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Unformatted text preview: J L V] Math. 224. Test 2
FALL 2002. Nov.14 David Gurarie Name “a 1. Modeling diﬁ'erential systems: describe system type (linear / nonlinear, coupled / uncoupled
etc.) and solution method (if available)
(a) Two—bin population growth model with age groups L1
f = 5 and children mortality m1
stabilize the pOpulation) = 10, L2 = 50; fertility
.01. (Extra: which fertility levels f would l x 13:“ 12.:
~xcwm} it ~ gag ~5m+i39€ [fr—~05“: I i
j Ml"? y» m “L 2: h “L {I a. f .. f2: " e 535:. w. 554%. (iii J2“ A — “‘M "5” 7 ‘ ADS MUM 637 fﬁ‘éﬁmﬁiag
.. —\ a i “a O 2 u f WW \‘2
ﬂ . =7— 4 :. I!
W“ 1 .0?“ 7.”? 4:1). (b) Predator—prey (Volterra—Lotka) model with linear growth for prey. Sketch phase—
plane and equilibria, describe equilibria types. ‘0“: E O ,_ 5:2..OICI (c1) Migration. Populations w, y of two states migrate, so that a fraction of :5 migrates
to y, and 6 fraction of y migrates to SE. Besides $~popu1ation grows at an annual
rate 7°, while y pOpulation decays at rate d. Set up the DE model of the system. 9 : ‘4) ‘P v] T’ ’64» L)
Di Cr )x Z! i}; :> A :
3 : ax  (5MB 3’ ‘” ~ (“on 7 S Sr) (J ma“) Q3 {3; nwaﬁawa 2. Consider migration problem 1(d) with coefﬁcients: a 2 .5, b = .5, d = .2 and 7" = 0. (a) Find the general solution, and IVP—solution for (£170, yo) = (0,1). l+’!,l)\+.i :O
A,L~ ~ 6 iz/ngﬂen (b) Sketch solution plots, and phase—plane plots. Explain the limiting behavior of the
system, as t ——> oo. M . » , ’1.
m 4 i; x (Q 4“ SW?) L34 ’7 ~ j— ” i (c) Solve the eigenvalue problem for 7* 2
solution curves for initial value (
Explain the difference in terms 0 .8, and use it to sketch the phase plot and
0,1). Compare solutions of part (b) and (c)
f bifurcations in parameter 7" (Extra credit for computing bifurcation value)
< “‘3 s‘ Howey /\ =”V5“e:
A v ' ’ '0: x o " b I {2  ... ) / :__,2 1'7
l6 —l7 3. Bifurcations of linear systems. Consider a family of linear systems with matrix 2
A: [ _1 (11] and parameter ~3Sa£3 stability diagram ( T : tr(A); D = det (b) Determine the stability type of equilibria in each range of parameter a (between bifurcation values), and describe the transition of equilibria—types at the each
bifurcation point f ‘ l 2' 2 a)
"WAR f" Dz>x ‘3) +( 1'
{ml“1mm ‘ I (0) Sketch phaseplots for typical values of a in different stability intervals and at —"’\
.f t. 1 AW "WWW _ x :
b1 urca 101%“ny :Eesz CL : 0 __ Kg: J
3: 3 ‘M
V :7 /
3.2 O I 0 4. Oscillator and sliding chain A (b) Explain the effect of damping on frequency of oscillations ﬂ (Extra for plotting ‘ functifgﬁédﬂ A Z
(“Ah W) :VHJ) 6 \Cr 0 Figure 1: W_...MM~ (C) Derive the Sliding Chain equation with attached mass: m):}§ = g (a: + m).
Solve it for initial value problem: a: (0) = :50; v (0) = 0. OW sEiIﬁEBn could b3
written as —m + Acosh ( ﬁt) (Find coefﬁcient K‘— ~ .— I'M—w~~‘.,.....—m.—..._...v h I ~~~~~~~~~~~~~~~ ‘8 I o a a; — ‘ //,, r :. z 1 _ F. u _' Q (d) Find Sliding time fork: 5; m = 3, 9 = lOiand initial :30 = 1 (Hint: cosh‘1 u 2
log (u + 1 /u2 + n._,i,i._.w.,_w.. 3 bx. mat) = ,8 ...
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This note was uploaded on 05/09/2008 for the course MATH 224 taught by Professor Hahn during the Spring '07 term at Case Western.
 Spring '07
 Hahn

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