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Unformatted text preview: 2. Show that the dynamics are qualatively unchanged when higher order terms are considered. 2 Firt Step Consider ˙ r = dμr + ar 3 , ˙ θ = ω + cμ + br 2 . Lemma 3.2.1 For∞ < μd a < 0 and μ suﬃciently small ( r ( t ) , θ ( t )) = ± rμd a , ² ω + ³ cbd a ´ μ µ t + θ ! is a periodic orbit. Lemma 3.2.2 The periodic orbit is 1. asymptotically stable for a < 0; 2. unstable for a > 2 Case 1: d > , a > . Case 2: d > , a < . Case 3: d < , a > . Case 4: d < , a < . 3 Remark: Supercritical and Subcritical bifurcation. 3 Second Step Poincar´ eAndronovHopf Bifurcation Consider the full normal form. Then, for μ suﬃciently small, cases 14 hold. Example: Consider ± ˙ x ˙ y ² = ± μ + 25 1 μ2 ²± x y ² + ± ( x 24 xy + 5 y 2 ) x ( x 24 xy + 5 y 2 ) y ² 4...
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 Fall '10
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 Leftwing politics, Normal Form, Dynamical systems, higher order terms, Hopf bifurcation, Purely imaginary eigenvalues

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