Consider the following map \begin{align*} T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\ (x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\theta+\frac{y}{2}\right)+0.01, \theta+\frac{3y}{4}+\sin^2\left(\pi\theta+\frac{y}{2}\right)+0.01\right) \end{align*} I have been doing some numerical simulations and it seems this system presents a chaotic attractor. Here there is an approximation of the attractor. The colored points are the fixed points of the map. According by the following result that can be found in the Katok and Hasselblat book the chaos in this system might be generated by a transverse intersection of stable and unstable manifold of the orange saddle point.

My Question: How do I check analytically that there is this intersection or how do I prove this map has chaotic behaviour?

Could anyone help me ?