Nonlinear Systems: Local Theory

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


In Chapter 1 we saw that any linear system
$$ \dot{x}=Ax $$
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
$$ \dot{x}=f(x) $$
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.


Nonlinear System Equilibrium Point Phase Portrait Unstable Manifold Local Theory 
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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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