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Nonlinear Systems: Bifurcation Theory

  • Lawrence Perko
Part of the Texts in Applied Mathematics book series (TAM, volume 7)

Abstract

In Chapters 2 and 3 we studied the local and global theory of nonlinear systems of differential equations
$$ \dot{x} = f(x) $$
(1)
with fC1(E) where E is an open subset of R n . In this chapter we address the question of how the qualitative behavior of (1) changes as we change the function or vector field f in (1). If the qualitative behavior remains the same for all nearby vector fields, then the system (1) or the vector field f is said to be structurally stable. The idea of structural stability originated with Andronov and Pontryagin in 1937. Their work on planar systems culminated in Peixoto’s Theorem which completely characterizes the structurally stable vector fields on a compact, two-dimensional manifold and establishes that they are generic. Unfortunately, no such complete results are available in higher dimensions (n ≥ 3). If a vector field fCl(E) is not structurally stable, it belongs to the bifurcation set in Cl(E). The qualitative structure of the solution set or of the global phase portrait of (1) changes as the vector field f passes through a point in the bifurcation set. In this chapter, we study various types of bifurcations that occur in C1-systems
$$ \dot{x}=f\left( {x,\mu} \right) $$
(2)
depending on a parameter μR (or on several parameters μR m ).

Keywords

Periodic Orbit Hopf Bifurcation Phase Portrait Bifurcation Diagram Pitchfork Bifurcation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Lawrence Perko
    • 1
  1. 1.Department of MathematicsNorthern Arizona UniversityFlagstaffUSA

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