An Algorithm of Real Root Isolation for Polynomial Systems with Applications to the Construction of Limit Cycles

By combining Wu’s method, polynomial real root isolation and the evaluation of maximal and minimal polynomials, an algorithm for real root isolation of multivariate polynomials is proposed. Several examples from the literature are presented to illustrate the proposed algorithm.

[1]  W. Wu.  ON THE DECISION PROBLEM AND THE MECHANIZATION OF THEOREM-PROVING IN ELEMENTARY GEOMETRY , 2008 .

[2]  Eduardo Sáez,et al.  Limit cycles of a cubic Kolmogorov system , 1996 .

[3]  Zhengyi Lu,et al.  Two limit cycles in three-dimensional Lotka-Volterra systems☆ , 2002 .

[4]  Wen-tsün Wu.  Numerical and Symbolic Scientific Computing , 1994, Texts & Monographs in Symbolic Computation.

[5]  Wang Dongming,et al.  A class of cubic differential systems with 6-tuple focus , 1990 .

[6]  Donald A. Drew,et al.  Differential equation models , 1983 .

[7]  James A. Yorke,et al.  Numerically determining solutions of systems of polynomial equations , 1988 .

[8]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[9]  Bruce W. Char,et al.  Maple V Language Reference Manual , 1993, Springer US.

[10]  Courtney S. Coleman,et al.  Hilbert’s 16th Problem: How Many Cycles? , 1983 .

[11]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[12]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[13]  G. E. Collins,et al.  Quantifier Elimination by Cylindrical Algebraic Decomposition — Twenty Years of Progress , 1998 .

[14]  Zhengyi Lu,et al.  Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle☆ , 2003 .

[15]  S. Smale.  The fundamental theorem of algebra and complexity theory , 1981 .

[16]  Jeremy Johnson,et al.  Algorithms for polynomial real root isolation , 1992 .