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Projective dynamics of homogeneous systems: local invariants, syzygies and the Global Residue Theorem

Part of: Nonlinear operators and their properties Arithmetic and non-Archimedean dynamical systems Theory of singularities and catastrophe theory Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  16 March 2012

Z. Balanov ,
A. Kononovich  and
Y. Krasnov
Z. Balanov
Affiliation:
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080USA (zalman.balanov@utdallas.edu)
A. Kononovich
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel (kononovich@math.biu.ac.il; krasnov@math.biu.ac.il)
Y. Krasnov
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel (kononovich@math.biu.ac.il; krasnov@math.biu.ac.il)
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Abstract

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We give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions.

Keywords

homogeneous polynomial systemsprojective dynamicssyzygieslocal invariantsGlobal Residue TheoremEuler–Jacobi Theorem

MSC classification

Secondary: 37P05: Polynomial and rational maps 47H10: Fixed-point theorems 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems 58K15: Topological properties of mappings 58K20: Algebraic and analytic properties of mappings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 577 - 589
Copyright
Copyright © Edinburgh Mathematical Society 2012

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