MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes. Roughly speaking, the paametes ae the unknowns in a system which ae not the state vaiables themselves. In applied poblems, they ae commonly associated with concete physical quantities fo example, the volume of a mixing tank, the length of a pendulum, the stiffness of a sping, the ate of a chemical eaction, etc. A key featue is that, while these values may not be pecisely known, thoughout the diffeential equation they ae assume to be fixed values. Fo example, econside ou logistic gowth model dp dt = P ( 1 P K ), P (0) = P 0 (1) whee P (t) is the population at time t, is the gowth ate, and K is the system s caying capacity. Fo this example, and K ae paametes. Depending on the specific modeling context (e.g. ae we modeling a colony of bacteia of a population of abbits?), the values of and K ae obviously going to be diffeent, but in each case we conside them to be fixed values (i.e. bacteia do not suddenly become abbits!). We saw last week, howeve, that this model can be solved egadless of the values we choose to get P (t) = KP 0 e t. (2) K P 0 + P 0 et This is petty emakable, but it was a pain to cay the paametes and K thoughout evey step of this deivation. We should wonde if thee is a simple way and, of couse, thee is. Conside the following agument. Instead of solving fo the the vaiables P and t, conside the new vaiables P = P, and τ = t. K 1
We want to ewite the oiginal DE (1) in tems of the new vaiables P and τ. Afte applying the chain ule, we see that we have d P dτ = d P dp dp = 1 K dt dt dτ [ P = P (1 P ). ( 1 P K Ou new diffeential equation in P and τ is )] 1 d P dτ = P (1 P ), P (0) = P0. (3) Remakably, the system (3) does not depend on the oiginal paametes and K! While the solution method is the same as fo the oiginal system (1), we no longe have to cay the paamete values though the pocess. Instead, we aive diectly at the solution P 0 e P τ (τ) = 1 P 0 + P 0 e. (4) τ It can be easily checked by substituting P (t) = P (t)/k, P0 = P 0 /K, and τ = t that educed solution (4) coesponds the ealie full solution (2). Thee is anothe subtle point to make about this pocess. Not only has this vaiable change simplified the amount of algeba we had to do, but it has also told us something fundamental about the behavio of the system: namely, it does not depend on and K. This is a diect consequence of (3) and consequently (4) not depending on and K. We should wonde whethe all systems can be manipulated in this way. That is, can we always emove the paamete-dependence though a vaiable substitution? It should take only a moment of thought to ealize this is an unealistic expectation in the physical wold, how things behave does depend on the undelying physical eality. A pojectile might escape the gavitation pull of the moon, but fall back down if fied fom Eath. The foce of gavity is diffeent in the two envionments, and the long-tem behavio of ou system eflects this. In fact, this tipping point between one type of behavio and anothe is the single lagest eseach aea within the study of dynamical systems. The applications ae fa-eaching, spanning the natual sciences, financial and economic foecasting, and engineeing. This aea is commonly called bifucation theoy, which we investigate now. 2
2 Motivating Example: Logistic plus Havesting Let s econside the logistic population gowth model with a slight modification. Instead of assuming the population gows in isolation (with intecompetition), let s assume that thee is an extenal facto which culls the population at a fixed ate. Fo example, imagine logges cleaing a foest who must meet a cetain quota, o fishemen who must catch a cetain numbe of fish to meet thei demands. The new model is ( dp dt = P 1 P ) p (5) K whee P (t) is the population at time t, is the gowth ate, K is the caying capacity, and p > 0 is the new havesting paamete. We now conside the esulting dynamical behavio. Unlike pevious systems, we will not compute the analytic solution (although it can be done with a little wok). Fo the logistic model without havesting, we saw that the qualitative behavio of the system did not depend upon the paametes and K. In paticula, the numbe of fixed points and thei stabilities emained the same and all solutions, egadless of (positive) initial condition P (0) = P 0 appoached the caying capacity K in the limit as time went to infinity. In ode to investigate whethe some simila paamete-fee dynamics is tue fo the logistic plus havesting model (5), we apply the vaiable tansfomations In this case, we have P = P, and τ = t. K d P dτ = d P dp dp = 1 K dt dt dτ [ P ( 1 P K = P (1 P ) p K. ) ] p 1 While we ae disappointed to see that we have been unsuccessful in eliminating all of the paametes fom the model, we should be happy to note we no longe need thee paametes to descibe the dynamics of the model. The model educes to d P dτ = P (1 P ) K, P (0) = P0 (6) 3
