Introduction to bifurcations Marc R. Roussel September 6, Introduction Most dynamical systems contain parameters in addition to variables. A general system of ordinary differential equations (ODEs) could therefore be written ẋ = f(x;k), where k is a set of parameters on which the equations, and thus their solutions, depend. If we solve a set of differential equations at different parameter values, we often find that, qualitatively, not much changes. However, in some models, we can find sets of parameter values which are close to each other but where the behavior of the model is in some way qualitatively different for one set or the other. For instance, a stable equilibrium point might have become unstable. We then say that the system has undergone a bifurcation. Bifurcations often change the attractors of a dynamical system. Informally, an attractor is a solution which is approached at long times. Stable equilibrium points are attractors, but they are not the only possibility. In today s lecture, we will engage in some simple numerical discovery exercises in which we will see a few important bifurcations and their consequences. We will be using the dynamical systems software xppaut (xpp for short). Full instructions for using this software will not be given here. You are directed to consult the xpp documentation for details. Andronov-Hopf bifurcations One very important kind of bifurcation is the Andronov-Hopf bifurcation (formerly known in the West as a Hopf bifurcation). In an Andronov-Hopf bifurcation, a stable focus becomes an unstable focus as a parameter is varied, and the attractor becomes a limit cycle. A limit cycle is an asymptotically stable, periodic solution which can be pictured as a closed curve in phase space. Limit cycles differ from conservative oscillations in mechanical systems in that the former have fixed shapes and sizes at given parameter values, while the corresponding quantities in a conservative mechanical oscillator depend on the total energy, i.e. on the initial conditions. There are two qualitatively different kinds of Andronov-Hopf bifurcations, sketched below. In a supercritical Andronov-Hopf bifurcation, the limit cycle grows out of the equilibrium point. In other words, right at the parameters of the Andronov-Hopf bifurcation, the limit cycle
has zero amplitude, and this amplitude grows as the parameters move further into the limit-cycle regime. Pictorially, think of it this way: In a subcritical Andronov-Hopf bifurcation, there is an unstable limit cycle surrounding the equilibrium point, and a stable limit cycle surrounding that. The unstable limit cycle shrinks down to the equilibrium point, which becomes unstable in the process. For systems started near the equilibrium point, the result is a sudden change in behavior from approach to a stable focus, to large-amplitude oscillations. Here is the corresponding picture: Because the stable limit cycle exists even when the equilibrium point is stable, if we imagine slowly varying a system parameter back-and-forth across the Andronov-Hopf bifurcation, we wouldn t expect to jump back to the equilibrium point at the same parameter value of the parameter from which this point lost stability. This is called hysteresis, and is associated with bistability, the fact that the system actually has two attractors over a range of parameters. In both cases, Andronov-Hopf bifurcations occur when an equilibrium point changes from being a stable to an unstable focus. We therefore detect Andronov-Hopf bifurcations through linear stability analysis. Example. The Brusselator is a historically important model of an oscillating chemical reaction. The Brusselator is an abstract model which was used to show that chemical systems could oscillate, but it does not describe any particular reaction. As you can guess from the There are additional technical conditions which are required to obtain an Andronov-Hopf bifurcation. However, these conditions are almost always realized in realistic systems of nonlinear ODEs. The interested reader can refer to any standard textbook on nonlinear dynamics for details.
foregoing discussion, it undergoes an Andronov-Hopf bifurcation. The model is k A X k B + X Y + D k X + Y X k X E In this model, A and B are assumed to be held constant in some way (buffering, continuous supply, etc.). D and E are assumed not to participate in any further reactions so that their concentrations are irrelevant. Accordingly, X and Y are the only variables. If we scale the variables and parameters by we get the dimensionless equations x = X k /k, a = A(k /k ) k /k, y = Y k /k, b = k B/k, τ = k t, ẋ = a bx + x y x, ẏ = bx x y. The equilibrium point is (x,y ) = (a,b/a). The characteristic equation for this equilibrium point is λ + λ(a + b) + a =. The eigenvalues are therefore λ ± = { } b a ± (b a ) a. It shouldn t be too difficult to convince yourself that the following are all possibilities:. a stable node. an unstable node. a stable focus. an unstable focus We concentrate first on the last two cases. We get a focus if the quantity under the square root is negative. The real parts of our two eigenvalues are then R(λ ± ) = (b a ).
An Andronov-Hopf bifurcation will occur when the real part of the eigenvalue changes sign, i.e. when it passes through zero. In this case, we expect to see the bifurcation when b = b AH = a +. For b < b AH, the equilibrium point is a stable focus, while it is an unstable focus for b > b AH. Is our bifurcation supercritical or subcritical? This kind of question can be answered by a simple numerical experiment. Here is the xpp input file for the Brusselator: # Brusselator ode file # Differential equations: x = a - b*x + xˆ*y - x y = b*x - xˆ*y # Initial conditions x() =.9 y() =.9 # Default values of the parameters: param a=, b= # Reserve lots of storage space # in case we want to get a long trajectory @ MAXSTOR= done Figure shows the behavior for b < b AH. As expected, we get a stable focus. If we pick a value of b just a bit above b AH, we get a small limit cycle which grows as we increase b (Fig. ). Another way to see this more clearly is to plot just the limit cycle, i.e. start plotting the trajectory only after a long time has elapsed so that the transient approach to the limit cycle is eliminated. Figure is an example of this procedure which shows clearly that the limit cycle grows as we move away from the bifurcation. Based on this behavior, we can conclude that the Andronov-Hopf bifurcation is supercritical. Because trajectories can t cross, planar systems can only have equilibria and limit cycles as attractors. To see more interesting behavior, we need to go to three dimensions. Dynamics in three dimensions In this section, we will study some of the possible behaviors of systems in three dimensions. We will again proceed by example.
