Do Dogs Know Bifurcations?

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Do Dog Know Bifurcation? Roland Minton Roanoke College Salem, VA 4153 Timothy J. Penning Hoe College Holland, MI 4943 Elvi burt uon the mathematical cene in May, 003.

1 Do Dog Know Bifurcation? Roland Minton Roanoke College Salem, VA 4153 Timothy J. Penning Hoe College Holland, MI 4943 Elvi burt uon the mathematical cene in May, 003. The econd author article "Do Dog Know Calculu?" [] introduced hi dog Elvi and Elvi ability to olve a claic otimization roblem. Peruchet and Gallego article "Do Dog Know Related Rate Rather Than Otimization?" [4] gave an alternative elanation of how dog (including their own) might olve the roblem. Elvi urriing reudiation of that elanation in [3] inired thi article. Here, we elore Elvi roblem-olving ability when he mut chooe between two qualitatively di erent otion. Such a ituation induce a bifurcation in hi otimal trategy. A a bonu, our analyi reveal a neat geometric roof of the arithmetic mean - geometric mean inequality. In the original roblem, Elvi i on the horeline and want to retrieve a ball thrown meter into the water and z meter downhore, a in Figure 1. Elvi run along the hore at eed r m/ to a oint y meter uhore from the ball, then wim to the ball at eed m/. Figure 1. The original roblem.

2 Author It i hown in [] that the total time to the ball i minimized with ^y = (r=) 1 (1) if z > ^y. Thi olution i remarkable becaue the otimal entry oint i indeendent of the ditance z. Alo, the ditance ^y i a linear function of. In [], Elvi actual entry oint for a large number of throw are reented. The catter lot of thee oint how a remarkable linear trend that cloely matche the line of otimal entry oint. Thu, it eem that Elvi i able to olve thi general roblem. In [4], Perruchet and Gallego tart with the function d(t) giving the ditance between Elvi and the ball. A he run along the horeline, the rate of change d 0 (t) i negative and increaing. The oition at which jd 0 (t)j reache the wim eed (that i, d 0 (t) = ) i hown to be eactly the otimal ^y in (1) above. Thi allow a di erent interretation of how Elvi get to the ball. Intead of ome internal calculation of (how far out in the water the ball i) and then y (where to enter the water), erha Elvi run along the hore until he ene that he could make better rogre to the ball by wimming. That i, intead of olving a global otimization roblem, erha Elvi i olving a local related rate roblem. Fortunately, Elvi ha rovided more clue about hi roblem-olving trategy. A noted in [3], when Elvi tart in the water and a ball i thrown a long ditance arallel to the hore, he rt wim to hore, then run along the hore, and nally wim back out to the ball. Thu, at leat in thi ituation, Elvi i aarently viewing the tak globally. However, hi behavior raie three new quetion. Firt of all, what are the oible otimal ath? Thi quetion i eaily anwered. For any ath, Elvi either reache the hore or tay in the water. If he tay in the water, then wimming directly to the ball will reult in the hortet time. If he reache the hore, then the otimal ath will involve wimming and running in traight line. Thu, the otimal ath will either be a traight wim to the ball (deignated S), or a ath (deignated SRS) coniting of three traight line. If the ball i thrown a hort ditance, S will be fater than SRS. The longer the throw, the more likely that SRS will be the quicker ath. The econd quetion i, what i the bifurcation oint at which the otimal trategy change from S to SRS? And third, doe Elvi change hi trategy at the otimal oint? That i, doe Elvi know bifurcation? The Swim-Run-Swim Problem Given the dicuion above, we comare the time of the S and SRS ath. To

3 Author 3 nd the otimal SRS ath, uoe Elvi tart 1 meter out in the water and race to a ball that i z meter downhore and meter out into the water, a in Figure. He rt wim ahore with eed m/ to a oint y 1 meter downhore, then run along the beach at eed r m/ to a oint y meter uhore from the ball, and nally wim out to the ball. Figure. The SRS roblem. The total time to reach the ball i given by T = 1 + y 1 + z y 1 y r + + y. () If we conider T a a function of y 1 and y, it reache a minimum when both artial derivative are zero. i = y i q i + y i for i = 1;. Setting each artial derivative equal to 0 and eliminating 1 r give y y 1 which are the coine of the angle in Figure 3. = 1 r y + y Figure 3. Equal angle. We conclude that angle in equal angle out! i = 0 for y i give ^y i = i (r=) 1 (3)

4 Author 4 for i=1,. The value for ^y coincide eactly with the olution of the original roblem, if z > ^y 1 + ^y. Uon re ection, thee reult are obviou. Thi i often the cae when uch a nice reult emerge. Think of the roblem in two art. Ste (i) i to go from a oint in the water to a ditant oint on the hore. Ste (ii) i to go from that oint on the hore to another ditant oint in the water. Since the original olution (1) i indeendent of z, te (i) and (ii) are indeendent. In fact, te (ii) i imly the original roblem. Further, the two te are equivalent with te (i) being te (ii) covered in revere. Therefore, the angle mut be equal. Thought of in a di erent way, ince (1) i indeendent of z, chooe z = ^y 1 + ^y o that there i no running at all. Then, analogou to light re ecting o a mirror or a billiard ball bouncing o a rail, the otimal ath ha angle in equal to angle out. Thi ugget a oible elanation of Elvi behavior. Perha Elvi ue a mall et of rule. For eamle, 1. If the ball i cloe, wim directly to it. (Elvi doe thi.). If the ball i far away, then (A) get out of the water and (B) olve the hore to ball roblem. The eerience gained olving (B) can hel in (A), ince angle in equal angle out. The correct angle might "feel" right. Notice that thi leave oen the quetion of how Elvi actually olve (B). Such an elanation i conitent with arti cial life model uch a Craig Reynold boid [5] and with contructal theory [1]. Bifurcation Point The net te i to comare the S and SRS trategie. Subtituting (3) into () give the total time for the otimal SRS ath, which can be written in the form T SRS = z r r= r. The time to wim directly to the ball i given by z T S = + ( 1 ). We want to nd all value of z for which T SRS = T S. Squaring the equation T SRS = T S, uing the quadratic formula to olve for z, and dicarding the etraneou olution give the critical value ~z = r 1. (4) (r=) 1