which depends only on the single combined paamete K = p/(k). We now conside what happens to the solutions of (6) as we change the value of K. We stat by detemining the fixed points as a function of K. We have d P dτ = 0 = P ( 1 P ) K = 0 = P 2 P + K = 0 = P = 1 ± 1 4 K. 2 We can see that, unlike the model with no havesting, the numbe of fixed points depends on the value of 1 4 K. We can intepet this fo both the educed model (6) and the oiginal model (5) as follows: 1. Two fixed points if K < 1 4 (equivalently, 0 < p < K 4 ) 2. One fixed point if K = 1 4 (equivalently, p = K 4 ) 3. No fixed points if K > 1 4 (equivalently, p > K 4 ) Clealy the points K = 1 4 and p = K 4 (fo the educed (6) and full (5) models, espectively) ae vey impotant points! Depending which side of this value we happen to be one, we may have eithe two o zeo fixed points. This value is called a bifucation value. To see how cossing this bifucation point affects the long-tem behavio of the system, we conside the diagam given in Figue 1. We can see that thee ae essentially fou cases: 1. If K = 0 (i.e. p = 0), the fixed point P = 0 is unstable and the fixed point P = 1 is stable. (This is the logistic model without havesting.) 2. If 0 < K < 1 4 (i.e. 0 < p < K 4 ), thee ae two stictly positive fixed points. The fist is unstable while the second is stable. 3. If K = 1 4 (i.e. p = K 4 ), thee is a single semi-stable fixed point, which is unstable to the left and stable to the ight. 4. If K > 1 4 (i.e. p > K 4 ), thee ae no fixed points and all solutions decease fo all time. All this talk of fixed points and stability, howeve, has a way of clouding what is eally going on. Let s etun to ou oiginal deivation of the system as a population gowth model with constant havesting. We make the following notes: 4
f(p) P zeo Reduced ~ K = 0 Full p = 0 zeo ~ 0 < K < 1 4 K 0 < p < 4 zeo K ~ = 1 4 p = K 4 zeo ~ K > 1 4 K p > 4 Figue 1: The stability diagam of the logistic model with an additional havesting tem p > 0. Fo p = 0 we have the logistic model. Fo 0 < p < K 4 we still have two fixed points, but a lowe effective caying capacity (the stable fixed point has moved to the left) and a egion nea zeo whee the flow is negative. Fo p = K 4 we have only a single semi-stable fixed point and a lage egion which leads to extinction. Fo p > K 4 we have no fixed points (i.e. all tajectoies eventually lead to extinction). Modeate havesting: Fo intemediate values of K (o p) we notice two things. The fist is that populations nea zeo decay. This makes sense since small populations may not be able to gow fast enough to ovecome the effects of havesting. The second thing to note is that the stable long-tem population level is less than the caying capacity in the unhavested model. This makes obvious sense since the havesting acts to cull the population. Ove-havesting: Once we supass the bifucation value, it no longe mattes what the initial population size is the havesting will always dive the population to zeo. This is a big deal! If we ae a logging company, we depend on the foest fo ou livelihood, but if we havest too agessively, the foest will be unable to eplenish itself at a sustainable level. Notice also that the outcome dops off discontinuously. At the bifucation value, the long-tem behavio of a lage system is well away fom zeo, but if we incease it even a little bit, the eventual outcome is extinction. Logging companies should invest good money in detemining how much havesting is too much! 5
What happens at zeo? We might wonde what happens at zeo since the model does not have a fixed point thee fo K > 0. In fact, if we ignoe physical eality, the model allows solutions to become negative. This case can be imagined as the logging company clea-cutting a foest and then continuing to havest tees. They still have thei demands to meet and so would still havest at the ate p, if they could, but obviously they cannot. We theefoe cease consideation of the model when P = 0. 