Dynamical Systems - Columbia University

1 DynamicalSystemsJoshuaWilde,revisedbyIsa b elTecu,TakeshiSuzukiandMar aJos Bo ccardiAugust13,2013 DynamicalSystemsaresystems,describ edbyoneormoreequations, ,thegrowthofap opulationcanb edescrib eundersto o dtob eeitherdiscrete(day1,day2etc.)orcontinuo us( ).Ifwetaketimetob econtinuous,dynamicalsystemswillb edescrib edbydi erentialequations-equationsthatinvolveth ederivative(theinstantaneouschange) ediscrete,dynamicalsystemswillb edescrib edbydi erenceequations-equationsrelatingthevalu eofavariableattimet+ okatb othcasesb erentialEquationsAdi erentialequationisanequationwhichinvolve sanunknownfunctionf(t) (t).Thenwedenotef (t)asdfdt(t)oras y(t).

2 Ageneraldi erentialequationisthenoftheform y=F(y(t),t)Thepurp oseofthisequationisnottosolveforthevaria blet,butrathertosolveforthefunctiony(t). (Sincethefunctionistheunknown,weuseyrath erthanftolab elit).InEconomics,di ,thechangeinthecapitalsto ckattimet, K(t),isafunctionofthecurrentcapitalsto ckK(t),thesavingrates,andthedepreciation rate : K(t) = (s )K(t)Thereforeitiscommontousetinsteadofx astheargumentoffandreferto for esofDi rstderivativef (t) : y=kyisa rstorderdi erenceequationsinceitonlyinvolvesonederi vativeoff(t). (n)(t) : y=kyisasecondorderdi erentialequationsinceitinvolvestwoderiva tivesoff(t).Alsoy(n)=kyisannthorderdi erentialequationisadi ,thedi erentialequationsmentionedthusfarhaveall b ,theequationy(n)=ky(x,t) tisnotordinarysincetherearetwoargumentsx andt,whiley(n)=ky(t) tisordinary(theonlyvariableist).

3 Allthedi erentialequationswewilllo okatwillb erentialequationislinearifitcanb ewrittenintheformy(n)+an 1(t)y(n 1)+ +a1(t)y=a0(t) erentialequationisonewheretheonlyo y=kyisautonomous(tentersonlythroughy(t)) while y=ky tisnot(tentersbyitself,outsideofy(t)). erentialEquationsDi erentialequationsaregenerallydi ,inthissectionofthecoursewewillexamineon ly rstorderlineardi erenceequations: y+p(t)y=q(t). ecialCasesNoticethatifp(t) = 0,thenwehaveasimpleintegrationproblemwea llknowhowtosolve y=q(t) y=Q(t) +C,whereQ(t)istheantiderivativeofq(t).No wletq(t) = 0,andp(t) = k R, y=ky dydt=ky(t).Rewritetheequationbydividingb othsidesbyy(t).(Weareassumingthaty(t)6= (t) = 0isobviouslyasolutionaswell.)

4 Thenwehave yy= othsideswithresp ecttotwehaveln(y) =kt+C y=ekt+C= ekt,where = (t)tob esomefunctionbutleaveq(t) = 0wecanfollowthesamestepsasintheexamplewi thp(t) = kab oveandgetln(y) =H(t) +C y=eH(t)+C= eH(t),whereH(t) = p(t)dtand = y+p(t)y=q(t), p(t)6= 0, q(t)6= neafunctionH(t)suchthatH(t) = p(t) ovedi erentialequationtoget yeH(t)+p(t)yeH(t)=q(t)eH(t)Noticethatbyt hechainrule,thatddt(yeH(t))= yeH(t)+p(t)yeH(t)whichisequaltotheleftha ndsideofthedi (yeH(t))=q(t)eH(t).Wenowintegrateb othsidestogetyeH(t)= q(t)eH(t)dt+C,andthenmultiplythroughbye H(t)togetthegeneralformofthesolution:y=e H(t){ q(t)eH(t)dt+C}.ExampleFindthesolutionoft heequation y=ay+ (t) = aandq(t) = (t) = adt= at+ ,wehavethaty=eat C{ be at+Cdt+D}y=eat C{ bae at+C+D}y= ba+Deat+Cy= ba+ eat,where = (0) = y= 0,y=.

5 Therefore,thegeneralsolutiontotheequatio nisy= (T) = y= ,y= ekT =yTe kTTherefore,thegeneralsolutiontotheequat ionisy=yTe kTekt=yTek(t T). (t) ciscalledasteadystateforadi ,considertheequationforthecapitalsto ckgivenab ove K(t) = (s )K(t)Weareeithersavingmorethatthedepreci ationofthecapitalsto ck,lessthatthevalueofthede-preciationoft hecapitalsto ck, ,thenwewillhavethesameamountofcapitalnex tp erio daswedothisp erio , ckwillremainthesameasthisyearinthethirdy ear, erentialequation y= ethesamethisyearandlastyear,wemusthaveth atydo esnotchange,or y= ,theonlyvalueofyforwhichthiscanhapp en(aslongasa6= 0)isy= 0,andsoy= ,one ndsthesteadystatestothedi erentialequation y=F(y,t)bysetting y=F(y,t)

