论文标题
Dud Canard：在$ \ Mathbb r^3 $中存在强的Canard Cycles
The dud canard: Existence of strong canard cycles in $\mathbb R^3$
论文作者
论文摘要
在本文中，我们通过II型的折叠鞍节和单数Hopf分叉的折叠鞍节点在$ \ Mathbb r^3 $中的CANARD极限周期的出生进行了严格描述。特别是，我们仅在分析案例中证明 - 对于所有$ 0 <ε\ ll 1 $，有一个周期性的轨道家族，出生于（单个）Hopf分叉，并延伸至$ \ Mathcal o（1）$ cycles，遵循折叠的Saddle-Node的强壮的Canard。我们的结果可以看作是$ \ mathbb r^2 $中牛排爆炸的扩展，但是与平面案例相比，$ \ mathbb r^3 $中的周期性轨道家族不是爆炸性的。因此，我们选择以$ \ mathbb r^3 $（``Dud Canard'''称为现象。证明的主要困难在于将HOPF循环与牛排周期联系起来，因为这些循环用不同的尺度描述。就像在$ \ Mathbb r^2 $中一样，我们使用爆炸来克服这一点，但由于其单一的性质，我们还必须弥补霍普夫分叉附近缺乏统一性的赔偿；这是限制$ε= 0 $的零HOPF分叉。在本文中，我们通过施加矢量场的分析性来做到这一点。这使我们能够证明存在不变的慢流形，这通常不是双曲线。
In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in $\mathbb R^3$ through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove -- in the analytic case only -- that for all $0<ε\ll 1$ there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to $\mathcal O(1)$ cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in $\mathbb R^2$, but in contrast to the planar case, the family of periodic orbits in $\mathbb R^3$ is not explosive. For this reason, we have chosen to call the phenomena in $\mathbb R^3$, the ``dud canard''. The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in $\mathbb R^2$, we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit $ε=0$. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic.
