# MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation

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1 MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter values could lead to vastly different long-term behavior of solutions. It could be the difference between a disease ing out or becoming an epidemic, or the difference between a population surviving or ing out due to predation. These changes in parameter values will be no less important for systems of multiple variables. Before considering a few examples, we make a few important notes about bifurcations in multiple dimensions: 1. As before, we will be primarily interested in the following: (a) changes in the number of fixed points; and (b) changes in the stability of fixed points. We will see that the primary determinants of bifurcations in multiple dimensions are the nullclines, since their intersections correspond to the fixed points. We will determine how the nullclines, and consequently the fixed points, change as our parameters vary. 2. In multiple dimensions, we still have the basic bifurcations we were introduced to when considering one-dimensional systems. That is, we will see: (a) saddle-node bifurcations when fixed points are created (or destroyed) from (into) nothing; (b) transcritical bifurcations when two curves of fixed points intersect and exchange stabilities; and (c) pitchfork bifurcations when there is both a creation (or destruction) or fixed points and a change in stability. There is one new major classification of bifurcation which will be qualitatively different than any of these: the Hopf bifurcation. This bifurcation occurs when there is the creation (or destruction) of the main new qualitative behavior we saw last week: limit cycles. 1

2 Saddle-Node Bifurcation r<0 r=0 r>0 Transcritical Bifurcation r<0 r=0 r>0 Pitchfork Bifurcation r<0 r>0 Figure 1: Characteristic vector field diagrams for the systems in (1) for the indicated values of r. Nullclines x = 0 (blue) and y = 0 (red) are shown. Example 1: Consider the following systems: (Saddle-node) (Transcritical) (Pitchfork) dt dt dt dt dt dt = r x 2 = y = rx x 2 = y = rx x 3 = y (1) 2

3 The vector field diagrams can be easily determined to have two or three qualitatively distinct possibilities, depending upon the value of r (see Figure 1). We are quickly able to correspond the qualitative feature of the bifurcation as being analogous to the corresponding one-dimensional bifurcations indicated. In particular, we should note two things: 1. The saddle-node bifurcation picture actually involves a saddle and a node! This is where the bifurcation gets its name even in onedimension where there is no such thing as a saddle. 2. We should not get too closely attached to the stabilities of the fixed points in Figure 1. It was easy to characterize the notion of a change in stability in one-dimension. There are significantly more possibilities in multiple dimensions. A stable node may become a saddle, or a stable spiral may become an unstable spiral, for instance. A consequence of this is that a saddle-node bifurcation may not necessarily involve a saddle and a node (see Example 2). Example 2: Consider the system dt = a + x y dt = x2 by where a, b R are parameters. We are going to consider what happens to the system as we vary the parameter a (keeping b = 1 constant) and what happens as we vary the parameter b (keeping a = 0 constant). Case 1: Vary a, keeping b = 1 constant: This gives the system dt = a + x y dt = x2 y. We need to consider the nullclines. We have and dt = 0 = y = x + a dt = 0 = y = x2. (2) 3

4 The nullcline corresponding to y = 0 never changes. The nullcline corresponding to x = 0 is just a line shifted up or down according to the value of a. It should be fairly easy to see that there are three qualitatively different cases for the value of a. 1. for a > 1/4, the curves intersect twice, giving rise to two fixed points (one a saddle, one various); 2. for a = 1/4, the curves intersect once, giving rise to one fixed points (degenerate); and 3. for a < 1/4, the curves never intersect, so the system has no fixed points. If we look at the vector fields plots for various values of a (Figure 2), we see the picture roughly corresponds to our one-dimensional saddle-node bifurcation. In fact, this is where the name comes from: the canonical case has a saddle and a sink node colliding and destroying each other. We can see, however, that the classification of the fixed points will have to be conducted on a case-by-case basis. (a) (b) (c) a=0.25 b=1 a=-0.25 b=1 a=-0.75 b=1 Figure 2: Vector field plots of the system in Example 1 for different values of a, holding b = 1. To determine the stability of the two fixed points when a > 1/4, we could determine the form of the fixed points as they depend upon a and then perform linearization. This would be a great deal of a work. A simpler method is to note that the system is Hamiltonian for all values of a and 4

