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. 
[![enter image description here][1]][1]
According by the following result that can be found in the Katok and Hasselblat book 
[![enter image description here][2]][2]
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 ?

 
  [1]: https://i.sstatic.net/EMUAl.png
  [2]: https://i.sstatic.net/abo3G.png