MEE 201 Advanced Dynamics. 11/15/2007. Homework set No. 5 Due Wednesday, 11/22/2007 in class. 1. Consider the standard map studied in class. I0 = I + δ sin(2 πϕ ) , ϕ0 = ϕ + I + δ sin(2 πϕ ) , Study the change of the phase space of the standard map as δ changes from 0 to 2 π . Speciﬁcally, plot the cases of δ = 0 . 2 / 2 π,0 . 6 / 2 /pi. 1 / 2 π and δ = 2 π . Describe your investigation. 2. Find the criterion for existence of 1 / 2 periodic orbit of the standard map (hint: study the second iterate of the map). Compare your analytical criterion with what you see n the numerical phase portrait for small δ < 1 / 2 π . What happens close to that trajectory as δ is increased? 3. Study the standard map with dissipation I0 = (1-γ ) I + δ sin(2 πϕ ) , ϕ0 = ϕ + I + δ sin(2 πϕ ) , where γ is small (in the range 0 ,0 . 5). Which changes do you expect in the phase portrait from the conservative case? Plot characteristic phase
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