Item
DaunisiEstadella, Pepus
MartÃn FernÃ¡ndez, Josep Antoni 

Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Jarauta Bragulat, Eusebio
Egozcue, Juan JosÃ© 

2008 May 29  
Evolution of compositions in time, space, temperature or other covariates is frequent
in practice. For instance, the radioactive decomposition of a sample changes its composition with time. Some of the involved isotopes decompose into other isotopes of the
sample, thus producing a transfer of mass from some components to other ones, but
preserving the total mass present in the system. This evolution is traditionally modelled
as a system of ordinary di erential equations of the mass of each component. However,
this kind of evolution can be decomposed into a compositional change, expressed in
terms of simplicial derivatives, and a mass evolution (constant in this example). A
rst result is that the simplicial system of di erential equations is nonlinear, despite
of some subcompositions behaving linearly.
The goal is to study the characteristics of such simplicial systems of di erential equa
tions such as linearity and stability. This is performed extracting the compositional dif
ferential equations from the mass equations. Then, simplicial derivatives are expressed
in coordinates of the simplex, thus reducing the problem to the standard theory of
systems of di erential equations, including stability. The characterisation of stability
of these nonlinear systems relays on the linearisation of the system of di erential equations at the stationary point, if any. The eigenvelues of the linearised matrix and the
associated behaviour of the orbits are the main tools. For a three component system,
these orbits can be plotted both in coordinates of the simplex or in a ternary diagram.
A characterisation of processes with transfer of mass in closed systems in terms of stability is thus concluded. Two examples are presented for illustration, one of them is a
radioactive decay Geologische Vereinigung; Institut dâ€™EstadÃstica de Catalunya; International Association for Mathematical Geology; CÃ tedra LluÃs SantalÃ³ dâ€™Aplicacions de la MatemÃ tica; Generalitat de Catalunya, Departament dâ€™InnovaciÃ³, Universitats i Recerca; Ministerio de EducaciÃ³n y Ciencia; Ingenio 2010. 

application/pdf  
Jarauta Bragulat, E.; Egozcue, J.J. â€™Compositional evolution with mass transfer in closed systemsâ€™ a CODAWORKâ€™08. Girona: La Universitat, 2008 [consulta: 13 maig 2008]. Necessita Adobe Acrobat. Disponible a Internet a: http://hdl.handle.net/10256/725  
http://hdl.handle.net/10256/725  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tots els drets reservats  
Models geomÃ¨trics
Equacions diferencials 

Compositional evolution with mass transfer in closed systems  
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