Description

• Title:
  Global and local structures of oscillatory bifurcation curves with application to inverse bifurcation problem
• Creator:
  Shibata, Tetsutaro
• Description:
  We consider the bifurcation problem $$ -u''(t) = \lambda (u(t) + g(u(t))), \quad u(t) > 0, \quad t \in I := (-1,1), \quad u(\pm 1) = 0, $$% where $g(u) = g_1(u) := \sin \sqrt{u}$ and $g_2(u) := \sin u^2 (= \sin (u^2))$, and $\lambda > 0$ is a bifurcation parameter. It is known that $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(g,\alpha)$. When $g(u) = g_1(u)$, this problem has been proposed in Cheng [4] as an example which has arbitrary many solutions near $\lambda = \pi^2/4$. We show that the bifurcation diagram of $\lambda(g_1,\alpha)$ intersects the line $\lambda = \pi^2/4$ infinitely many times by establishing the precise asymptotic formula for $\lambda(g_1,\alpha)$ as $\alpha \to \infty$. We also establish the precise asymptotic formulas for $\lambda(g_i,\alpha)$ ($i = 1,2$) as $\alpha \to \infty$ and $\alpha \to 0$. We apply these results to the new concept of inverse bifurcation problems.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2017 Topological Methods in Nonlinear Analysis
• Date:
  2017.09.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 50, No 2 (December 2017); 603 - 622;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.032/13024 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2017.032/13025
• Subject:
  Oscillatory bifurcation;
  global and local structure of bifurcation curves;
  Primary;
  34C23;
  Secondary;
  34F10
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.032
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other