A geometric approach to exponentially small splitting: Zero-Hopf bifurcations of arbitrary co-dimension

Abstract

In this paper, we present a geometric approach to exponentially small splitting in zero-Hopf bifurcations of arbitrary co-dimension. In further details, we consider a family of problems that generalizes the third order Michelsen/Kuramoto-Sivashinsky-type equations $\epsilon^{2(\kappa-1)} f'''+f'={Q}(f)$, where ${Q}$ is an arbitrary real polynomial with $\kappa=\operatorname{degree}Q\ge 2$ simple real roots. For $\epsilon=0$, the system has $(\kappa-1)$-many heteroclinic connections and we describe the exponentially small splitting for each connection for all $0<\epsilon\ll 1$ under a separate nondegeneracy condition. In particular, we find that the $j$th-splitting is of the form $\epsilon^{-\frac{3\kappa}{2}}\exp\left({-\epsilon^{1-\kappa}T^j}\right)(C^j+\mathcal O(\epsilon))$, where $T^j>0$ can be calculated explicitly and be interpreted as the blowup time of special unbounded solutions of the $\epsilon=0$-limiting system in imaginary time $f'=iQ(f)$. Our approach extends a similar geometric method developed by the present author for the generic zero-Hopf bifurcation of co-dimension two, which does not rely on explicit time-parametrizations of the unperturbed heteroclinic connections and their singularities in the complex plane. Instead, we work exclusively in the complexified phase space and relate the exponentially small splitting to the lack of analyticity of center-like invariant manifolds of associated generalized saddle-nodes.