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 $ε^{2(κ-1)} f'''+f'={Q}(f)$, where ${Q}$ is an arbitrary real polynomial with $κ=\operatorname{degree}Q\ge 2$ simple real roots. For $ε=0$, the system has $(κ-1)$-many heteroclinic connections and we describe the exponentially small splitting for each connection for all $0<ε\ll 1$ under a separate nondegeneracy condition. In particular, we find that the $j$th-splitting is of the form $ε^{-\frac{3κ}{2}}\exp\left({-ε^{1-κ}T^j}\right)(C^j+\mathcal O(ε))$, where $T^j>0$ can be calculated explicitly and be interpreted as the blowup time of special unbounded solutions of the $ε=0$-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.

Figures (6)

• Figure 1: Illustration of our approach (for $\kappa=3$). We blow up $(x,y,z,\epsilon)=(0,0,0,0)$ to a complex 3-sphere $\mathbb S^3$, see (\ref{['eq:bu']}), which we here indicate in the $(\operatorname{Re}(x),\operatorname{Im}(x),\epsilon)$-projection. On top of the sphere, we use the coordinates $(x_2,y_2,z_2)$ to describe the existence of the saddle-foci $(x_2,y_2,z_2)=\mathbf{q}^{j}(\epsilon)\to (q^j,0,0)$ as $\epsilon\to 0$, and their one-dimensional invariant manifolds as graphs $(y_2,z_2)=m_2^{j}(x_2,\epsilon)$ over fixed large compact domains $x_2\in \mathcal{X}^{j}\subset \mathbb C$ for all $0<\epsilon\ll 1$. Here $j\in \{1,\ldots,\kappa\}$. The orange curves are orbits of $x_2'=Q(x_2)$, $x_2\in \mathbb C$, (on the Poincaré sphere,) and $\mathcal{X}^j$ is in the basin of attraction for $q^j$ for the forward flow for $j$ odd (backward flow for $j$ even, respectively). On the other hand, the purple curves are orbits of $x_2'=iQ(x_2)$, $x_2\in \mathbb C$. A central objective of our approach is to extend the unstable and stable manifolds of $q^{j}$ by working in the coordinates $(r_1,y_1,z_1,\epsilon_1)$, see (\ref{['eq:hatx1']}), so that they can be compared with the unperturbed invariant manifolds that are defined as graphs over $x\in S^{l}$, $l\in \{1,\ldots,2(\kappa-1)\}$, within $\epsilon=0$. Our results show that if the unperturbed manifolds are non-analytic, then there is an $\mathcal{O}(1)$-splitting of the unstable and stable manifolds $\mathbf{W}^{u/s}(\mathrm{q}(\epsilon))$ for $x\in S^j\cap S^{j+1}$, $\vert x\vert>0$ small enough, as $\epsilon\to 0$ (for at least one $j\in \{1,\ldots,\kappa-2\}$). Finally, we integrate the differences $(\Delta y_2^j,\Delta z_2^j):=(m_2^{j+1}-m_2^{j})(x,\epsilon)$ along the special trajectories $\mathcal{H}^j$ (in purple) of $x_2' = iQ(x_2)$ to the point $p^j:=\mathcal{H}^j \cap \{\operatorname{Im}(x_2)=0\}\in (q^{j+1},q^j)\subset \mathbb R$. In this step, we view the equation for the difference as a slow-fast system with a normally hyperbolic critical manifold, which we then treat by GSPT (through a Fenichel-type normal form).
• Figure 2: The slow-fast dynamics of (\ref{['eq:x2y2z2fast']}) for $\kappa=4$. The layer problem (\ref{['eq:layer']}) is in green whereas the reduced problem (\ref{['eq:reduced']}) is in blue. The $x_2$-axis is the critical manifold, which is normally elliptic. In red we show stable and unstable manifolds of the perturbed points $\mathbf{q}^1(\epsilon)$ and $\mathbf{q}^2(\epsilon)$ for $0<\epsilon\ll 1$. We are concerned with the asymptotic splitting of $\mathbf{W}^{\sigma^j}(\mathbf{q}^j(\epsilon))$ and $\mathbf{W}^{\sigma^{j+1}}(\mathbf{q}^{j+1}(\epsilon))$ within some fixed section $\{x_2=p^j\}$ with $p^j\in (q^{j+1},q^j)$.
• Figure 3: Phase portraits of $x_2'=Q(x_2)$ on the Poincaré hemisphere $\mathbb H^2$ for $\kappa=2$ (a) and $\kappa=3$ (b). The points $q^3<q^2<q^1$ denote the roots of $Q$. The sets $\breve {\mathcal{X}}^j$ (green and purple) in (a) are compact neighborhoods of $\breve q^j$ within the basins of attraction for the forward/backward flow of $x_2'=Q(x_2)$. These sets are relevant for Proposition \ref{['prop:Wuspj2']}.
• Figure 4: Phase portraits of $x_2'=iQ(x_2)$ on the Poincaré hemisphere $\mathbb H^2$ for $\kappa=2$ (a) and $\kappa=3$ (b). The points $p^2<p^1$ define the sections $\{x_2=p^j\}$ where the separation of the invariant manifolds are measured.
• Figure 5: The sectors $S^l$, $l\in \{1,\ldots,2(\kappa-1)\}$, in Lemma \ref{['lem:Sj']} for $\kappa=2$ (a) and $\kappa=3$ (b). The sectors $S^l$ are centered along $\breve e^l$. On the hand, $\breve h^l$ corresponds to a $\frac{\pi}{2(\kappa-1)}$ rotation of $\breve e^l$ counter-clockwise. It will be important that $S^{l}\cap S^{l+1}$ (with $2\kappa-1\equiv 1$) is an open sector centered along $\breve h^l$ with opening $2\chi>0$.

Theorems & Definitions (46)

• Theorem 1.1
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Corollary 2.5
• proof