期刊
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
卷 74, 期 12, 页码 4012-4035出版社
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.na.2011.03.030
关键词
Separable Hamiltonian systems; Phase portrait; Bifurcation diagram
资金
- MICINN/FEDER [MTM2009-06973]
- Generalitat de Catalunya [2009SGR-859]
- MEC/FEDER [MTM2008-03437]
We study a generalization of potential Hamiltonian systems (H(x, y) = y(2) + F(x)) with one degree of freedom; namely, those with Hamiltonian functions of type H(x, y) = F(x) + G(y), which will be denoted by X-H. We present an algorithm to obtain the phase portrait (including the behaviour at infinity) of X-H when F and G are arbitrary polynomials. Indeed, from the graphs of the one-variable functions F and G, we are able to give the full description on the Poincare disk, therefore extending the well-known method to obtain the phase portrait of potential systems in the finite plane. The fact that the phase portraits can be fully described in terms of the two one-variable real functions F and G allows, as well, a complete study of the bifurcation diagrams in complete families of vector fields. The algorithm can be applied to study separable Hamiltonian systems with one degree of freedom, which include a vast amount of examples in physical applications. (C) 2011 Elsevier Ltd. All rights reserved.
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