1 Differential Equation Investigations using Customizable

Transcription

1 Differenial Equaion Invesigaions using Cusomizable Mahles Rober Decker The Universiy of Harford Absrac. The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for invesigaing he graphs of firs and second order differenial equaions, as well as he graphs of funcions, parameric curves and daa poins. Wih hese programs one can dynamically change parameers or iniial condiions, and see he resuls immediaely in muliple views (phase or ime plos). This paper focuses on aciviies ha can be used in a firs course in differenial equaions. Mahles which can be used o invesigae mass-spring sysems, pendulum sysems, and populaion growh models are presened. Conceps from differenial equaions which are addressed include linear and nonlinear beas, bifurcaion via he race-deerminan plane, and he Poincare map for periodic firs-order equaions.. Inroducion: The mass-spring sysem The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for invesigaing he graphs of firs and second order differenial equaions, as well as he graphs of funcions, parameric curves and daa poins. Wih hese programs one can dynamically change parameers or iniial condiions, and see he resuls immediaely in muliple views simulaneously (phase and ime plos). If a compuer algebra sysem is available, for eample Maple, he resuls can hen be impored ino Maple for furher refinemen here, and for repor wriing. When a mahle iniially opens up i is argeed oward he invesigaion of a paricular opic in a firs course in differenial equaions. However, because he programs are cusomizable, hey can hen be changed o invesigae a relaed (or unrelaed) opic, and he changes can be saved. For eample, one apple has been designed by he auhor o sudy he sandard damped mass-spring differenial equaion m + c + k =. () Second-order equaions mus be enered as firs-order sysems, so his equaion is enered in he form d/d = y, dy/d = c m y k m. The mahle opens up wih a phase plo and wo ime plos (see Figure ). One can se muliple iniial condiions by double clicking (or pressing he Keep IC buon) in any of he hree views. Clicking and dragging allows he user o immediaely see he changes in he soluion curve as he iniial condiion is varied. Any of he hree parameers m, c, k can be adjused by yping in new values or by using he slider; changes are seen immediaely and smoohly as in an animaion. Thus one can observe he ransiion from no damping, o underdamping, o overdamping dynamically. One can race all hree curves simulaneously (similar o a graphing calculaor, bu wih hree views) o see how he hree views are relaed. Righ-clicking and dragging a recangle creaes a zoom bo.

2 Figure : Mass-spring sysem m + c + k = In order change he apple o one ha represens a mass-spring sysem wih a driving force m + c + k = a sin(w) () one simply adds he erm +a sin(w) o he second differenial equaion o ge he sysem d/d = y, dy/d = c m y k m + a sin(w). See Figure. To invesigae he phenomena of beas and resonance, i is necessary o adjus some of he seings in he funcion window (he value of ma is changed o, and he and y iniial condiions are se o and ) and he range seings in he graph windows (from and o and for and y, and from o for ). The damping parameer c mus be se equal o. Now when he parameer w is slowly varied beween abou.5 and.5, one sees firs he emergence of beas (a around w =.7), hen one observes he beas geing longer and larger, evenually urning ino resonance a w = (and back again ino beas as w increases beyond ). When sudens sar o play around wih ineracive sofware, and see hings like he emergence of beas, hey ofen sar o ask quesions. They may ask why beas firs appear a abou w =.7; his is a good opic for undergraduae invesigaion/research (for a discussion of how o calculae when beas emerge, see [3]).

3 Figure : Driven mass-spring sysem m + c + k = a sin(w). Follow-up: The driven pendulum An ecellen follow-up projec (or demonsraion by he eacher) is o invesigae similar behavior for a damped, rigid pendulum m + c + k sin() = a sin(w) (3) (for a pendulum he parameers m, c, k do no have precisely he same inerpreaion as for a mass-spring sysem, bu o keep he analogy ransparen we keep he same parameer names). The equaion sysem now becomes = y, y = c m y k m sin + a sin(w) (one has only o change he second equaion for he sysem in Figure ). By choosing a = one has an apple for a free pendulum, and for a we ge a driven pendulum. One ineresing aciviy is o coninuously vary (using a slider) he forcing frequency w of a driven, undamped pendulum, given by y + sin(y) = a sin(w), and observe he resul in hree views a once (phase plo and ime plos). This ime he resuls are a bi more comple han for he mass-spring sysem. Wheher or no beas are observed as w varies is dependen on he value of he ampliude a of he forcing erm. For a =.5 we do see beas emerge, bu somehing new happens 3

