MATH 345 Study Guide - Midterm Guide: Saddle Point, Phase Plane, Phase Portrait
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Be sure that this examination has three pages. 11 page of notes may be used. [25: consider the one-dimensional equation for (t) in the phase circle s 1 = r/(2 z), given by where r r is a parameter. = f ( , r) = r sin sin2 , (1) (a) observing in (1) that f (0, r) = 0 for all r, use linear stability analysis at the. Xed point = 0 (mod 2 ) to verify that the stability changes as r is increased through rc = 0. (b) what type of bifurcation (saddle-node, transcritical, or pitchfork) occurs at ( , rc) Find the normal form of this bifurcation. (hint: an alternate expression for the vector eld, f ( , r) = r sin 1. 2 + 1 (c) plot the set z = {( , r) s 1 r : f ( , r) = 0 } (horizonal axis r, vertical axis ).
