Computer Science
Scientific paper
Mar 1982
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1982phdt........23p&link_type=abstract
Thesis (PH.D.)--THE UNIVERSITY OF TEXAS AT AUSTIN, 1982.Source: Dissertation Abstracts International, Volume: 43-03, Section: B,
Computer Science
1
Scientific paper
The Lindstedt method is generalized for conservative systems with two degrees of freedom, especially systems with resonance. The frequency corrections (that characterize the method for one degree of freedom) are supplemented with homogeneous algebraic equations relating the components of motion in different directions. Recursive algorithms are developed. The "equations of condition for periodicity" appear in a very lucid form when the Lindstedt method is used. In a given resonant problem, a complete enumeration is possible for the different modes of nonlinear periodic oscillation (in the neighborhood of the equilibrium); this will include both "normal modes" and "resonant modes.". By an appropriate modification, the algorithm for exact resonance is applied to near resonance; the structural stability of resonance phenomena is investigated. The Birkhoff normal form is developed (in action -angle variables) by the method of Lie series transforms. Although the methods are completely different in conception, the periodic solutions obtained from the normal form always satisfy the recursive equations of Lindstedt. Expansion of the normal form in the neighborhood of periodic manifolds allows the development of formal series for the stability indices. For non-resonant problems, it is shown that the normal modes are always stable (in the neighborhood of the equilibrium); if resonance is present, stability depends on the individual dynamical system. For resonant systems, the expansion is made possible by transformations that isolate the critical angle. Different forms of such transformations are appropriate to different families of periodic solutions. The 1:1 resonance is used as the example, especially for the Contopoulos system and the Henon-Heiles system. Six families of periodic solutions are found for the former, and eight families for the latter.
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