6 = y=b+ y= b,andthesteadystatevalueofthesolutionisy = erentialequationandtheb erentialequation y(t) =F(y(t))withthevalueofthefunctionyontheh orizontalaxisandthechangeiny, y,ontheverticalaxis.(Notethatwecannotdra waone-dimensionalphasediagramforanon-aut onomousdi erentialequationssinceinthatcase ychangeswithy(t)andwitht.)Anyp ointatwhichthegraphintersectsthehorizont alaxis,thatis,atwhich y= 0, ointatwhichthegraphof y=F(y)isab ovethehorizontalaxis, yisp ointatwhichthegraphof y=F(y)isb elowthehorizontalaxis, ehaviorofy:Thearrowswillp ointtotherightonanysegmentonwhichthegrap hisab ovetheaxisandtotheleftonanysegmentthatis b ,letusconsiderthesimpledi erentialequation y= :Caseone:a > , >0,thenwheny >0, y > <0,then y <0, ,wecanseethatiftheequationstartsatanyp ointotherthany0= 0,thesystemwilldivergetonegativein nityorp ositivein ,y=y0eat,andlett.

7 Ify0>0,theny ,andify0<0,theny .Casetwo:a < , ,theonlysteadystateisaty= <0,thenwheny >0, y < <0,then y >0, ,wecanseethatiftheequationstartsatanyp oint,itwilleventuallyconvergetoy= ,y=y0eat,andlett .Ify0>0,theny 0,andify0<0,theny 0also(rememb er,a <0). inthedomainasymptoticallystableif r >0andB(y ,r) domainsuchthatifwehaveasaninitialp ointanyy B(y ,r),thesystemwillconvergetoy ,thesystemwasstablewhena <0,butunstablewhena > :Ifthearrowsp ointtowardsthesteadystatefromb othsides, :Let y=F(y).IfdF(y)dy|y <0,thenthesteadystatey isstableifdF(y)dy|y >0,thenthesteadystatey isunstableExampleLet y=y2 4y+ :Let y= =y2 4y+ 3 = (y 1)(y 3).

8 Therefore,wehavetwosteadystates,y= 1andy= ,takethederivativeof y=y2 4y+ 3withresp ecttoytogetd ydy= 2y 1,wehaved ydy|y = 2(1) 4 = 2<0,thereforey= 3,wehaved ydy|y = 2(3) 4 = 2>0,andwehavethaty= erentialEquationsConsiderthegeneraltwo-e quationsystemofdi erentialequations: x(t) =F(x(t),y(t),t) y(t) =G(x(t),y(t),t)Theselo okliketwosingledi erentialequations,buttheproblemisthatyap p earsintheequationfor xandxapp earsintheequationfor efore,wecan ndthesteadystateofthesystembysettingb oth x= 0and y= x=ex 1 1and y= oththeseequationsequalto0yields x= 0 ex 1= 1 x= 1 y= 0 ye= 0 y= x=x+ 2yand y=x2+ oththeseequationsequalto0yields x= 0 x= 2y y= 0 y= x2 x= 2( x2) x(1 2x) = 0 x={0,12} y={0, 14}Therefore,thetwosteadystatesare(x,y) ={(0,0),(12, 14)}.

9 X=e1 x 1and y= (2 y) oththeseequationsequalto0yields x= 0 e1 x= 1 x= 1 y= 0 (2 y)e= 0 y= 2 MathCamp7 PhaseDiagramsIfwehaveatwodimensionalsyst emasintheexampleab ove,wecandrawphasediagramwithxononeaxisa ndyontheother(unlikeintheonedimensionalc ase,wenolongerplot xor yonanaxis).Wethendrawinthecurveinx,y-spa cealongwhich x= 0andthecurvealongwhich y= ehaviorofthesystem, ndthesignsof xand yineachofthesegmentsoftheplanedividedbyt he x= 0and y= ,if x >0and y <0, erentialequation y=f(y),wecouldtestwhetherthesteadystatew asstablebycheckingwhetherdf(y)dy y < ,thenthedi erentialequationsismorecomplicated, etheJacobianmatrixofthesystemofdi : y isstableifandonlyifalleigenvaluesofJ(y )arenegativeorhavenegativerealparts.

10 Y isunstableifandonlyifsomeeigenvalueofJ(y )isp ositiveorhasp hassomepureimaginaryorzeroeigenvaluesand nop ositiveeigenvalues,thenwecannotdetermine thestabilityofthesteadystatebylo x=ex 1 1and y= ez= (x,y) = (1,0).TheJacobianofthesystemis(ex 10yexex)(z) =(1 00e), othofthesearep ositive, x=x+ 2yand y=x2+ (x,y) ={(0,0),(12, 14)}.TheJacobianofthesystemis(122x1).Whe nz= (0,0),thenwehavetheJacobian(1 20 1)whichimpliesthattherep othofthesearep ositive,(0,0) (12, 14),thenwehavetheJacobian(1 21 1)8 DynamicalSystemsandsolvingfortheeigenval uesyields = 1 ositive,(12, x=e1 x 1and y= (2 y) ez= (x,y) = (1,2).TheJacobianofthesystemis( e1 x0(2 y)ex ex)(z) =( 100 e),whichimpliesthattheeigenvaluesofthesy stemare 1and othofthesearenegative, erentialEquationsConsiderthelinearsystem ofdi erentialequations: x=a11x+a12y y=a21x+a22ywhichcanb eexpressedas x=Ax,wherex= (x,y) , x= ( x, y) ,andA=(a11a12a21a22).)