Find a system with only two saddle points such that there are no trajectories that connect them

Question

I'm looking for an example of a dynamical system with only two fixed points, both saddles such that there are no trajectories that connect them. Is this even possible? A picture would be sufficient (no need for analytic solutions).

Answer 1

Picture two "X"s (the crossings are the stationary points) next to eachother. Extend the inner arms either straight up or straight down vertically to infinity, and let the outer arms extend outward to infinity. Put directions on the arms consistent with saddle points, such that the region in between them has a upward (or downward) flow.

$\begin{array}{ccccccccccccc} \searrow&&&&&\uparrow&&\uparrow&&&&&\swarrow\\ &\searrow&&&&\uparrow&&\uparrow&&&&\swarrow\\ &&\searrow&&\nearrow&&&&\nwarrow&&\swarrow\\ &&&\times&&&&&&\times\\ &&\swarrow&&\nwarrow&&&&\nearrow&&\searrow\\ &\swarrow&&&&\uparrow&&\uparrow&&&&\searrow\\ \swarrow&&&&&\uparrow&&\uparrow&&&&&\searrow \end{array}$

Answer 2

In Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifuracations of Vector Fields, under Global Bifurcations, they have a section called Saddle Connections and describe when a trajectory joins two saddle points or forms a loop containing a single saddle point.

They provide several examples, but the first is what I believe what you are looking for and you can experiment with. The system is:

$$ x' = \mu + x^2 - x y \\ y' = y^2 - x^2 -1$$

The analysis shows that we can get some interesting dynamics and qualitative differences for the saddle point connections. Here are the three phase portraits in order of $\mu = \{-0.1, 0, 0.1\}$.

Notice the very interesting change in the separatrix for the negative and positive case (either of which I believe satisfies your question). The zero case (middle phase portrait) does not.