Limits of stability for an area-preserving polynomial mapping

Details Limits of stability for an area-preserving polynomial mapping Limits of stability for an area-preserving polynomial mapping

Nov 1982

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1982cemec..28..295b&link_type=abstract

Celestial Mechanics, vol. 28, Nov. 1982, p. 295-317.

Mathematics

10

Computer Techniques, Optimization, Polynomials, Systems Stability, Transformations (Mathematics), Variational Principles, Eigenvalues, Matrices (Mathematics), Theorems

Scientific paper

The Henon-Heiles mapping x-prime = x + a(y - y cubed), y-prime = y - a(x-prime - x-prime cubed) is studied in order to locate the unstable regions of the (x,y) plane. It is found that when this mapping is put into the normal form, it is a typical twist mapping. The criteria of Moser (1971) are used in obtaining an upper limit to the size of a stable region around the origin, and this limit is found to decrease to zero as the value of the parameter 'a' increases toward 2.0. Direct calculation for a = 1.99, however, shows that there is a fairly large region inside x = 0.412, y = 0, from which escape from near the outer boundary requires at least 160 mappings. The region of high stability is thus seen to be much larger than any region of absolute stability predicted by the KAM (Moser, 1971) theorem. A general survey is made of instability regions for the parameter value a = 1.0, the survey being carried out to the extent allowed by a computer with 18-decimal-place accuracy. For all the x-axis fixed points (of the above mapping) deemed to be representative and significant, both the locations and variational matrix traces are calculated.

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