Smooth invariant closed curve

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

The smooth case corresponds, in particular, to a small time-periodic perturbations of an autonomous system possessing a homoclinic loop to a saddle-node equilibrium (see the previous section). Indeed, for a constant time shift map along the orbits of the autonomous system the equilibrium point becomes a saddle-node fixed point and the homoclinic loop becomes a smooth closed invariant curve, but the transversality of to F is, obviously, preserved under small smooth perturbations. 

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

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