Verification of bifurcation diagrams for polynomial-like equations

Korman, Philip; Li, Yi; Ouyang, Tiancheng

doi:10.1016/j.cam.2006.11.031

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Chapter 2 Definitions and Background1

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“…Changes in the sign of f (u) lead to multiple positive solutions of (2.1). All of the results cited in this chapter are based on the papers [7], [8], [9], [12], [13], [14], [15].…”

Section: Chapter 2 Definitions and Backgroundmentioning

confidence: 99%

Bifurcation theory for a class of second order differential equations

Correa¹,

Li²

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We consider the existence of positive solutions of the nonlinear two pointas the parameter λ varies through positive values. Every solution u(x) is an even function, and when it exists, it is uniquely identified by α = u(0). We study how the number of solutions changes when the parameter varies, i.e. we will be focusing on the locations of bifurcation points.The authors P. Korman, Y. Li and T. Ouyang ( "Computing the location and the direction of bifurcation", Mathematical Research Letters, accepted ), prove that a necessary and sufficient condition for α to be a bifurcation point iswhere F (α) = α 0 f (u) du. We will prove that G(α) has vertical asymptotes at α = b, α = 1 and at any point α ∈ (0, 1) for which α 0 f (u) du = 0. We will use the asymptotic behavior of G to estimate intervals where G(α) = 0, that is, intervals where there is no bifurcation point. Abstract Approved:Thesis Supervisor

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“…Changes in the sign of f (u) lead to multiple positive solutions of (2.1). All of the results cited in this chapter are based on the papers [7], [8], [9], [12], [13], [14], [15].…”

Section: Chapter 2 Definitions and Backgroundmentioning

confidence: 99%

Bifurcation theory for a class of second order differential equations

Correa¹,

Li²

View full text Add to dashboard Cite

show abstract

Verification of bifurcation diagrams for polynomial-like equations

Cited by 1 publication

References 13 publications

Bifurcation theory for a class of second order differential equations

Bifurcation theory for a class of second order differential equations

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