We begin by studying the non-autonomous ordinary differential equation
$$ \dot x = f\left( {t,x} \right) $$
where U ⊂ R×Rd is open, and (t, x) ∈ U, and f: U → Rd is continuous. For a curve φ we let Dφ denote its domain, which is an interval. By a solution φ we always will mean a “non-continuable” solution; i.e., there is no solution Ψ (defined on any interval DΨ) such that DΩ ⊂ DΩ, DΩ ≠ Dφ, and Ω ≡ φ on Dφ. We are also interested in studying other systems with non-uniqueness including the contingent equation \( \dot x \in \Omega \left( {t,x} \right), \) , and the control equation \( \dot x = g\left( {t,x,u} \right) \) for u ∈ Λ, where various additional conditions may be put on the integrable function u(•) or on the set g(t, x, Λ). In general we are interested in the set S of curves or solutions (with varying domains) which represent some kind of non-autonomous dynamical system without uniqueness.


Curve Space Contingent Equation Rest Point Mathematical System Theory Autonomous Differential Equation 
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© Springer-Verlag Berlin Heidelberg 1969

Authors and Affiliations

  • James A. Yorke

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