less than co*. The schematic ^ g u re 4.2. relates the phenomena at co=0.85 and (3=0.1 and shows the paths represented by Poincar6 coordinate xL under variation of parameter F. To examine the stability transition of the n= 1 path in this constant co section recall that the product of the mapping eigenvalues is a constant given by
equation (4.21). The eigenvalues are therefore either real or complex and constrained to lie on a circle of radius ywhere VAiA2 = yand centred on the origin in the complex Argand space.
The local solution path starts at S°, an attracting spiral point, and develops into S1 an attracting spiral cycle. At point a, S1 becomes a directly attracting node cycle where the complex eigenvalues of the spiral cycle become real and positive. At fold A, Aj (j may be 1 or 2) penetrates the unit circle at +1 and the local solution path is following a directly unstable saddle D1, (where r for resonant distinguishes this from the hill top saddle D1). At fold B the path re-stabilizes as A; re-enters the unit circle at +1. Then A„ where 0=1,2), become complex at b, pass completely around the circle of radius y to give an inversely attracting node with real negative mapping eigenvalues between c and C. At C, Aj passes out of the unit disc at -1, and this event is a supercritical flip bifurcation into a stable n= 2 subharmonic. A recursive sequence of flip bifurcations leads to the creation of a chaotic attractor. This chaotic solution subsequently is the subject of a global bifurcation event, a blue sky catastrophe or boundary crisis, at E which results in a phase space no longer containing an attracting solution.
Meanwhile, the inversely unstable n- 1 solution continues to the fold G where it turns back to become the directly unstable hill-top saddle cycle D 1. Before doing so, however, it is clear from the constraints on the eigenvalues that a reverse flip
*Figures 4.3,4.4 and 4.5 represent numerically followed /z=l paths at co=0.6, 0.7 and 0.8. Graphs of the Poincare variables f e y ,) and the parameter Fare recorded. There is a growth of the hysteresis region A-B while a reduction in distance to the final fold G. 2Figure 4.6 represents the f e F ) path for n- 1 and n - 2 solutions at co = 0.85, (3 = 0.1. In this diagram two folds in the n- 2 subharmonic are observed. As with n= 1 the prescribed sequence of admissible flip fold events must be followed. Thus further Feigenbaum cascades and reverse cascades, issuing from a 2 to 4 period doubling bifurcation, are present. The completed cascades may not be present at co = 0.85 but may be present at a slightly differing co parameter value. As is described in figure 4.1 the n=2 solution curve continues to the reverse flip at g, if not as simply as implied in this schematic diagram. Figure 4.6 starts to hint at the incredible complexity of any phase space slice at some fixed F.
4.2.5 Confluence of flips and folds and Recursive series of cusps.
The confluence of fold and flip lines is one of the most surprising details to emerge from the numerical studies of equation (4.18). This is hightened by the theoretical limitation on the eigenvalues which actually precludes a flip fold intersection in a co-dimension two event.1. Figures 4.3, 4.4 and 4.5: Local solution path under variation of F parameter for escape equation.
As co is decreased, the F coordinates of the cascade C-D-E are lowered until at co2, Fa=Fe; this is illustrated in !figures 4.7 and 4.8. For co < co2 there is now no stabilization of the jump from A, and the fold line A becomes the escape boundary as indicated by the dot screen.
As co is decreased below co2, C-D-E retreats below the fold B, all seeming to merge with the Melnikov curve of the homoclinic tangency M, at co * 0.6. The related numerical procedures experience severe difficulties in this region, (see section 4.2.2 ) but the AUTO package has managed to follow the fold B to co » 0. The Flip bifurcation C has not been followed below co « 0.6 but it is clear from the theoretical restraint on the eigenvalues that the fold B and flip C cannot touch or cross. Hence the resultant locus of B below co » 0.6 represents a bound on C. Numerical studies have shown the two bifurcation loci C and B asymtotically approaching each other.
The scenario in figure 4.1, with its cusp and associated flip boundary, is repeated at dimishing scales at lower forcing frequencies (figure 4.8). Two extra cusps of the fundamental n= 1 response surface emulate the behaviour around the main cusp P. Each flip line passing transversly between the folds, signals the onset of a period doubling cascade to chaos and escape. The recurrent tendency for flip and fold lines to run together at this scale compounds already severe numerical problems. This would in effect produce large changes in eigenvalues thus inevitably
large second partial derivatives which would, as section 4.2.2 suggests, further degrade the accuracy of a local bifurcation path following procedure described in this chapter.
This hierarchy of cusps and flips is analogous to the fine structure in the bifurcation set of the Duffings equation.