Differential Equations and Dynamical Systems pp 65-178 | Cite as

# Nonlinear Systems: Local Theory

Chapter

## Abstract

In Chapter 1 we saw that any linear system has a unique solution through each point x where

$$ \dot{x}=Ax $$

(1)

_{0}in the phase space**R**^{ n }; the solution is given by x(*t*)=*e*^{ At }x_{0}and it is defined for all*t*∈**R**. In this chapter we begin our study of nonlinear systems of differential equations$$ \dot{x}=f(x) $$

(2)

**f***: E →***R**^{n}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 x_{0}∈*E*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 x_{0}where**f**(x_{0})=0 is typically determined by the behavior of the linear system (1) near the origin when the matrix*A=D*f(x_{0}), the derivative of**f**at x_{0}. We also discuss some of the ramifications of these theorems for two-dimensional systems when det*D***f**(x_{0})*≠*0 and cite some of the local results of Andronov et al. [A-I] for planar systems (2) with det*D***f**(x_{0})=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