Abstract
In Chapter 1 we saw that any linear system
has a unique solution through each point x0 in the phase space R n ; the Solution is given by x(t) = eAtx0 and it is defined for all t ∈ R. In this chapter we begin our study of nonlinear systems of differential equations
where f: E → Rn and E is an open subset of Rn. We show that under certain conditions on the function f, the nonlinear system (2) has a unique solution through each point x0 ∈ 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 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.
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© 1991 Springer-Verlag New York, Inc.
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Perko, L. (1991). Nonlinear Systems: Local Theory. In: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0392-3_2
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DOI: https://doi.org/10.1007/978-1-4684-0392-3_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-0394-7
Online ISBN: 978-1-4684-0392-3
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