Lamerey diagram

If li < 0, then the fixed point at the origin is stable for // < 0 and attracts all trajectories from any small neighborhood. When > 0, the fixed point becomes imstable and the stable period-two point bifurcates from it see Fig. 11.4.1 illustrating the corresponding Lamerey diagrams. 

Let us now consider the case A < 0 in more detail. First, let us examine the associated Lamerey diagram (Fig. 13.2.6) with A < 0 when i/ > 1 (A -h7 < 0). Just as in the orientable case, a stable fixed point exists for // > 0 indeed, the map T is decreasing and contracting, so the interval [0, a p) = T(0)] is mapped into the inside itself by T, which implies the existence of a unique stable fixed point on this interval by virtue of the Banach contraction mapping principle. 

Fig. 13.2.6. Lamerey diagram of the Poincare map for the non-orientable separatrix loop with (Tq < 0.

Fig. 13.2.7. Lamerey diagram for the non-orientable separatrix loop to a seuidle with (Tq > 0.

Fig. 13.3.2. Lamerey diagram corresponding to the double separatrix loop.

The Lamerey diagrams for this map are shown in Fig. 13.6.5. The right-hand side of the map is either monotonically increasing (when —n/2 < (j) < tt/2), or monotonically decreasing (when 7t/2 < < < Sir/2). Such maps may have only... 

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