Semi-algebraic description of the equilibria of dynamical systems

Authors:
Changbo Chen;Marc Moreno Maza
Affiliations:
The University of Western Ontario, London, Canada;The University of Western Ontario, London, Canada
Venue:
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
Year:
2011

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Abstract

We study continuous dynamical systems defined by autonomous ordinary differential equations, themselves given by parametric rational functions. For such systems, we provide semi-algebraic descriptions of their hyperbolic and non-hyperbolic equilibria, their asymptotically stable hyperbolic equilibria, their Hopf bifurcations. To this end, we revisit various criteria on sign conditions for the roots of a real parametric univariate polynomial. In addition, we introduce the notion of comprehensive triangular decomposition of a semi-algebraic system and demonstrate that it is well adapted for our study.