# hw7 - produced as a function of μ ω and sketch the...

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Homework 7 Cover Sheet Name: Space below is for the instructor. Problem Score

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CDS 140a: Homework Set 7 Due: Wednesday, November 28, 2007. Describe the bifurcation at the origin for the following system ˙ x = 2 x + 3 y + ǫx + y 3 ˙ y = 2 x 3 y + x 3 Describe the bifurcation at the origin for the following system ˙ θ = θ + v 2 + ǫv 2 ˙ v = sin( θ ) Show that ˙ x = (1 + μ ) x y + x 2 xy ˙ y = 2 x y + x 2 undergoes a supercritical Poincar´ e-Andronov-Hopf bifurcation at μ = 0. Consider the simple pendulum with linear dampling δ and constant forcing β : ˙ θ = v ˙ v = sin( θ ) δv + β (1) Here, ( θ, v ) S 1 × R 1 . Show that a saddle-node bifurcation occurs at β = 1 provided δ n = 0; allow δ to take either sign. You will need to compute a center manifold and the reduced one-dimensional system. Draw the bifurcation diagram and phase potraits for the system in the case δ > 0 and β > 1. Consider the system ˙ x = ωy ˙ y = ωx μy + ( x + δy ) z ˙ z = z x 2 2 y 2 (2) Show that the system undergoes a Poincar´ e-Andronov-Hopf bifurcation as μ passes through 0 with ω n = 0. Determine the stability type of the limit cycles
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Unformatted text preview: produced as a function of μ, ω and sketch the bifurcation diagrams for all topo-logically distinct cases. You will need to compute a suitable approximation to the center manifold and apply the Hopf formula for the coeFcient a in the normal form. ±or this system, show that there is a range of values for δ for which “formal” fast system approximation z ≈ − x 2 − 2 y 2 gives incorrect result. R g ω θ Figure 1: Bead on rotating hoop • Consider the system consisting of a bead on a frictionless rotating hoop as depicted in Figure 1. The equations of motion for the system are ˙ θ = v ˙ v = − g R sin( θ ) + ω 2 sin( θ ) cos( θ ) (3) Show that a pitchfork bifurcation occurs at a particular value of the parameter ω . Compute the bifurcation value and draw the bifurcation diagram....
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hw7 - produced as a function of μ ω and sketch the...

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