Deciding Hopf bifurcations by quantifier elimination in a software-component architecture
The paper investigates the phenomenon of Hopf bifurcations for autonomous, deterministic ordinary differential equations with parameterized polynomial vector fields. The authors make use of ``quantifier elimination'' -- a symbolic method which has been developed and implemented by the second author, avoiding Lie-symmetry methods like in the book of \textit{P. J. Olver} [Applications of Lie groups to differential equations. Paperback ed. (English) Graduate Texts in Mathematics. 107. New York, NY: Springer. xxviii, 513 p. (2000; Zbl 0937.58026)] or differential Galois theory like that of \textit{M. F. Singer} [Computer algebra and differential equations, Colloq., Comput. Math. Appl., 3-57 (1988; Zbl 0713.12005)] as well. The used techniques rely on Hurwitz determinants, applying the theory of sub-resultant sequences, of Groebner bases and quantifier elimination algorithms. A series of examples from existing literature illustrates the use of Java-based architectures to evaluate the related algebraic conditions on Hopf bifurcations. It is worth noting that the technique of quantifier elimination on real closed fields is due to \textit{A. Tarski} [A decision method for elementary algebra and geometry. 2nd ed. (English) Berkeley. University of California Press. III, 63 p. (1951, Zbl 0044.25102)].
scientific article; zbMATH DE number 1513056
scientific article