So far we have looked at two kinds of steady states—equilibrium state and periodic state—and their stability. In case of equilibrium states, we have seen that they can be identified either by solving the steady-state condition,
f x( )=0, or by simulating the dynamical system, x f x= ( ). However, as you may have realized after studying about stability, the method of simulating forward in time will only lead to stable steady states. Likewise, in case of periodic states, they may be located either by solving for the periodic con- dition, x t T( + )=x t( ), or by time simulation of the dynamical system; how- ever, in the latter case, again only stable periodic states can be obtained.
We have seen that a dynamical system, f x( )=0, may have multiple steady states—equilibria and periodic ones—and that some of them may be stable and others unstable. Therefore, the notion of stability should be applied only to a steady state, and not to the dynamical system as a whole.
Homework Exercise: Consider the dynamical system below, called the van der Pol oscillator:
x−µ(1−x x x2) + =0 (3.22) with μ > 0. Cast Equation 3.22 in the dynamical system form,
x f x u x x x u= ( , ), =[ , ], =µ, and determine its steady states and their stability by any of the methods discussed previously. Depending on your choice of the parameter μ, the phase portrait may appear as in Figure 3.11.
Notice that the equilibrium state at (0, 0) is unstable and there is a single, isolated periodic state that appears to be stable. This is an example of a limit cycle.
We have seen one example of a dynamical system with a varying param- eter (Figure 3.4) where the location as well as stability of the steady states can change with changing values of the parameter. Thus, in general, for a dynamical system of the form, x f x U = ( , ), where U is a vector of param- eters, what is of interest is an entire family of steady states, such as x*, and changes in stability, if any, as one or more of the parameters U are varied.
At critical values of the parameters, there may be a qualitative change in the number and/or nature of stability of the steady states. These critical points are called bifurcation points and the corresponding phenomenon is identified as a particular kind of bifurcation. For example, in Figure 3.4, at the critical value of γ = 1, a single equilibrium state bifurcates into three, with the equilibrium state x*=0 changing from stable to unstable. Thus,
γ = 1 is a bifurcation point and the bifurcation phenomenon in question is labeled as a pitchfork bifurcation.
Change in stability can be identified by tracking the eigenvalues in case of equilibrium states and the Floquet multipliers in case of periodic states with varying parameter. Passage of an eigenvalue from the left to the right half-plane or a Floquet multiplier going out of the unit circle would signify onset of instability in case of equilibrium or periodic states, respectively.
Bifurcations are commonly observed in a wide range of dynamical sys- tems in varied fields such as chemical engineering, electronics, biology, eco- nomics, and, of course, in aircraft flight dynamics. Most dynamical systems encountered in practice, both natural and man-made, are nonlinear and multiparameter, and several bifurcation phenomena occur in these systems.
Sometimes, a bifurcation phenomenon can lead to a more desirable or effi- cient operating state for a system, though more often than not it represents a loss of safety/control or a breakdown of the system’s regular operation.
A bifurcation analysis maps all possible steady states of a dynamical system and their stability, and various occurrences of bifurcations as one or more parameters are varied. Such plots are often called bifurcation dia- grams. Figure 3.4 is an example of a bifurcation diagram. If it is desired to operate the rotating hoop with bead system at the steady state θ = 0, then the bifurcation diagram in Figure 3.4 reveals that the system must be operated at a value of γ less than 1. On the other hand, if the objective is to raise the bead to some angle θ on the hoop as it rotates, then the cor- responding value of γ > 1 can be read from Figure 3.4 as well.
–4 –4 –3 –2 –1 0 1 2 3 4
–3 –2 –1
x
0 1 2 3 4
x
FIGURE 3.11 Phase portrait of the van der Pol oscillator.
Depending on the nature of the bifurcation, the post-bifurcation behavior of the dynamical system can vary. Since each bifurcation event is accompanied by a change in stability, either of an equilibrium state or a periodic state, a catalog of different types of possible bifurcations can be built by observing the various ways in which the eigenvalues or Floquet multipliers can cause instability. In general, there are only two distinct ways for an eigenvalue to cross the imaginary axis in the complex plane to induce a bifurcation. Likewise, there are only three clearly different crossings of the unit circle by the Floquet multiplier. These are depicted in Figure 3.12.
At a stationary bifurcation, a steady state gives rise to one or more steady states of the same type (equilibrium or periodic, as the case may be) and there is an exchange of stability, or a pair of steady states annihilate each other. We shall examine these cases in more detail shortly. Stationary bifurcations are marked by an eigenvalue at the origin in case of equilib- rium states or a Floquet multiplier at “1” in case of periodic states.
At a Hopf bifurcation, a pair of complex conjugate eigenvalues or Floquet multipliers, as the case may be, cross over as indicated in Figure 3.12. In case of equilibrium states, a Hopf bifurcation gives rise to a periodic state (limit cycle) whereas for a periodic state, it creates a quasi-periodic state. When viewed in the Poincaré plane, the periodic orbit of course appears as a fixed point and the quasi-periodic state is a closed orbit about that fixed point.
