A geometric approach to inner problems associated with exponentially small splitting phenomena in local bifurcations

In this paper, we study analytic nonlinear partial differential equations for $\mathbf y=\mathbf y(x,\theta)\in \mathbb C^n$ of the form $x^2 \mathbf y'_x(1+\mathcal O(x)) + \mathbf y'_\theta+x \mathbf A \mathbf y = \mathcal O(x^2)$, $()'_z=\frac{\partial}{\partial z}$, $z=x,\theta$, with $x\in \mathbb C$ and $\theta\in \mathbb C /(2\pi \mathbb Z)$ denoting the independent variables.

We show that solutions are $1$-sums (with respect to $x$) of Fourier series with coefficients of type Gevrey-$1$.

The motivation for studying these equations is that so-called inner problems, associated with two-dimensional formal connections in unfoldings of local bifurcations, can be brought into this form (by looking for invariant manifolds in blowup coordinates).

In the present paper, we give two examples: The zero-Hopf bifurcation and the resonant Hopf-Hopf bifurcation.

Importantly, these inner problems are given by the unperturbed problem (i.e. at the bifurcation) and the invariant manifolds are expressed directly in phase space (through blowup coordinates associated with the local bifurcation).

This contrasts the Lazutkin-based formulation of inner problems which is based upon a blowup of singularities (with respect to complex time) of approximate solutions; for local bifurcations this requires (artificial) coordinate transformations.

We solve the PDE by extending the Banach-convolution-algebra-approach to Borel-Laplace by Bonckaert and De Maesschalck (2008) to account for analytic Fourier series.

In further details, we apply the Borel transform with respect to $x$ (keeping $\theta$ fixed) and solve the resulting equation in an appropriate Banach space of Fourier series with coefficients that have exponential growth in the Borel plane.

The solutions of the PDE are then obtained through application of the Laplace transformation.