Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

Abstract

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.

Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

“…Proposition 4.31 in [16] states that the monodromy operator acting on the solution Φ j (x, ε) decomposes into the Stokes operator multiplied, from the right, by the classical monodromy operator acting on branch of F (x, ε). In [15] Theorem 32, Klimeš expresses in a remarkable way the acting of the monodromy operators on analytic extension of the solutions of the perturbed equation to the whole Ω 1 (ε) ∪ Ω 2 (ε) by the monodromy matrices M j (ε), unfolded Stokes matrices St j (ε) and the matrices e π i(Λ+Q/x j ) , j = L, R. His formulas have been deduced provided that there is an agreement of the matrices F (x) and F (x, ε) on the right intersections Ω R and Ω R (ε). In the next proposition we reformulate (without giving a proof) his formulas, provided that the above agreement is on the left intersections (see for details and proof [15]).…”

“…In [15] Theorem 32, Klimeš expresses in a remarkable way the acting of the monodromy operators on analytic extension of the solutions of the perturbed equation to the whole Ω 1 (ε) ∪ Ω 2 (ε) by the monodromy matrices M j (ε), unfolded Stokes matrices St j (ε) and the matrices e π i(Λ+Q/x j ) , j = L, R. His formulas have been deduced provided that there is an agreement of the matrices F (x) and F (x, ε) on the right intersections Ω R and Ω R (ε). In the next proposition we reformulate (without giving a proof) his formulas, provided that the above agreement is on the left intersections (see for details and proof [15]). Note that these relations are in concordance with the definition of the monodromy around x = ∞ for both equations.…”

“…Remark 5.2. In fact Theorem 32 in [15] states that on the lower sector Ω 2 (ε) the monodromy matrixM R (ε) is expressed as…”

“…Thus along this very particular sequence of values of ε the monodromy matrices convergent. As a consequence of the decomposition theorem of Lambert and Rousseau (Proposition 4.31 in [16]), a theorem of Klimeš (Theorem 32. in [15]) and our decomposition theorem (Theorem 4.9) we point which part of the monodromy matrices are the so called unfolded Stokes matrices. This result allows us to connect by a limit √ ε → 0 the pointed part of the monodromy matrices and the Stokes matrices of the initial equation (see Theorem 5.5).…”

“…This result allows us to connect by a limit √ ε → 0 the pointed part of the monodromy matrices and the Stokes matrices of the initial equation (see Theorem 5.5). Problems of this kind have been considered in the already classical works of Glutsyuk [4], Ramis [21], Zhang [33], Duval [3], Schäfke [25] as well as in recent ones: Glutsyuk [5], Lambert and Rousseau [8,16,17,18], Slavyanov and Lay [27], Klimeš [12,13,14,15], Remy [24]. Our work is closer to the works of Glutsyuk [4,5] and Lambert and Rousseau [16,17] where the authors introduce a small parameter that splits the irregular point at the origin of a linear system (not a scalar equation) into two finite Fuchsian singularities.…”