Show that the fixed point of the map fx y 3x 6y 04y is a s

Show that the fixed point of the map f(x, y) = (3x + 6y, 0.4y) is a saddle and find its stable and unstable manifolds. Draw the manifolds and show graphically the behavior of the orbits.

Solution

A point x which is mapped to itself under a map G, so that x = G(x). Such points are called invariant points or fixed points. Stable fixed points are called elliptical. Unstable fixed points,corresponding to an intersection of a stable and unstable invariant manifold, are called hyperbolic(or saddle). Points may also be called asymptotically stable(superstable).