5 Author 5 For z < ~z, the fatet route i to wim directly to the ball. For z > ~z, the fatet route i the SRS ath found above. The value ~z i called a bifurcation oint, ince the nature of the otimal olution change at thi value. Thi reult lead to ome intereting inight. Firt, there i no bifurcation oint if > r. If wimming i fater than running, then wimming directly to the ball i alway otimal. Second, if < r and r, then ~z i very large. For horter ditance z, the mall advantage that running rovide cannot comenate for having to wim the etra ditance to hore. Finally, a r! 1 in equation (4), ~z! 1. For the hyical roblem with large value of r, the otimal trategy i to wim the hortet ditance oible getting to hore and back to the ball. That i, a r! 1, the hae of the otimal SRS ath will form three ide of a traezoid, a in Figure 4. Figure 4. The SRS ath. The hyical roblem alo hel u determine the length of the to of thi traezoid. A r! 1, the running time along the beach aroache zero, o the total SRS time equal the time to wim 1 + meter. At the bifurcation oint, the S and SRS time are equal, o the S ath mut alo have length 1 + meter, a in Figure 5. Check thi out geometrically! Figure 5. A mean triangle. Figure 5 how u that given two oitive number, 1 and, 1 1 ( 1 + ). Thu, in the roce of analyzing otimal retrieval trategie, we have dicovered a icture roof for the relationhi between the geometric mean and arithmetic mean

6 Author 6 of two number. The gure alo reveal that equality hold only when 1 =. How intereting that by thinking rather intuitively about a roblem in the hyical world, we dicover truth about the mathematical world. In the cae that a ball i thrown arallel to the hore, = 1. Here, the equation T SRS = T S imli e and the bifurcation oint i r= + 1 ~z = r= 1. (5) The Bifurcation Eeriment Since Elvi already revealed hi willingne to bifurcate [3], the remaining quetion i whether he bifurcate at the correct oint. To anwer thi, the econd author and two undergraduate reearch tudent took Elvi to the ame Lake Michigan beach where the rt eeriment [] wa done. Taking the average of everal timed trial, Elvi running eed wa etimated to be 6.39 m/ and hi wimming eed to be 0.73 m/. However, once Elvi actually tarted chaing the ball, hi running eed lowed down coniderably to an average of 3.0 m/. (Thi reduced eed wa likely a combination of being tired from wimming to hore and jut enjoying a lazy July afternoon.) Uing equation (5), we ee that the otimal bifurcation oint i then ~z = :56. The econd author tood 4 meter out in the water with Elvi and threw a ball variou ditance, but landing about 4 meter from the hore. One tudent meaured the ditance of the throw and the other recorded Elvi choice. The reult are in Table 1. Trial number z (m) Strategy 1 16:5 SRS 9:5 S 3 14: S 4 15:1 SRS 5 15 SRS 6 1 S 7 7:8 S 8 11:6 S 9 18: SRS Table 1. The data ugget everal concluion about Elvi choice. Firt, it eem that

7 Author 7 there i a bifurcation. He conitently alie the SRS trategy for longer ditance and wim directly to the ball for horter one. Secondly, there may or may not be a well-de ned bifurcation oint. If there i, it eit omewhere between 14 m and 15 m for thi eamle, but without doing many more trial to narrow down the oint and to how conitency, one cannot be ure. Latly, if there i a wellde ned bifurcation oint, it i not where it hould be. According to equation (5), the bifurcation hould be at 10.4 m. Elvi bifurcation ditance wa (a diaointing) 4 m larger. Thu it might be concluded that Elvi know bifurcation qualitatively, but not quantitatively. Let we be too hard on Elvi, though, it hould be remembered that dog, like all of u, learn from eerience and that bifurcation by their very nature make learning di cult. Bifurcation force a choice uon u, and we often do not have the oortunity to go back and try both otion. But, in forcing the choice, bifurcation add interet to mathematic and richne to life. A the oet Robert Frot wrote, "Two road diverged in a wood, and I - I took the one le traveled by, And that ha made all the di erence." Reference [1] Adrian Bejan and Gil Merk, ed., Contructal Theory of Social Dynamic, Sringer, 007. [] Timothy J. Penning, Do dog know calculu?, College Math. J. 34 (003) [3] Timothy J. Penning, Reone, College Math. J. 37 (006) 19. [4] Pierre Perruchet and Jorge Gallego, Do dog know related rate rather than otimization?, College Math. J. 37 (006) [5] Craig Reynold, Flock, herd and chool: a ditributed behavioral model, Comuter Grahic: Proceeding of SIGGRAPH 87 (1987) 5-34.

Figure 1 Siemens PSSE Web Site

Figure 1 Siemens PSSE Web Site Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of