y.6...8.6.. b=.9.6.8...6 x Figure : A trajectory in the phase plane for the Brusselator with a =, b =.9. y y.8.6.6.....8.8.6.6... b=.. b=...6.8...6.8 x.6.8...6.8 x Figure : Trajectories for two values of b just above the Andronov-Hopf bifurcation with a =.
y.. b=..8 b=..6 b=...6.8...6 x Figure : Brusselator limit cycles obtained for a =. Example. The autocatalator is another abstract chemical reaction model: k P A k c P + C A + C k u A B k A + B B k B C k C D The concentrations of A, B and C are the only variables in this model, the other concentrations being either fixed or irrelevant. With the transformations a = A k k u /k, µ = k cp/k, b = B k /k, κ = k k /(k k c ) k /k u, c = C k k /(k uk ), δ = k u/k, τ = k u t, σ = k u /k, 6
the rate equations become ȧ = µ(κ + c) ab a, ḃ = σ ( ab + a b ), ċ = (b c). δ The xpp input file for this model is as follows: # Autocatalator.ode a = mu*(kappa+c) - a*bˆ - a b = (a*bˆ + a - b)/sigma c = (b-c)/delta param mu=., kappa=6, delta=e-, sigma=e- # The variables range over several orders of magnitude, so # it s convenient to plot, and log(c) aux la = aux lb = aux lc = log(c) # In order to avoid problems with the logs, # start from a point other than (,,). a() = b() = c() = # This system is stiff, so we need # an appropriate integrator. @ METHOD=stiff # The time scale of the oscillations is really fast, # and the spikes are really sharp and high, so we need # to adjust both the integration step size and # the maximum variable value allowed. @ DT=e-, BOUNDS=e @ MAXSTOR= done This system has a supercritical Andronov-Hopf bifurcation, as shown in Figs. and. We can confirm the Andronov-Hopf bifurcation by having xpp compute the eigenvalues. At µ =., xpp reports Eigenvalues: -.6 + i 9.6677 -.6 + i -9.6677-9.8897 + i. 7
.8 µ=..6.. -. -. -.6 -.8 -.7 -.7 -.7 -.69 -.69 -.68 -.68 Figure : Stable focus in the autocatalator for µ =., κ = 6, δ =. and σ =...... µ=. -. -. -. -. -.8 µ=.9 -.78 -.76 -.7 -.7 -.7 Figure : Autocatalator limit cycles. All parameters are as in Fig., except for µ, as noted in the figure. 8
µ=. -7-6 - - - - - Figure 6: Period doubling in the autocatalator. All parameters are as in Fig., except µ =.. while at µ =., we get Eigenvalues:.76 + i 9.9976.76 + i -9.9976-9.8989 + i. The real parts of the complex eigenvalues have changed sign, indicating an Andronov-Hopf bifurcation. We determine that the bifurcation is supercritical by examining how the system behaves near the bifurcation. In this case, a very small limit cycle is born at the bifurcation. This limit cycle grows as we continue to increase µ (Fig. ). If we continue to increase µ, the limit cycle doubles up (Fig. 6). We say that the system has undergone a period-doubling bifurcation, so called because the period the time it takes for the trajectory to repeat itself roughly doubles as we pass through this bifurcation since we now have to go around the loop twice before coming back to the same point. Note also that the trajectory appears to cross itself. This is an artifact of the two-dimensional projection of a three-dimensional system, but it does show why period-doubling bifurcations can t happen in two dimensions. If we increase µ further, the limit cycle undergoes additional period-doubling bifurcations (Fig. 7). Period doublings can and often do continue ad infinitum. Note that the values of µ at which the bifurcations occur get closer and closer. It therefore becomes difficult to find parameter values at which the higher periods appear, although they should be there. If you think about the logical end result of period doubling, it should be an attractor with an infinite period, i.e. one which doesn t actually repeat itself. There are different ways 9
µ=. µ=. -7-6 - - - - - -7-6 - - - - - Figure 7: Additional period doublings in the autocatalator. All parameters are as in Fig., except for µ which has the indicated values. in which we can draw curves in three-dimensional space which don t repeat themselves, but period doubling leads to a very particular type of curve known as a strange attractor. Strange attractors have a number of, well, strange geometric properties, chief among which is that they are fractal, i.e. they have properties which are consistent with a fractional spatial dimension. In other words, they are neither curves nor surfaces nor volumes, but something in between. In any event, we observe a strange attractor in our model at µ =. (Fig. 8). From our perspective, the most important property of systems with strange attractors is that they are usually chaotic. Chaotic systems are mainly characterized by their sensitive dependence on initial conditions. This means that, if you start two systems off with similar (but not identical) initial conditions, they will tend to drift apart. After a little while, they won t be doing the same thing at all. Figure 9 demonstrates this effect. In both cases shown, two trajectories were started from the initial points (., 7,.) and (., 7,.), which is near the attractor. In the limit-cycle regime, even when the limit cycle is complicated, these two systems just chase each other around the limit cycle, maintaining, on average, a constant distance. On the other hand, chaos causes the two systems to rapidly lose synchrony.
µ=. - -8-7 -6 - - - - - Figure 8: Strange attractor of the autocatalator. All parameters are as in Fig., except µ =.. - µ=. µ=. t - t Figure 9: Illustration of sensitive dependence on initial conditions. The graph on the left shows the log b vs t time series for two systems started near the period-8 attractor shown in Fig. 7, while that on the right shows the same thing for the chaotic attractor shown in Fig. 8. Note the rapid desynchronization in the latter case.