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Part of the book series: Texts in Applied Mathematics ((TAM,volume 7))

Abstract

In Chapter 2 we saw that any nonlinear system

$$ \dot x = f\left( x \right), $$
((1))

with ∈ C 1(E) and E an open subset of Rn, has a unique solution ø t (x0), passing through a point x0E at time t = 0 which is defined for all tI(x0), the maximal interval of existence of the solution. Furthermore, the flow ø t of the system satisfies (i) ø0(x)=x and (ii) ø t+s (x)=ø t s (x)) for all x ∈ E; and the function ø(t, x)=ø t (x) defines a C 1-map ø: Ω→ E where Ω, = {(t, x) ∈ R × E|tI(x)}.

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© 2001 Springer Science+Business Media New York

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Perko, L. (2001). Nonlinear Systems: Global Theory. In: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0003-8_3

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  • DOI: https://doi.org/10.1007/978-1-4613-0003-8_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6526-9

  • Online ISBN: 978-1-4613-0003-8

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