A Short Course in Ordinary Differential Equations by Qingkai Kong

Author:Qingkai Kong
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(ii) is a 2π-periodic solution with the orbit given by the circle r = 1.

(iii)When 0 < r(0) < 1, then 0 < r(t) < 1 for all due to Lemma 4.1.1. Moreover, r′(t) > 0 except when θ = 0 [mod π], where the orbit is perpendicular to the x 1-axis. This implies that r(t) is strictly increasing. Thus there is no closed orbit inside the circle r = 1. Also, since , we see that θ(t) is strictly decreasing and θ(t) → −∞ as t → ∞.

(iv)When r(0) > 1, then as above, r(t) > 1 for all and r(t) is strictly decreasing. Thus there is no closed orbit outside the circle r = 1. Also, since as r → 1 uniformly for , we see that θ(t) → −∞ as t → ∞ for r close to 1.

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