On the Complete Integrability of the Raychaudhuri Differential System in R 4 and of a CRNT Model in R 5

• Ferragut, Antoni ^[1] ; Valls, Claudia ^[2]
  1. [1] Universitat Jaume I

     Universitat Jaume I

     Castellón, España

     
  2. [2] Universidade Técnica de Lisboa
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 17, Nº 1, 2018, págs. 291-307
• Idioma: inglés
• DOI: 10.1007/s12346-017-0230-7
• Enlaces
  • Texto completo
• Resumen

  • We study the Darboux integrability of two differential systems with parameters: the Raychaudhuri equation (a relativistic model in R4) and a chemical reaction model in R5. We prove that the first one is completely integrable and that the first integrals are of Darboux type. This is the first four-dimensional realistic non-trivial model which is completely integrable with first integrals of Darboux type and for which for a full Lebesgue measure set of the values of the parameters the three linearly independent first integrals are rational. For the second one, we find all its Darboux polynomials and exponential factors and we prove that it is not Darboux integrable.

• Referencias bibliográficas
  • 1. Banaji, M.: P matrix properties, injectivity, and stability in chemical reaction systems. SIAM J. Appl. Math. 67, 1523–1547 (2007)
  • 2. Banaji, M., Craciun, G.: Graph-theoretic approaches to injectivety and multiple equilibria in systems of interacting elements. Commun....
  • 3. Banaji, M., Craciun, G.: Graph-theoretic approaches for injectivety and unique equilibria in general chemical reaction systems. Adv. Appl....
  • 4. Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Darboux integrability and the inverse integrating factor. J. Differ. Equ. 194, 116–139...
  • 5. Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks I: the injectivity property. SIAM J. Appl. Math. 65,...
  • 6. Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks II: the species reaction graph. SIAM J. Appl. Math....
  • 7. Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks: semiopen mass action systems. SIAM J. Appl. Math....
  • 8. Dasgupta, A., Nandan, H., Kar, S.: Kinematics of deformable media. Ann. Phys. 323, 1621–1643 (2008)
  • 9. Feinberg, M.: Lectures on chemical reaction networks (1980)
  • 10. Feliu, E., Wiuf, C.: Preclusion of switch behavior in networks with mass-action kinetics. Appl. Math. Comput. 219, 1449–1467 (2012)
  • 11. Feliu, E., Wiuf, C.: Simplifying biochemical models with intermediate species. J. R. Soc. Interface 10, 20130484 (2013)
  • 12. Ferragut, A., Gasull, A.: Searching Darboux polynomials. Acta Appl. Math. 430, 167–186 (2015)
  • 13. Ghose, A., Guha, P., Khanra, B.: Determination of elementary first integrals of a generalized Raychaudhuri equation by the Darboux integrability...
  • 14. Kar, S., Sengupta, S.: The Raychaudhuri equations: a brief review. Pramana 69, 49–76 (2009)
  • 15. Llibre, J., Zhang, X.: On the Darboux integrability of polynomial differential systems. Qual. Theory Dyn. Syst. 11, 129–144 (2012)
  • 16. Raychaudhuri, A.: Relativistic cosmology. I. Phys. Rev. 98(4), 1123–1126 (1955)
  • 17. Valls, C.: Invariant algebraic surfaces for generalized Raychaudhuri equations. Commun. Math. Phys. 308, 133–146 (2011)
  • 18. Valls, C.: Analytic first integrals for generalized Raychaudhuri equations. J. Math. Phys. 52, 103502 (2011)
  • 19. Valls, C.: Darbouxian integrals for generalized Raychaudhuri equations. J. Math. Phys. 52, 032703 (2011)