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Nonlinear Systems: Local Theory

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

Abstract

In Chapter 1 we saw that any linear system
$$ \dot{x}=Ax $$
(1)
has a unique solution through each point x0 in the phase space R n ; the solution is given by x(t)=e At x0 and it is defined for all tR. In this chapter we begin our study of nonlinear systems of differential equations
$$ \dot{x}=f(x) $$
(2)
where f: E → Rn and E is an open subset of R n . We show that under certain conditions on the function f, the nonlinear system (2) has a unique solution through each point x0E defined on a maximal interval of existence (α, β) ⊂ R. In general, it is not possible to solve the nonlinear system (2); however, a great deal of qualitative information about the local behavior of the solution is determined in this chapter. In particular, we establish the Hartman-Grobman Theorem and the Stable Manifold Theorem which show that topologically the local behavior of the nonlinear system (2) near an equilibrium point x0 where f(x0)=0 is typically determined by the behavior of the linear system (1) near the origin when the matrix A=Df(x0), the derivative of f at x0. We also discuss some of the ramifications of these theorems for two-dimensional systems when det Df(x0) 0 and cite some of the local results of Andronov et al. [A-I] for planar systems (2) with det Df(x0)=0.

Keywords

Nonlinear System Equilibrium Point Phase Portrait Unstable Manifold Local Theory 
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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