Quasiminimal attractors

The quasiminimal attractors correspond to the limit points of the union of the curves Cs. 

For each admissible pair (p, s) of infinite sequences, there exists a curve C(p, s) corresponding to the existence of a quasiminimal attractor with the rotation number / (p) = /3(s). The curve C(p, s) is found from... 

In the other remaining cases, quasiminimal attractors do not appear. In the case > 0 and A2 (see Fig. 13.7.7), there exist cycles only with the codes 1, 2 and 21, A = 1,2,... (here 1 " denotes the word consisting of k ones ), and the parameter plane is partitioned into a countable number of regions by the curves Ci, C2, C12, C 2ifc and 0121 (A = 1,2,...). Note that these curves accumulate onto the negative /i2 semi-axis where the separatrix Fi forms a simple homoclinic loop and the separatrix F2 tends to the loop as t —> +00. 

If a P-trajectory is unstable in the sense of Lyapimov, this should imply that each trajectory from its quasiminimal set is also unstable. Nevertheless, this set can be attractive as a whole, and in this case it may be a mathematical image of the complex oscillatory process that we have been seeking. However, a clear understanding of the necessity of having such attractors in nonlinear dynamics came years later, only in the seventies. 

Therefore, if a structurally stable system has an attractive quasiminimal set — a strange attractor, then periodic orbits will be dense in it. 

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