5 then compute the Hamiltonian function H(x, y) = x3 3 + xy y2 2 + ay. Combined with the vector field diagram, we can now easily see that the fixed points on the left branch of the parabola corresponds to saddles, while the fixed points on the right branch correspond to centers. Case 2: Vary b, keeping a = 0 constant: This gives the system The nullclines are given by dt = x y dt = x2 by. (3) and dt = 0 = y = x dt = 0 = y = 1 b x2. We notice that the nullcline corresponding to x = 0 never changes. The nullcline corresponding to y = 0 is always a parabola centered at (0, 0). The parameter b controls how steep it is (and in the case b = 0 corresponds to line y = 0). We have the following qualitatively different cases for the value of b. 1. for b < 0, the curves intersect twice, once at (0, 0) and once to the left (one a source, one a saddle); 2. for b = 0, the curves intersect only once, at (0, 0) (degenerate); and 3. for b > 0, the curves intersect twice, once at (0, 0) and once to the right (one a saddle, one various). We will once again have to be careful when it comes to classifying the fixed points. The system is not Hamiltonian for most values of b so we will not have the luxury of appealling to a conserved quantity. Instead, we determine the fixed points explicitly and perform linear stability analysis. We have that x = 0 implies y = x and y = 0 implies x = by 2 so that by 2 y = y(b y) = 0. It follows that y = 0 or y = b so that the fixed 5

6 points are (0, 0) and (b, b). This confirms are intuition that there are two fixed points whenever b 0. To determine how the stability of these fixed points changes as b is varied, we perform linear stability analysis. We have [ ] 1 1 Df(x, y) = 2x b so that and Df(0, 0) = [ b [ 1 1 Df(b, b) = 2b b ] (4) ]. (5) It can be quickly checked the eigenvalues of (4) are λ = 1 and λ = b. It follows that the fixed points is (0, 0) is a source if b < 0, a saddle if b > 0, and an undetermined borderline case if b = 0. The eigenvalues of (5) are slightly more complicated. We can determine that λ = (1 b) ± (1 b) 2 4b. 2 We can analyze this by cases. Notice, first of all, that we will need more than just two cases in order to accommodate the sign of (1 b) 2 4b. Note that this quadratic has roots at and : 1. If b < 0, we have that 0 < (1 b) 2 4b and 0 < 1 b < (1 b) 2 4b so that there will is one positive eigenvalue and one negative eigenvalue. It follows that (b, b) is a saddle in this region. 2. If 0 < b < 3 2 2, we have that 0 < (1 b) 2 4b and 0 < (1 b) 2 4b < 1 b so both eigenvalues are real-valued and positive. It follows that (b, b) is a source in this region. 3. If < b < 1, we have that (1 b) 2 4b < 0 and 0 < 1 b so that both eigenvalues are complex and have positive real part. It follows that (b, b) is a spiral source in this region. 4. If 1 < b < , we have that (1 b) 2 4b < 0 and 1 b < 0 so that both eigenvalues are complex and and have negative real part. It follows that (b, b) is a spiral sink in this region. 5. If b > 3+2 2, we have that 0 < (1 b) 2 4b and 1 b < (1 b) 2 4b < 0 so that both eigenvalues are real-valued and negative. It follows that (b, b) is a sink in this region. 6

