Vector Field Curves - How to Add 4 Curves Between Manifolds

[Dustinsfl]

\begin{tikzpicture}[scale = 1.5]
\draw (0,0) circle (1cm);
\draw[-] (1,0) -- (-1,0);
\draw[->] (1.2,0) -- (.5,0);
\draw[->] (-1.2,0) -- (-.5,0);
\draw[-] (.907107,.907107) -- (-.907107,-.907107);
\draw[<-] (.907107,.907107) -- (.307107,.307107);
\draw[<-] (-.907107,-.907107) -- (-.307107,-.307107);
\end{tikzpicture}

So I am trying to add in 4 curves. One between the 4 manifolds of the saddle. How can I do this?

 

[NateD]

\begin{tikzpicture}[scale = 1.5]\draw (0,0) circle (1cm);\draw[-] (1,0) -- (-1,0);\draw[->] (1.2,0) -- (.5,0);\draw[->] (-1.2,0) -- (-.5,0);\draw[-] (.907107,.907107) -- (-.907107,-.907107);\draw[<-] (.907107,.907107) -- (.307107,.307107);\draw[<-] (-.907107,-.907107) -- (-.307107,-.307107);\draw[dashed] (0.5,0.87) to[out=30,in=-30] (0.87,0.5);\draw[dashed] (-0.5,0.87) to[out=150,in=-150] (-0.87,0.5);\draw[dashed] (0.5,-0.87) to[out=-30,in=30] (0.87,-0.5);\draw[dashed] (-0.5,-0.87) to[out=210,in=150] (-0.87,-0.5);\end{tikzpicture}