Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system. (English) Zbl 1100.34022

Authors’ summary: We consider the Lotka-Volterra equations \[ \dot x = x(1 + ax + by), \quad y = y(-\lambda+cx+dy), \]
with \(\lambda\) 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 \(\lambda= p/q\) with \(p+q\leq 12\) and \(\lambda = n/2, 2/n\) with \(n\in\mathbb N\), 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 \(\lambda\). The work on normalizability is more tentative. We give some sufficient conditions for this via monodromy groups, and give a complete classification when \(\lambda = 0\). We also investigate in detail the case \(\lambda = 1\), 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.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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