4 also. For a mass-spring sysem, he beas have he same shape for w less han he resonan value of w = as hey do for w greaer han w =. For he pendulum, however, he beas for w less han he resonan value are more rounded han hose of he corresponding mass-spring sysem, and for w greaer han he resonan value hey are nearly diamond shaped. See Figure 3 (Figure 3 was creaed using he Copy Maple commands feaure of he mahle, and hen pasing he commands ino Maple). By resonan value in his case a=.5, w=.98 a=.5, w=.9 a=.5, w= a=.5, w= a=.5, w= a=.5, w= Figure 3: Beas in he pendulum equaion + sin() = a sin(w) we mean he value of w ha resuls in a maimal bea period; his value is no longer w = bu somewhere around w =.9. For a slighly larger value of a, chaos breaks loose. Choosing a =.5, and again varying w, we see he emergence of beas, wih a ransiion o chaoic behavior, and a reurn o beas. The shape of he beas is similar o when a =.5. See Figure 3. Finally, if we move a up o a =., we find no beas a all in he range.5 < w <.5 (primarily chaos). While all of his behavior can be observed wih any sofware capable of graphing differenial equaions, when one is using an ineracive/dynamic ool, hese behaviors leap ou a he user as she/he plays wih he sofware. I is fair o say ha he auhor would no have noiced mos of wha is described above wihou such a ool. Also, he sae of he mahle can be saved, and reurned o laer. Thus wha sars ou as a single mahle, can be cusomized (by he eacher, or by he suden) o several ools for invesigaing muliple phenomena..3 Bifurcaion in he undriven pendulum Anoher aciviy is o coninuously vary he amoun of damping for a free pendulum my + cy + k sin(y) =, and ry o idenify bifurcaion values. For his we use a differen mahle; in addiion o a phase plo of he differenial equaion, here is a race-deerminan plane graph (for deails of he race-deerminan plane see []). The fied poins are ploed in he phase plane, and he corresponding sabiliy of each fied poin is ploed in he racedeerminan plane (he green poin in he race-deerminan plane gives he sabiliy of he green poin in he phase plane, and similarly for he yellow poins). See Figure 4. 4

5 Afer an iniial eploraion of various iniial condiions, wih a fied value of he damping coefficien, a complee phase porrai can be generaed. Then he damping coefficien can be increased coninuously; one observes he approimae poin a which he sable fied poins change from spiral sinks o sinks. In he window ha represen he phase porrai of he differenial equaion, one sees he soluion curve sop spiraling/oscillaing; a he same ime one sees in he race-deerminan plane ha he yellow poin is crossing he parabola deerminan = 4 race (ha is, he race-deerminan poin is crossing ino he sink region from he spiral sink region). This is he siuaion show in Figure 4. Figure 4: Phase porrai and race-deerminan for he pendulum equaion m + c + k sin() =.4 Populaion growh and he Poincare map The las aciviy shows how a relaively advanced concep, he Poincare map, can be made accessible o undergraduaes hrough he use of mahles. A model which is ofen invesigaed in a firs course in differenial equaions is he logisic populaion growh equaion = a( /b). Here, is he populaion size, a represens he growh rae when he populaion is small, and b represens he maimum susainable populaion (and hence he 5

6 limi of he populaion as ). If we apply his model o a populaion of fish, hen o model he effecs of fishing we could add a erm o represen he number of fish, c, removed per uni ime (say, per year). The new equaion would be = a( /b) c (4) Finally, if we wan o represen seasonal fishing, we could muliply c by a seasonaliy erm cos(π) (minimum fishing when = or =, and maimum fishing when = /). The new equaion is = a( /b) c( cos(π)). (5) Equaion 4 can be sudied by sudens using he ools of fied poins and sabiliy analysis; since i is quadraic in, here are eiher,, or fied poin soluions. A mahle designed o sudy his equaion is shown in Figure 5. In addiion o he window Figure 5: Poincare map for for = a( /b) c a he bifurcaion poin which conains he graph of he differenial equaion, here is a window which shows a rough Poincare map. The Poincare map akes an iniial value from a differenial equaion = ( ) o is value ( + period ), period unis laer, where for Equaion 4, period is equal o. Thus, if an iniial poin ges mapped o he same poin a =, we 6

7 know ha = is a consan soluion (fied poin). Hence whenever he Poincare map crosses he line y =, he differenial equaion has a fied poin. Now i is possible o esimae he value of c for which here is eacly one fied poin. This is he bifurcaion value of c for which he number of fied poins changes from o. One simply uses he slider o change he value of c unil he Poincare graph becomes angen o he line y = (as seen in Figure 5). Equaion 5 does no have any fied poins, as i is nonauonomous. I is, however, periodic in wih period. In such a siuaion, he Poincare map can be used o look for periodic soluions, raher han fied poins. As wih fied poins, he inersecions of he Poincare map wih he line y = represen he eisence of periodic soluions. Such soluions are hard o find wihou he use of he Poincare map, bu afer coming o undersand how he Poincare map picks ou he fied poins of an auonomous equaion, i is concepually fairly easy o see how i can be used o pick ou he periodic soluions of a periodic differenial equaion. The search for periodic soluions o various polynomial and polynomial-like differenial equaions wih periodic coefficiens is an acive area of research (see []). Wih he use of ineracive/dynamic sofware, undergraduaes gain access o a opic which is boh closely relaed o sandard undergraduae opics, and o curren mahemaical research. [] Benardee, D., V. Noonburg and B. Pollina. Periodic Soluions and Bifurcaions of a Periodically Harvesed Logisic Equaion. Amer. Mah. Monhly, acceped, 6. [] Blanchard, P., R. Devaney, and G. Hall.. Differenial Equaions, e. Brooks/Cole. [3] Decker, R., and V. Noonburg. In preparaion. Differenial Equaions for Scieniss and Engineers. 7