hopf_2005

# hopf_2005 - 2 Show that the dynamics are qualatively...

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Hopf Bifurcation CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction Consider ˙ w = g ( w, η ) . where w R n , η R p . Suppose it has a fixed point at ( w 0 , η 0 ), i.e., g ( w 0 , η 0 ) = 0. Moreover, its linearized equation ˙ ξ = D w g ( w 0 , η 0 ) ξ has two purely imaginary eigenvalues with the remaining ( n - 2) eigenvalues having nonzero real parts. Center Manifold Theorem tells us that the orbit structure near ( w 0 , η 0 ) is determined by the associated equation on the two-dimensional center manifold, which can be written in the following form ˙ x ˙ y = α ( μ ) - ω ( μ ) ω ( μ ) α ( μ ) x y + f 1 ( x, y, μ ) f 2 ( x, y, μ ) where λ ( μ ) = α ( μ ) ± ( μ ) (with λ (0) = ± (0) are the eigenvalues of the linearized equation. Normal Form was found to be ˙ r = α ( μ ) r + a ( μ ) r 3 + O ( r 5 ) , ˙ θ = ω ( μ ) + b ( μ ) r 2 + O ( r 4 ) . 1

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Remark: Notice that d = d (Re λ ( μ )) | μ =0 . Hence, for d > 0, the eigenvalues cross from the left half-plane to the right half-plane as μ increases and, for d < 0, the eigenvalues cross from the right half-plane to the left half-plane as μ increases. Two Steps. 1. Neglect the higher order terms and study the resulting “truncated” normal form.

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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 )) = ± r-μd a , ² ω + ³ c-bd 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´ e-Andronov-Hopf Bifurcation Consider the full normal form. Then, for μ suﬃciently small, cases 1-4 hold. Example: Consider ± ˙ x ˙ y ² = ± μ + 2-5 1 μ-2 ²± x y ² + ± ( x 2-4 xy + 5 y 2 ) x ( x 2-4 xy + 5 y 2 ) y ² 4...
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• Fall '10
• list
• Left-wing politics, Normal Form, Dynamical systems, higher order terms, Hopf bifurcation, Purely imaginary eigenvalues

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