Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. (English) Zbl 0965.65137
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 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 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 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)].
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 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)].
Reviewer: Henri Schurz (Minneapolis)
MSC:
| 65P30 | Numerical bifurcation problems |
| 68W30 | Symbolic computation and algebraic computation |
| 37M20 | Computational methods for bifurcation problems in dynamical systems |
Keywords:
Hopf bifurcation; ODEs with polynomial vector fields; quantifier elimination; symbolic methods; symmetry methods; Groebner basis methods; Hurwitz determinants criterion; Java-based software architectureReferences:
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