polynomial or rational function example of homoclinic orbit $f:S^3 \to S^3$?

Question

I'd like to understand the definition of homoclinic orbit. Can we have a polynomial map from the three-sphere to itself that exhibits a such a fixed point. We could have a map $f:S^3 \to S^3$ with a fixed point $f(x_0)=x_0$ with an orbit auch that for some $x\in S^3$ we get $$ \lim_{n \to \pm \infty} f^{n}(x)= x_0$$ This could be a perturbations of a rotation or a map $(x,y,z,w) \to (f_1,f_2,f_3,f_4) \in \mathbb{R}P^3$ where each of the $f$ are rational functions and we rescale. Technically projective space and the 3-sphere are different manifolds.