chemical reaction is expressed as: k1
k2
(1)
A + A ⇄ A + B, B → P k−1
This classical mechanism has been studied by many authors, and its mathematical properties have been explored in [1]. Fraser [2] has used it as an example in his work on the dynamical systems approach to chemical kinetics. Calder [3] studies several properties about stability, unicity, concavity and asymptotic behaviour. * L. Bayón [email protected] 1
Department of Mathematics, University of Oviedo, Oviedo, Spain
2
Department of Physical and Analytical Chemistry, University of Oviedo, Oviedo, Spain
13
Vol.:(0123456789)
Journal of Mathematical Chemistry
There exist different modifications to the basic scheme (see [4–6]). We study the following innovative version of the LM: k1
k2
(2)
A + A ⇄ A + B, 2B → P k−1
where k1 , k−1 , and k2 are the reaction rate constants. Application of the law of mass action gives the system of ordinary (non linear) differential equations that the concentrations of A, B and P must verify:
da = k−1 ab − k1 a2 d𝜏 db = k1 a2 − k−1 ab − 2k2 b2 d𝜏 dp = k2 b2 d𝜏
(3)
where 𝜏 is time and we the initial conditions are: a(0) = a0 , b(0) = 0 , p(0) = 0 (traditionally, the complex and product are not initially present). The following conservation law holds: (4)
a(𝜏) + b(𝜏) + 2p(𝜏) = a0 After simplifying, we obtain the planar reduction of system (3):
da = k−1 ab − k1 a2 d𝜏 db = k1 a2 − k−1 ab − 2k2 b2 d𝜏
(5)
which, rescaling the variables as follows:
t = k2 𝜏,
x=
k1 a, k2
y=
k1 b, k2
𝜀=
k−1 , k1
𝜎=
k2 k1
(6)
becomes the dimensionless planar system:
dx = 𝜀xy − x2 dt dy = x2 − 𝜀xy − 2𝜎y2 dt
(7)
which has a single equilibrium point at (0, 0) whose Jacobian matrix is null. This is a degenerate singularity, for which the classical methods are unsuitable. We study the dynamical properties of system (7) using the blow-up technique [7]. This technique allows us to study the long-term behavior of the reaction and t
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