7 What happens is fairly clear at the bifurcation value b = 0. If we look at the vector field plots for various values of b near zero (Figure 3), we see that the situation corresponds roughly to a one-dimensional transcritical bifurcation. (a) (b) (c) a=0 b=-1 a=0 b=0 a=0 b=1 Figure 3: Vector field plots of the system (3) for different values of b near the transcritical bifurcation b = 0, holding a = 0. Near the bifurcation value b = 1, things are less clear. The stability of the fixed points changes, but independently of the creation or destruction of fixed points. To see what is really happening, we investigate the vector diagram in Figure 4. We can clearly see the stability switch as b passes through the bifurcation value of b = 1, but there is also something else which is interesting about this picture. At the bifurcation value b = 1 the system is Hamiltonian which can be used to show that the fixed point (1, 1) is a nonlinear center which has a homoclinic orbit around it which extends from the saddle point (0, 0). Bifurcations which create and then destroy homoclinic orbits are called homoclinic bifurcations. 2 Hopf Bifurcations Consider the system dt = x + ay + x2 y dt = b ay x2 y. This is the reduced glycolysis model from last week where X corresponds to ADP and Y was F 6P. We showed that (6) had a limit cycle for the 7 (6)

8 (a) (b) (c) a=0 b=0.8 a=0 b=1 a=0 b=1.2 Figure 4: Vector field plots of the system (3) for different values of b near the homoclinic bifurcation b = 1, holding a = 0. In (a) the fixed point (b, b) is unstable, in (b) it is stable but not asymptotically stable (and surrounded by a homoclinic orbit), and (c) it is asymptotically stable. parameter values a = 1/2 and b = 1/10. We now consider the question of what can happen for other parameter values. Recall our argument for the existence of a limit cycle when a = 1/2 and b = 1/10. Since the system only had a single fixed point, which was unstable, in order to construct a trapping region R we only had to show that that solutions were bounded. It followed by the Poincaré-Bendixson Theorem that solutions had to approach some sort of periodic orbit. We now want to see how many of these pieces remain in place as we vary a and b. In particular, we want to consider: 1. How the stability of the fixed point(s) change as a and b; and 2. Whether solutions are still bounded. If either of these points fail, we may have reconsider whether the system permits limits cycles. To consider the first point, we start by determining how the fixed point(s) depend upon the parameters a and b. The system (6) has nullclines given by dt = 0 = y = x a + x 2 and dt = 0 = y = b a + x 2. 8

9 Intersecting these two curves gives fixed point ( ( x, ȳ) = b, b a + b 2 It is important to note at this point that there is always exactly one fixed points. This is a relief since we do not have to look for saddle-node, transcritical, or pitchfork bifurcations. The linearization of (6) is given by [ ] 1 + 2xy a + x 2 Df(x, y) = 2xy a x 2 which leads to Df ( b, ) b a + b 2 = ). b 2 a a + b 2 a + b 2 2b2 a + b 2 a b 2. (7) We will consider the eigenvalues of the general matrix (7) in a moment. For illustrative purposes, however, we start by consider a specific case. Let s consider the parameters values a = b = 1 so that the fixed point is given by ( x, ȳ) = (1, 1/2). We have ( Df 1, 1 ) [ ] 0 2 = which has eigenvalues λ 1,2 = 1±i. Since the real part of the eigenvalues are negative, it follows that the fixed point is a stable spiral. We are no longer able to use the Poincaré-Bendixson Theorem to conclude the existence of limit cycles. In fact, we suspect from the vector field plot that solutions approach the fixed point (1, 1/2) asymptotically (see Figure 5). That is to say, the limit cycle has disappeared. We want to give further consideration to exactly what has happened to produce these different results. We can see that the nullclines do not change appreciably as a > 0 and b > 0 are varied. The curve for x = 0 always goes through (0, 0), rises, then approaches y = 0 as x. The curve for y = 0 goes through (0, b/a) then approaches y = 0 as x. This always divides the positive portion of the (x, y)-plane into four regions which produces swirling behavior. This raises the question of how, if the vector field diagram has not changed appreciably, what exactly has changed. We have determined that 9