3 One-Dimensional Bifucations In the logistic plus havesting model (5) we saw that the system undewest a change in the numbe of fixed points as a esult a change in the havesting paamete p. We say that a system undegoes a bifucation when it passes though one of these cases into anothe and that the point whee it changes is the bifucation value (in this case, p = K/4). The logistic plus havesting model (5) exhibits a special kind of bifucation known as a saddle-node bifucation. In fact, howeve, thee ae moe qualitatively distinct types of bifucations which a one-dimensional system can undego. In paticula, it is possible fo not only the numbe of fixed points to change but also thei stabilities! We biefly now intoduce the canonical foms fo the aleady studied saddle-node bifucation, and also the tanscitical and pitchfok bifucations. A key featue of bifucation analysis is the constuction of a bifucation diagam which we will illustate though the canonical examples. 3.1 Saddle-Node Bifucation The canonical fom fo a system with a saddle-node bifucation is dx dt = x2 (7) whee is a fee paamete. We can quickly analyze the numbe and stability of fixed points by constucting the vecto field diagam (factoing x 2 = ( x)( x) whee possible). We have the following cases: 1. If < 0, thee ae no fixed points and all tajectoies flow left. 2. If = 0, thee a single semi-stable fixed point at x = 0. Again, all tajectoies flow left. 6
3. If > 0, thee ae two fixed points, x = and x = the fist of which is unstable while the second is stable. This should seem familia: it was exactly the set of cases we deived when we wee consideing the logistic plus havesting model. We would like to captue this infomation gaphical and it tuns out that, fo one-dimensional systems, thee is a vey convenient way to do this. The pictue we want is that of a bifucation diagam, which is constucted as follows: 1. Set up an gid whee the hoizontal axis coesponds to the bifucation paamete (in this case, ) and the vetical axis coesponds to the state vaiable (in this case, x). 2. Daw a solid line though the cuve of stable fixed points as they depend upon (in this case, the cuve is x = ). 3. Daw a dotted line though the cuve of unstable fixed points as they depend upon (in this case, the cuve in x = ). If we ae in doubt about steps 2. and 3., we can pick a few values of and connect the stable and unstable fixed points to one anothe. Fo the canonical system (7) we have the bifucation diagam given by Figue 2. Bifucation Diagam Vecto Field x < 0 = 0 > 0 Figue 2: Bifucation diagam fo the system (7). The solid line coesponds to the cuve of stable fixed points x = while the dotted line coesponds to the cuve of unstable fixed points x =. 7
3.2 Tanscitical Bifucation The canonical fom fo a system with a tanscitical bifucation is dx dt = x x2. (8) We pefom the same analysis as fo the system (7), whee we facto x x 2 = x( x). We quickly obtain the following thee cases: 1. If < 0 then thee ae two fixed points with x = < 0 being unstable and x = 0 being stable. 2. If = 0 then thee is a single semi-stable fixed point at x = 0. All tajectoies flow left. 3. If > 0 then thee ae two fixed points with x = > 0 being stable and x = 0 being unstable. What we notice with this example is that, while the numbe of fixed points does not change on eithe side of the bifucation value = 0, the stability of the fixed points does. The cuve of fixed points coesponding to x = goes fom unstable to stable, while the (tivial) cuve x = 0 goes fom being stable to unstable. The bifucation diagam fo this tanscitical bifucation is given by Figue 3. Bifucation Diagam Vecto Field x < 0 = 0 > 0 Figue 3: Bifucation diagam fo the system (8). The lines of equilibia x = 0 and x = exchange stabilities at the tanscitical bifucation value = 0. 8
3.3 Pitchfok Bifucation The canonical fom fo a system with a pitchfok bifucation is dx dt = x x3. (9) We pefom the same analysis as befoe. Fo this system thee ae only two cases: 1. If 0 then thee is a single stable fixed point x = 0. 2. If > 0 thee ae two stable fixed points, x = and x =, and an unstable fixed point at x = 0. In a sense, pitchfok bifucations ae like a saddle-node and tanscitical bifucation in one. The lines of fixed points x = 0 undegoes a stability switch fom stable to unstable at the same time as two new lines of fixed points ae ceated. The bifucation diagam fo the system (9) is given in Figue 4. Bifucation Diagam Vecto Field x < 0 > 0 Figue 4: Bifucation diagam fo the system (9). The line of fixed points x = 0 undegoes a stability switch as two new lines of stable fixed points (x = and x = ) ae bithed at the pitchfok bifucation value = 0 Pitchfok bifucations may futhe be divided into two cases: supecitical and subcitical. In supecitical pitchfok bifucations, a stable banch of fixed points loses stability as the new banches appea, while fo subcitical pitchfok bifucations, an unstable banch gains stability at the banching event. The distinction between these two cases will not facto significantly in ou analysis, but it is a vey impotant featue in applications. 9
