Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T21:34:49.624Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 ([email protected])
A. Kononovich
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel ([email protected]; [email protected])
Y. Krasnov
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel ([email protected]; [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1.Atiyah, M. and Bott, R., Lefschetz fixed point formula for elliptic complexes, II, Applications, Annals Math. 88 (1968), 451–491.CrossRefGoogle Scholar
2.Balanov, Z. and Krasnov, Y., Complex structures in real algebras, I, Two-dimensional commutative case, Commun. Alg. 31 (2003), 4571–4609.CrossRefGoogle Scholar
3.Berenstein, C. A., Vidras, A. and Yger, A., Analytic residues along algebraic cycles, J. Complexity 21 (2005), 5–42.Google Scholar
4.Coleman, C. S., Systems of differential equations without linear terms, in Nonlinear differenrial equations and nonlinear mechanics, pp. 445–453 (Academic Press, 1963).CrossRefGoogle Scholar
5.Gasull, A. and Torregrosa, J., Euler–Jacobi formula for double points and applications to quadratic and cubic systems, Bull. Belg. Math. Soc. Simon Stevin 6 (1999), 337–346.CrossRefGoogle Scholar
6.Gay, C. A. R., Vidras, A., Yger, A. and Berenstein, C. A., Residue currents and Bezout identities, Progress in Mathematics (Birkhäuser, 1993).Google Scholar
7.Jacobi, C. G. J., Theoremata nova algebraica circa systema duarum aequationum inter duas variabiles propositarum, J. Reine Angew. Math. 14 (1835), 281–288.Google Scholar
8.Krasnoseľskii, M. and Zabreiko, P., Geometric methods of nonlinear analysis (Springer, 1984).Google Scholar
9.Krasnov, Y., Differential equations in algebras, in Hypercomplex analysis (ed. Sabadini, I., Shapiro, M. and Sommen, F.), Trends in Mathematics, pp. 187–205 (Birkhäuser, 2009).Google Scholar
10.Krasnov, Y., Kononovich, A. and Osharovich, G., On a structure of the fixed point set of homogeneous maps, Discr. Contin. Dynam. Syst., in press.Google Scholar
11.Markushevich, A. I., Theory of functions of a complex variable, 2nd edn (American Mathematical Society, Providence, RI, 2005).Google Scholar
12.Mawhin, J. and Ward, J. R. Jr, Guiding-like functions for periodic or bounded solutions of ordinary differential equations, Discrete Contin. Dynam. Syst. 8 (2002), 39–54.Google Scholar
13.Shafarevich, I., Basic algebraic geometry, I (Springer, 1994).Google Scholar
14.Šilov, G., Integral curves of a homogeneous equation of the first order, Usp. Mat. Nauk 5 (1950), 193–203 (in Russian).Google Scholar
15.Tsygvintsev, A., On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations, J. Phys. A 34 (2001), 2185–2193.Google Scholar