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Degree Theory for Orbits of Prescribed Period of Flows with a First Integral
Author(s) -
Dancer E. N.,
Toland J. F.
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-60.3.549
Subject(s) - mathematics , degree (music) , differentiable function , mathematical proof , dynamical systems theory , context (archaeology) , function (biology) , pure mathematics , mathematical analysis , geometry , paleontology , physics , quantum mechanics , evolutionary biology , acoustics , biology
A new degree function, defined for flows which have a continuously differentiable first integral, counts, algebraically, the number of orbits of fixed period τ in a set Ω. The degree takes account of the number of such orbits and of the order of their isotropy group. The context in which it is defined, namely when a dynamical system has a first integral, is one where the Fuller index is always trivial. In passing we give a rudimentary account of generic bifurcation theory for orbits of fixed period of dynamical systems which have a first integral. The paper is in two parts. The first gives a reasonably self‐contained account of the principles involved in the definition of the degree function and of the consequent degree theory, which should be accessible to a wide audience including those with an interest in applications. Part 2 is a highly technical detailed account of the proofs that all the claims made in Part 1 are valid.