4 Examples Example 1: Daw the bifucation diagam fo the following system with > 0, identifying and labeling all bifucation values: dx dt = x ex + 1. (10) Solution: Although it may not seem like it yet, thee is actually a subtle diffeence between this system and the pevious few. To see this, conside detemining the fixed points. We want to solve x e x + 1 = 0 (11) fo vaious values of R. It should not take much convincing to ealize that we ae not going to be able to aive at any simple fom x = f() by simply manipulating the expession. Ou method fo this system will be geometic athe than algebaic. We may not know how to solve the expession, but we know what the individual functions x, e x, and 1 look like. So we instead daw the pictue. In this case, we will beak the expession into two pats, f(x) = x and g(x) = e x 1, and ecognize that the solutions of (11) coespond to the intesection points of these two cuves (see Figue 5). 0 < < 1 = 1 > 1 * * * * * Blue: f(x) = x Geen: g(x) = e x - 1 Red: fixed points Figue 5: Compaison of the functions f(x) = x and g(x) = e x 1 fo vaious values of. The intesection points coespond to the fixed points of the system (10). The stabilities ae detemined by consideing which function is highe/lowe than the othe, which detemines the diection of the vecto field. We can easily convince ouselves that thee ae thee cases: 10
1. If 0 < < 1, the cuves meet twice, once fo x < 0 and once at x = 0. Futhemoe, we can see which of the two cuves is lage so that we can detemine that the fixed point with x < 0 is unstable and x = 0 is stable. (Fo instance, f(x) = x < e x 1 = g(x) fo x > 0 so that x e x + 1 < 0.) 2. If = 1, the cuves meet tangentially at the point x = 0 and nowhee else. This fixed point is semi-stable and all tajectoies move to the left. 3. If > 1, the cuves meet twice, once fo x = 0 and once fo x > 0, with x = 0 being unstable and x > 0 stable. Even though we ae not able to detemine explicit foms fo the cuves of stable and unstable fixed points, we can still identify that thee is a tanscitical bifucation at = 1. The bifucation diagam is given by Figue 6). Bifucation Diagam Vecto Field x 0 < < 1 = 1 > 1 Figue 6: Bifucation diagam fo the system (10). Even though we cannot explicitly identify the cuves of fixed points, we can captue thei qualitative flavo and identify the tanscitical bifucation at = 1. Example 2: Daw the bifucation diagam fo the following system ove the ange > 1 2 : dx = x sin(x). (12) dt Solution: As in the pevious example, we appeal to geometic intuition athe than attempting to detemine an analytic fom fo the steady state cuve. We beak the poblem into the functions f(x) = x and g(x) = 11
sin(x) and ecognize that the intesection points between these two cuves coespond to the fixed points of (12). In Figue 7 we can see that thee ae thee cases (although the final two do not make a qualitative diffeence fo the dynamics). 1/2 < < 1 = 1 > 1 * * * * * Blue: f(x) = x Geen: g(x) = sin(x) Red: fixed points Figue 7: Compaison of the functions f(x) = x and g(x) = sin(x) fo vaious values of. The intesection points coespond to the fixed points of the system (12). We quickly ecognize that the system (12) undegoes a pitchfok bifucation at the value = 1 whee the thee fixed points coalesce into one. To detemine the stabilities, we conside which of the functions f(x) = x o g(x) = sin(x) lies on top of the othe in the given egions. Afte a little wok, we should agee that: 1. If 1/2 < < 1, thee ae two oute fixed points which ae unstable, and a fixed point at x = 0 which is stable. 2. If 1, thee is a single unstable fixed point at x = 0. The esulting bifucation diagam is given in Figue 8. We can clealy see that the system undegoes a pitchfok bifucation at = 1, and that this bifucation is subcitical since the unstable banch gains stability as the new banches of fixed points appea. Notice, howeve, that this occus in the diection as deceases. It is woth noting that moe inteesting behavio can be obseved if we expand the ange of the paamete fom 1/2 to 0. A fulle exploation of the behavio of the system ove the missing ange 0 1/2 is left fo homewok. 12
Bifucation Diagam Vecto Field x 1/2 < < 1 > 1 Figue 8: Bifucation diagam fo the system (12). Even though we cannot explicitly identify the cuves of fixed points, we can captue thei qualitative flavo and identify the subcitical pitchfok bifucation at = 1. 13