examples of growth of perturbations in chemical organizations
We will examine several simple examples of chemical systems where we start with one species of molecules (or a closed subset of species) then intoruduce a small perturbation and evolve the system using mass action dynamics. We want to know whether this perturbation will grow and, if so, at what rate. Ultimately, we would like to link the behavor to some feature of the reaction system, perhaps related to Rosen’s theory of M-R systems.
To get started, consider a trivial case, . The system of equations which describes this system is:
It is easy enough to solve this system. We begin by noting that , hence . Substituting this back in to the second equation, we conclude that
This equation can readily be solved to yield the implicit solution
which can be solved to produce the explicit solution
Looking at the solution, we see that it starts out at and grows towards as . This is as we expect — as time goes on, whatever A’s there are left react with B’s to turn into B’s until we are left with nothing but B’s.
If we suppose that, at the initial time , there is only a tiny proportion of B’s, i.e. , then we may make an expansion of the fraction and conclude that grows exponentially for small values of :
We can also come to this conclusion by bounding without solving the equation first. For a simple equation like this which is readily solved, this is hardly needed but, for larger, more complicated equations, it becomes important and this simple example can serve as a illustration of the general technique.
By the existence theorem, there exists a positive real number and a function such that and satifies the differential equation. Since , by continuity there exists a positive real number and a function which satisfies the same differential equation with the same initial condition. Furthermore, we assume that is maximal.
Starting with this condition and doing some algebra, we conclude that
Now, so, by the mean value theorem, we conclude that
We can ensure that the bound on is satisfied if the condition is met, which amounts to demanding that where .
|Title||examples of growth of perturbations in chemical organizations|
|Date of creation||2014-05-31 15:55:20|
|Last modified on||2014-05-31 15:55:20|
|Last modified by||rspuzio (6075)|