Invariant circles, computation

Beyond its applicability in the simplification of the computations, the stroboscopic representation greatly simplifies the recognition of patterns in the transient response of periodically forced systems. A sustained oscillation appears as a finite number of repeated points, while a quasi-periodic response appears as an invariant circle (see Figs. 3, 4, 6 and 9). 

There exists a quantity (the rotation number) which is independent of the particular stroboscopic section we take and characterizes the entire torus. For a radial parameterization of the invariant circle the rotation number is defined—and may be computed—as the limit... 

In our computation of invariant circles of maps the main cost lies in the... 

An alternative formulation for the torus-computing algorithm is to solve for an invariant circle along with a nonlinear change of coordinates that makes the action of the stroboscopic map conjugate to a rigid rotation on the circle. This is equivalent to the parameterization... 

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

Aronson, D. G., Chory, M. A., Hall, G. R. and McGehee, R. P., 1982, Bifurcations from an invariant circle for two-parameter families of maps of the plane a computer assisted study. Comm. Math. Phys. 83, 303-354. 

Iooss, G., Ameodo, A., Coullet, P. and Tresser, G, 1980, Simple computations of bifurcating invariant circles for mappings. Lect. Notes Math. 898,192-211. 

Numerical computation of invariant circles of maps (with I.G. Kevrekidis, L.D. Schmidt, and S. Pelikan). Physica 16D, 243-251 (1985). 

FIGURE 10 Change of the torus and the angular function with W o. (a) A succession of computed sections of invariant tori for various values of >/ >o (Brusselator, a = 0.0072). The centre point is indicated by (+). (b, c) The occupancy of the converged Jacobian for (u/gjo = 1.186667 and 1.3, respectively. The bumps on some of the circles are artifacts of the mesh and are associated with the almost vertical parts of the nonzero band. They can be eliminated by mesh adaptation. 

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