Bifurcation diagrams for singularly perturbed system.
By Matteo Franca. 2012
55 citations recorded by the source corpus.
We consider a singularly perturbed system where the fast dynamic of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system is 1-dimensional and it admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. In this setting Battelli and Palmer proved the existence of a unique trajectory (˜x(t,",�), ˜ y(t,",�)) homoclinic to the slow manifold. The purpose of this paper is to construct curves which divide the 2-dimensional parameters space in different areas where (˜x(t,",�), ˜ y(t,",�)) is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.
Subfields: Computer Networks and Communications, Public Health, Environmental and Occupational Health, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b7546e993021ab241af5207335d371a12ac6003c
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In Computer Networks and Communications
In Public Health, Environmental and Occupational Health
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Citations in the corpus
- Non-negative solutions of systems of ODEs with coupled boundary conditions (2012, methodology): “In [9] Cui and Sun study, via fixed point index theory, the existence of one positive solution for the system u00ðtÞ þ f1ðt;uðtÞ; vðtÞÞ 1⁄4 0; t 2 ð0;1Þ; v 00ðtÞ þ f2ðt;uðtÞ;vðtÞÞ 1⁄4 0; t 2 ð0;1Þ; uð0Þ 1⁄4 0; uð1Þ 1⁄4 b121⁄2v ; vð0Þ 1⁄4 0; vð1Þ 1⁄4 b221⁄2u ; ð3Þ”
- Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions (2013, background): “In particular we are interested in systems that present a coupling in the boundary conditions (BCs); this type of problems have been studied in [5, 6, 7, 10, 11, 32, 36, 47] and often occur in applications, for example when modelling the displacement of a suspension bridge subject to nonlinear contr”