Browsing by Autor "F. Calogero"

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Isochronous rate equations describing chemical reactions
(Institute of Physics, 2010) F. Calogero; F. Leyvraz; M. Sommacal
We consider systems of ordinary differential equations in the plane featuring at most quadratic nonlinearities. It is known that, up to linear transformations of the variables, there are only four systems for which the origin is an isochronous center, that is, for which all orbits in the vicinity of the origin are periodic with the same, fixed period. On the other hand, if, after an affine transformation, the system's coefficients satisfy certain positivity requirements, these systems can be interpreted as kinetic equations for chemical reactions. Here we show that, for two of these four isochronous systems, it is possible to find an affine transformation such that the transformed system obeys all these positivity conditions. For the third we can show that this is not possible, whereas for the fourth the issue remains to some extent open. Hence for the two cases mentioned above these systems may be interpreted as kinetic equations describing isochronous chemical reactions.
Isochronous Systems, the Arrow of Time, and the Definition of Deterministic Chaos
(Springer Science+Business Media, 2010) F. Calogero; F. Leyvraz
Novel rate equations describing isochronous chemical reactions
(Springer Science+Business Media, 2010) F. Calogero; F. Leyvraz; M. Sommacal
Oscillatory and isochronous rate equations possibly describing chemical reactions
(Institute of Physics, 2009) F. Calogero; F. Leyvraz
We study a simple mathematical model that can be interpreted as a description of the kinetics of the following four reactions involving the two chemicals U and W: (i) U + U double right arrow U with rate alpha, (ii) U + W double right arrow U with rate beta, (iii) W + W double right arrow U with rate gamma and (iv) W + W double right arrow W + W + W with rate delta + 2 gamma. The model can be generally solved by quadratures, and in the special case beta = 2 alpha, explicitly in terms of elementary functions. We focus on the case characterized by the two inequalities gamma beta(2) > alpha delta(2) and 2 beta gamma > delta(2), and we show that in this case the solutions vanish asymptotically at large times. But if a constant decay with rate theta of chemical U is added, then a nonvanishing equilibrium configuration arises. Moreover, for arbitrary strictly positive initial conditions, the solutions remain bounded. They either tend asymptotically (in the remote future) to this nonvanishing equilibrium configuration, or are periodic, or tend to a limit cycle. Indeed, we find that this system goes through a standard supercritical Hopf bifurcation at an appropriate value of the parameters. Another interesting case arises when, in addition to the original reaction, a negative constant term is added to the equation corresponding to chemical U, corresponding to siphoning out a constant amount of chemical U per unit time, independent of its concentration. A very remarkable feature of this (possibly not very realistic) model is the following: in the special case beta = 2 alpha, we again find an explicit solution in terms of elementary functions, which oscillates at a fixed frequency, independent of the initial condition. In other words, it is an isochronous system. If beta not equal 2 alpha, however, no periodic orbits exist, implying that the nature of the bifurcation at beta = 2 alpha is rather peculiar.