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10 20 30 (c)RK4 time plot
24 (d) RK4phase plane plot
FIGURE 2O9 Solution theLoiko-Volterro Euler's for model method fime-series {b)phose-plone ond ond plots, {o) (cJ plots RK4 meihod time-series ld)phcse-plone ond
predator peak lags the prey Also, observe that the processhas a fixed period-that is, it repeatsin a set time for The phase-plane representation the accurateRK4 solution (Fi9209Q indicates that the interaction between the predator and the prey amounts to a closed counterclockwise orbit Interestingly, there is a resting or critical point at the center of the orbit The exact location of this point can be determined by setting Eqs (2049) and (2050) to steadystate (dy ldt : dx /dt : 0) and solving for (x, y) = (0, 0) and (cld, alb) The formeris the trivial result that if we start with neither predatorsnor prey, nothing will happenThelatter is the more interesting outcome that if the initial conditions are set at x:cld and : a /b, the derivativeswill be zero, andthe populationswill remain constant ! Now, let's use the sameapproachto investigatethe trajectoriesof the Lorcnz equations with the following parameter values:a = 10,b = 8I 3,and r : 28 Employ initial conditions
Lorenz model "{ versus r
5 0 0 1r,' - : : 5
-5 -10
-20 FIGURE20IO Time-domoin representotion versusfor fheLorenz of x t equotions solidtimeseries for the The is initiol conditions 5, 5)Thedoshed is wheretheinitiol line forr is perturbed condttion {5, slighily i s 0 0 r, 5 , 5 )
of"r:)':z:5andintegratefrom/:0to20Forthiscase,wewillusethefourth-order RK methodto obtain solutionswith a consrant time stepof h :003125 The results are quite different from the behavior of the Lotka-Volterra equationsAs in Fig 2010,the variabler seems be undergoing almostrandompatternof oscillations, to an bouncing around from negative values to positive values The other variablesexhibit similar behavior However, even though the pattems seem random, the frequency of the oscillation and the amplitudesseemfairly consistent An interestingfeatureof suchsolutionscan be illustratedby changingthe initial condition for "r slightly (from 5 to 5001)The resultsare superimposed the dashedline in as Fig 2010Although the solutionstrack on eachother for a time, after about / : l5 they diverge significantlyThus, we can see that the Lorenz equationsare quite sensitiveto their initial conditionsThe term chaotic is used to describesuch solutionsIn his original study, this led Lorenz to the conclusionthat long-rangeweatherforecastsmight be impossible!
(a) y versusr (b) z versus-r
10 n -10
45 40 35 30
(c) versus!
45 40
25 20
10 5 -20
0 "r
10 5 -40
represeniot;on fhe Lorenz Phose-plone for (oJ eqLrotions rr, (bl r:, ond (cJyl proiections
The sensitivityof a dynamicalsystemto small perturbations its initial conditions of is sometimescalled the butte(Iy ffictThe idea is that rhe flapping of a butterfly'swings might induce tiny changesin the atmospherethat ultimately leads to a large-scale weather phenomenon like a tornado Although the time-series plots are chaotic, phase-planeplots reveal an underlying structure Because we are dealing with three independent variables, we can generate projectionsFigure 2011 showsprojectionsin the xy,xz, and the yz planesNoticehowa structure is manif-est when perceivedfrom the phase-plane perspectiveThe solutionforms orbits around what appearto be critical points Thesepoints are called strangeattractorsin the jargon of mathematicians who study suchnonlinearsystems Beyond the two-variableprojections, MATLAB's p1or3 function providesa vehicle to directly generate three-dimensional a phase-piane plot:
>> plot3 (y(: ,L) ,y(:,2) ,y(:,2) ) >> xlabel('x' ) ;ylabel('y' ) ;z1abe} ('2, ) ;grid
intt res
As was the casefor Fig 201i, the three-dimensional plot (Fig2012) depictstrajecrories cycling in a definite pattem arounda pair ofcritical points As a final note, the sensitivity of chaotic systemsto initial conditions hasimplications for numericalcomputations Beyond the initial conditionsthemselves, differentstepsizes or different algorithms (and in some cases,even different computers)can introducesmall differencesin the solutionsIn a similar fashion to Fig 2010, thesediscrepancies will eventuallylead to large deviationsSomeof the problemsin this chapterand in Chap 21 are designedto demonstrate this issue
(a) (b) (c) (d)
30 20 10
l\l tI l-I I
_10 -20
I rl
t-t_ t-
- 4 0 -20
FIGURE 2O12 Three-dimensionol phose-plone representoiion Lorenz forlhe equolions generoted MATLABs with
p}ot 3 iunction
201 Solve the following initial value problem over the interval frorn / : 0 to 2 where r'(0) : l Display all your resultson the samegraph dr'
dt -:\,/--l l\
202 Solve the following problem over the interval from -r : 0 to I usinga stepsizeof 025 where-l (0) - l Display all your resultson the sameglaph
-:,1r -" r I I )\ r'r'v/ /i;
(a) (b) (c) (d)
Analytically Using Euler'smethodwith /i : 05 and 025 Using the midpointmethodu'ith ft : 05 Usingthe fburth-order methodwith i : 05 RK
(a) (b) (c) (d) (e)
Analytically Using Euler'smethod Using Heun's methodwithout iteration UsineRalston's method Using the fburth-orderRK method
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