Proof of the Transversality for the Standard Map

Yoshihiro Yamaguchi¹ and Kiyotaka Tanikawa²

¹2-4-14, Kokubunjidai-chuo, Ichihara, Chiba 290-0193, Japan
²National Astronomical Observatory, Mitaka 181-8588, Japan
E-mail address: chaosfractal@icloud.com; tanikawa.ky@nao.ac.jp

(Received January 18, 2021; Accepted April 10, 2021)

Abstract. We consider the standard map. The stable and unstable manifolds of the saddle fixed point are proved to intersect transversely at the primary homoclinic point u for any parameter value. For the proof, we use the particular objects called the dominant axis (DA) and subdominant axis (SD), and symmetric periodic orbits that have orbital points on these axes. The periodic orbit named 1/q-BE has the orbital point z_k at the intersection point of DA and SD. Let ξ_k be the slope of SD at z_k. Take a sequence of z_k accumulating at u as k → ∞. We prove that the slope ξ_k monotonically decreases to the slope ξ_u(u) of the unstable manifold at u (the monotone inclination property). Using Ushiki's theorem, the hyperbolic region (HR) is constructed. It is proved that the orbital point z_k in HR is a saddle point with reflection. Using the monotone inclination property and the properties of z_k in HR, the transversality at u for any value of a (> 0) is proved.

Keywords: Standard Map, Stable and Unstable Manifolds, Transversality, Ushiki's Theorem, Dominant and Subdominant Axes