zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system. (English) Zbl 1100.34022

Authors’ summary: We consider the Lotka-Volterra equations

x ˙=x(1+ax+by),y=y(-λ+cx+dy),

with λ a nonnegative number. Our aim is to understand the mechanisms which lead to the origin being linearizable, integrable or normalizable. In the case of integrability and linearizability, there is a natural dichotomy. When the system has an invariant line other than the axes, then the system is integrable and we give necessary and sufficient conditions for linearizability in this case. When there is no such line, then the conditions for linearizability and integrability are the same. In this case, we show that the monodromy groups of the separatrices play a key role. In particular for λ=p/q with p+q12 and λ=n/2,2/n with n, the origin is linearizable if and only if the monodromy groups can be shown to be linearizable by elementary arguments. We give 4 classes of these conditions, and their duals, in terms of the parameters of the system, and conjecture that these, together with two exceptional cases of Darboux linearizability, are the only integrability mechanisms for rational values of λ. The work on normalizability is more tentative. We give some sufficient conditions for this via monodromy groups, and give a complete classification when λ=0. We also investigate in detail the case λ=1, with a+c=0. Much of our ideas here are based on recent work on the unfolding of the Ecalle-Voronin modulus of analytic classification. In particular, we give examples of “half- normalizable” systems as well as an experimental example of a “transcritical bifurcation” of the functional moduli associated to the critical point.

MSC:
34C05Location of integral curves, singular points, limit cycles (ODE)
References:
[1]L. Cairo, H. Giacomini andJ. Llibre,Liouvillian first integrals for the planar Lotka-Volterra system, Rend. Cir. Mat. Palermo,52 (2003), 389–418. · Zbl 1194.34014 · doi:10.1007/BF02872763
[2]C. Christopher, P. Mardešić andC. Rousseau,Normalizable, integrable and linearizable points in complex quadratic systems in C2, J. Dynam and Control Syst.,9 (2003), 311–363. · Zbl 1022.37035 · doi:10.1023/A:1024643521094
[3]J. Ecalle,Les fonctions résurgentes, Publications mathématiques d’Orsay, 1985.
[4]A. Fronville, A. Sadovski andH. Żolek,Solution of the 1 resonant center problem in the quadratic case, Fundamenta Mathematicae,157 (1998), 191–207.
[5]A. A. Glutsyuk,Confluence of singular points and nonlinear Stokes phenomenon, Trans. Moscow Math. Soc.,62 (2001).
[6]S. Gravel andP. Thibault,Integrability and linearizability of the Lotka-Volterra system for λ, J. Differential Equations,184 (2002), 20–47. · Zbl 1054.34049 · doi:10.1006/jdeq.2001.4128
[7]H. Hukuhara, T. Kimura and T. Matuda,Équations différentielles ordinaires du premier ordre dans le champ complexe, Math. Soc. of Japan (1961).
[8]Y. Ilyashenko,Nonlinear Stokes phenomena, in Nonlinear Stokes phenomena, Y. Hyashenko editor, Advances in Soviet Mathematics, vol. 14, American Mathematical Society (1993).
[9]Y. S. Ilyashenko andA. S. Pyartli,Materialization of Poincaré resonances and divergence of normalizing series, J. Sov. Math.31 (1985), 3053–3092. · Zbl 0575.34037 · doi:10.1007/BF02107555
[10]V. Kostov,Versal deformations of differential forms of degree α on the line, Functional Anal. Appl.,18 (1984), 335–337. · Zbl 0573.58002 · doi:10.1007/BF01083698
[11]C. Liu, G. Chen andC. Li,Integrability and linearizability in the Lotka-Volterra systems, J. Differential Equations,198 (2004), 301–320. · Zbl 1046.34075 · doi:10.1016/S0022-0396(03)00196-7
[12]A. Lins Neto,Algebraic solutions of polynomial differential equations and foliations in dimension two, in Holomorphic Dynamics, Lect. Notes in Math.135 (1988), 192–232.
[13]P. Mardešić, R. Roussarie andC. Rousseau,Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms, Moscow Mathematical Journal,4 (2004), 455–502.
[14]J. Martinet andJ.-P. Ramis,Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math., Inst. Hautes Etud. Sci.55 (1982), 63–164. · Zbl 0546.58038 · doi:10.1007/BF02698695
[15]J.-F. Mattei andR. Moussu,Holonomie et intégrales premières, Ann. Scient. Éc. Norm. Sup., 4 e série13 (1980), 469–523.
[16]J. Moulin-Ollagnier,Liouvillian integration of the Lotka-Volterra system, Qualitative Theory of Dynamical Systems,3 (2002), 19–28. · Zbl 1054.34003 · doi:10.1007/BF02969331
[17]R. Pérez-Marco andJ.-C. Yoccoz,Germes de feuilletages holomorphes à holonomie prescrite, Astérisque,222 (1994), 345–371.
[18]C. Rousseau,Normal forms, bifurcations and finiteness properties of vector fields, inNormal forms, bifurcations and finiteness properties of vector fields, Proceedings of NATO Advanced Study Institute Montreal 2002, Kluwer, 2004, 431–470.
[19]C. Rousseau,Modulus of analytic classification for a family unfolding a saddle-node, to appear in Moscow Mathematical Journal.
[20]S. M. Voronin,Invariants for points of holomorphic vector fields on the complex plane, in The Stokes phenomenon and Hilbert’s 16th problem (Groningen, 1995), World Sci. Publishing, River Edge, NJ (1996), 305–323.
[21]J.-C. Yoccoz,Théorème de Siegel, nombres de Brjuno et polynômes quadratiques, Astérisque231 (1995), 1–88.