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By John Earman
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Additional info for A primer on determinism
If the family of solutions of the family of initial value problems y˙ = g(φ(y, ξ)), y(0) = 0, with parameter ξ ∈ R , is given by ρ : R × Rn → R, then ψ, deﬁned by ψ(t, ξ) = φ(ρ(t, ξ), ξ) is the ﬂow of the diﬀerential equation x˙ = g(x)f (x). n Proof. By deﬁnition ψ(0, ξ) ≡ ξ, and by the chain rule d ψ(t, ξ) = g(ψ(t, ξ))f (ψ(t, ξ)). 11) by a reparametrization of time. In the most important special cases the function g is constant. If its constant value is c > 0, then the reparametrization of the diﬀerential equation x˙ = cf (x) by ρ = ct results in the new diﬀerential equation dx = f (x).
8 Manifolds In this section we will deﬁne the concept of a manifold as a generalization of a linear subspace of Rn , and we will begin our discussion of the central role that manifolds play in the theory of diﬀerential equations. 34 1. Introduction to Ordinary Diﬀerential Equations Let us note that the fundamental deﬁnitions of calculus are local in nature. For example, the derivative of a function at a point is determined once we know the values of the function in some neighborhood of that point.
What are the properties of the ﬂow of grad H? More generally, for a smooth function G : Rn → R (maybe n is odd), let us deﬁne the associated gradient system x˙ = grad G(x). Because a conservative force is the negative gradient of a potential, many authors deﬁne the gradient system with potential G to be x˙ = − grad G(x). The choice of sign simply determines the direction of the ﬂow. Prove the following statements: (a) A gradient system has no periodic orbits. (b) If a gradient system has a rest point, then all of the eigenvalues of its linearization at the rest point are real.