2021 Volume 36 Issue 1 Pages 15-24
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 zk at the intersection point of DA and SD. Let ξk be the slope of SD at zk. Take a sequence of zk accumulating at u as k → ∞. We prove that the slope ξk monotonically decreases to the slope ξu(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 zk in HR is a saddle point with reflection. Using the monotone inclination property and the properties of zk in HR, the transversality at u for any value of a (> 0) is proved.
