Bifurcation near Equilibrium

  • Shui-Nee Chow
  • Jack K. Hale
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 251)


In this chapter, we discuss various types of dynamic behavior when a bifurcation arises from the existence of a simple eigenvalue. More specificially, we consider an equation
$$ Cx + N(x,\mu ) = 0 $$
in a Banach space X for μ in a Banach space E, N(0, 0) = 0, N(0, 0)/x = 0 under the assumption that the linear operator C has zero as a simple eigen-value. The method of Liapunov—Schmidt gives a scalar bifurcation function G(a, μ) defined for (a, μ) in neighborhood of (0, 0) ∈ ℝ × E. Suppose Cx + N(x, μ)is the vector field for an evolutionary equation
$$ \tfrac{{dx}}{{dt}} = Cx + N(x,\mu ) $$


Banach Space Periodic Solution Periodic Orbit Equilibrium Point Stability Property 
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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Shui-Nee Chow
    • 1
  • Jack K. Hale
    • 2
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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