10 Figure 5: Vector field plot of the example system with a = b = 1. Trajectories appear to asymptotically converge toward the unique fixed point at (x, y) = (1, 1/2). somewhere between the parameter values a = 1/2, b = 1/10 and a = b = 1 the single fixed points has undergone a switch from instability to stability. Since solutions are always trapped near the fixed point, solutions must go somewhere. That is, something must be attracting solutions. In one case, it is the fixed points; in the other, it is a stable limit cycle. A bifurcation which gives rise to a limit cycle is called a Hopf bifurcation. A Hopf bifurcation is called: 1. supercritical when a spiral sink loses stability and gives rise to a stable limit cycle; 2. subcritical when a spiral source gains stability and gives rise to an unstable limit cycle. In both cases, the basic structure of a Hopf bifurcation is that spiral node locally changes stability and, if the global behavior does not change, the stability must be passed from the fixed point to an unseen limit cycle. Conditions for the existence of Hopf bifurcations exist but are beyond the scope of this course, although it is often sufficient to determine when the complex eigenvalues go from Re(λ) < 0 to Re(λ) > 0 as the various parameters are changes. Our approach will be to invoke the Poincaré-Bendixson Theorem to argue for the existence of the required limit cycles. 10

11 We have seen that the qualitative behavior is different for two particular sets of parameter values, but we are also interested in when the change in behavior occurs. To answer this, we need to consider the general structure of (7). Determining the eigenvalues of (7) is cumbersome but fortunately not required. We can determine the stability by determining the trace of the matrix. We have tr ( Df ( b, )) b a + b 2 = b2 a a + b 2 (a + b2 ) = b2 a (a + b 2 ) 2 a + b 2 = b4 + b 2 2ab 2 a 2 a a + b 2. We have that tr(a) > 0 and tr(a) < 0, respectively, are the conditions for instability and stability of the fixed point so that we are interested in the solving b 4 +b 2 2ab 2 a 2 a = 0. We must give up on any reasonable hope for explicitly determining the desired values relationship between a and b explicitly. We can, however, plot this curve implicitly in the (a, b)-plane to get a sense of which parameter values permit limit cycles and which do not (see Figure 6). x x x x x x x x x x x Limit cycles! Hopf bifurcations Figure 6: Bifurcation diagram of the system (6) with the region permitting limit cycles shaded. The curve separating the filled in and empty region represents the curve where Hopf bifurcations occur. Such a diagram is called a bifurcation diagram. We recall that in 11

12 one-dimension our bifurcation diagrams included more information. In particular, we had the fixed points explicitly appearing. Here we only draw the regions in parameter space and label the regions with different qualitative behaviors. We notice immediately from Figure 6 that the region permitting limit cycles is very small. This should, perhaps, not be so surprising. Limit cycles are somewhat exotic and we should not expect them to be the norm. Nevertheless, we can often determine their existence by consider when stable spirals locally lose stability but remain globally stable. To complete the argument, we should show that there is a bounded trapping region for all values of a and b. This will guarantee that, when we lose stability of the fixed points, we do not escape to infinity and so are guaranteed by the Poincaré-Bendixson Theorem that a limit cycle exists. In fact, we have alrea done all of the argumentation. We saw that the nullclines always divided the region into four regions (see Figure 7). It can be easily determined that solutions move: 1. up in region 1 (R1); 2. to the right in region 2 (R2); 3. down in region 3 (R3); and 4. to the left in region 4 (R4). As before, the only region we have to make any general considerations at all is in region 3. We want to show that solutions not only move down, but do so in such a way as to meet the line leading to region 4 before x become infinite. We again consider the function Along solutions we have that L(x, y) = x + y. dl dt = dt + dt = b x so that, so long as x > b, we are traveling downhill relative to L(x, y) (see Figure 7). This is sufficient to construct the trapping region required from the Poincaré-Bendixson theorem. It follows that, whenever the fixed point is unstable, we necessarily have limit cycles. 12

13 b a R3 b a+b 2 0 R2 R1 b R4 Figure 7: (Somewhat) generic vector field plot for (6). Solutions may not pass through the red lines from the inside to the outside, so that it constitutes a trapping region. 13

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