Blow up in Polynomial Differential Equations

Authors: Rudolf Csikja, Janos Toth

Abstract:

Methods to detect and localize time singularities of polynomial and quasi-polynomial ordinary differential equations are systematically presented and developed. They are applied to examples taken form different fields of applications and they are also compared to better known methods such as those based on the existence of linear first integrals or Lyapunov functions.

Keywords: blow up, finite escape time, polynomial ODE, singularity, Lotka–Volterra equation, Painleve analysis, Ψ-series, global existence

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333943

References:

[1] L. A. Beklemisheva, ''Classification of polynomial systemswith respect to birational transformations,'' Differentsial'nye Uravneniya, vol. 14, no. 5, pp. 807-816, 1978.
[2] L. Brenig, M. Codutti, and A. Figueiredo, ''Numerical inte-gration of dynamical systesm using Taylor expansion,'' in IS-SAC '96: International Symposium on Symbolic and Algebraic Computation, ed. Y.N. Lakshman, ACM, New York, 1996, pp.189-195.
[3] L. Brenig, and A. Goriely, ''Universal canonical forms for time-continuous dynamical systems,'' Phys. Rev. A, vol. 40, no. 7, pp.4119-4122, Oct. 1989.
[4] A. Constantin, ''Global existence of solutions for perturbed differential equations,'' Annali di Matematica Pura e Applicata,vol. 168, no. 1, pp. 237-299, Dec. 1995.
[5] K. L. Chung, Markov chains with stationary transition porbabilities, (2nd ed.), Berlin, Heidelberg, New York: Springer Verlag, 1967.
[6] R. Csikja-s homepage: http://www.math.bme.hu/˜csikja/blowup code.zip.
[7] J.Deaak; J. Toth; B., Vizvari, ''Mass conservation in complex chemical mechanisms,'' Alkalmazott Matematikai Lapok vol. 16,(1-2) pp. 73-97, 1992 (in Hungarian).
[8] P. Erdi, and J. Toth, Mathematical models of chemical reactions: Theory and applications of deterministic and stochastic models, Princeton: Princeton University Press, 1989.
[9] W. M. Getz, and D. H. Jacobson, ''Sufficiency conditions for finite escape times in systems of quadratic differential equations,'' J. Inst. Math. Applics., vol. 19, pp. 377-383, 1977.
[10] P. Glendinning, Stability, instability and chaos: An introduction to the theory of nonlinear differential equations, Cambridge Texts in Applied Mathematics, Cambridge: Cambridge University Press, 1994.
[11] A. Goriely, A brief history of Kovalevskaya exponents and modern developments, 2000.URL citeseer.ist.psu.edu/goriely00brief.html
[12] A. Goriely, ''Painleve analysis and normal forms theory,'' Physica D, vol. 152-153, pp. 124-144, 2001.
[13] R. Kowalczyk, ''Preventing blow-up in a chemotaxis model,''Journal of Mathematical Analysis and Applications vol. 305,(2), pp. 566-588, 2005.
[14] L. Perko, Differential equations and dynamical systems, Berlinetc.: Springer Verlag, 1996.
[15] S., Schuster; T. Höfer, ''Determining all extreme semi-positive conservation relations in chemical-reaction systems-A testcriterion for conservativity,'' Journal of the Chemical Society,Faraday Transactions vol. 87, (16), pp. 2561-2566, 1991.
[16] J. Toth, and P. Erdi, ''Models, problems and applications of formal reaction kinetics,'' A kemia ujabb eredmenyei, vol. 41,pp. 227-350, 1978, (in Hungarian).
[17] J. Toth, G. Li, H. Rabitz, and A. S. Tomlin, ''The effect of lumping and expanding on kinetic differential equations,'' SIAMJ. Appl. Math., vol. 57, no. 6, pp. 1531-1556, 1997.