Im (λ)
Re (λ)
Im (λ)
Re (λ)
Im (λ)
Re (λ)
Im (λ)
Re (λ)
Im (λ)
Re (λ)
Stationary bifurcation
Hopf bifurcation
Period doubling bifurcation
FIGURE 3.12 Types of bifurcations depending on the movement of eigenvalues (left) and Floquet multipliers (right).
In addition, for periodic states, a period-doubling bifurcation occurs when the Floquet multiplier crosses the unit circle at “−1.” There is no equivalent of this bifurcation in case of equilibrium states. At a period- doubling bifurcation, typically, a periodic state of period T loses stability and a new stable periodic state of period 2T emerges. On the Poincaré plane, the original periodic state appears, as always, as a single fixed point and the new bifurcated period−2T orbit alternates between two points.
3.3.1 Stationary Bifurcations of Equilibrium States
As seen in Figure 3.12, a stationary bifurcation of an equilibrium state is marked by a single real eigenvalue passing through the origin from the left to the right half complex plane. So, clearly, the equilibrium state in question transitions from stable to unstable. However, depending on other conditions (besides the single zero eigenvalue), the form and nature of the various equi- librium states, and hence the type of bifurcation, may be different. There are three main types of bifurcation of equilibrium states that arise under the condition of a single real eigenvalue crossing the origin in the complex plane. These are sketched in Figure 3.13 and discussed below.
3.3.1.1 Saddle-Node Bifurcation
Consider a single stable equilibrium state that with a varying parameter μ has a real eigenvalue that tends to the origin. At a critical value of the parameter μ, this eigenvalue crosses the origin and then moves into the right half-plane. This critical value corresponds to a saddle-node bifurca- tion and the shape of the curve of equilibrium states with parameter μ must appear as shown in Figure 3.13. It is remarkable that this curve is always locally quadratic around the critical point—it cannot take any other shape. Saddle-node bifurcations are observed in an amazingly large num- ber of diverse dynamical systems and they appear identically every time.
There are two ways of understanding the changes that occur at a sad- dle-node bifurcation. One is to follow the curve of equilibrium states along the arc-length and observe the change in the sign of the eigen- value, as we did above. The other is to vary the parameter μ and observe the change in the system dynamics. In case of the saddle-node bifurca- tion in Figure 3.13, for a negative value of μ, there are two equilibrium states—one stable and the other unstable. Perturbations lead away from the unstable state and toward the stable one, as marked in Figure 3.13.
As μ tends to the critical value, the two states approach each other and coalesce—at the critical point, there is only one steady state with a zero
eigenvalue that is neither clearly stable, nor clearly unstable. Trajectories approaching from negative values of the state variable tend to the (criti- cal) steady state whereas for positive values of the state variable, trajec- tories tend to diverge from the steady state. Beyond the critical value of the parameter μ, there are no equilibrium states in the vicinity of the bifurcation point. Trajectories now pass unimpeded as indicated by the arrow in Figure 3.13 for negative μ. Thus, from this point of view, at a saddle-node bifurcation, a pair of equilibrium states collide and annihi- late each other. Consequently, the dynamics on either side of the critical point are vastly different.
Imagine a dynamical system with a saddle-node bifurcation with its state on the stable branch in Figure 3.13. Let the parameter μ be varied slowly so that the state moves along the stable branch in a quasi-steady manner toward the critical point. Once the parameter goes beyond the critical point, there are no steady states in the vicinity and the system must then move along the upward arrow in Figure 3.13. It is as if the system stepped off a cliff and plunged downward to oblivion (except the arrow points upward in Figure 3.13).
Transcritical bifurcation
μ
μ μ
μ Saddle-node bifurcation
Supercritical pitchfork bifurcation Subcritical pitchfork bifurcation x
x˙ = μ + x2 x˙ = μx + x2
x˙ = μx – x3 x˙ = μx + x3 x x
x
FIGURE 3.13 Three main types of stationary bifurcations of equilibrium states—
stable ones are shown in full line, unstable ones are dashed.
In real-life physical systems though, there is usually another branch of steady states that the system can move to when it steps off a saddle- node bifurcation point. In that case, the system is said to “jump” from one branch of steady states to another. Jump phenomenon accompanied by hysteresis is fairly common in several dynamical systems. Figure 3.14b shows an example of a pair of saddle-node bifurcations, resulting in a jump phenomenon. Imagine moving along the upper stable branch until the critical point S1 is reached. The system then jumps to the lower stable branch as marked by the downward arrow and proceeds further to the right in Figure 3.14b. However, from this state, as the parameter condi- tions change to the left, the system follows the lower stable branch until the other saddle-node point at S2 where it jumps up, back to the upper sta- ble branch. In this manner, a hysteretic response is created. Between the saddle-node points marked by S1 and S2, the system has multiple steady states (which by itself is not uncommon for nonlinear systems) and the branch the system occupies depends on the direction of change in the parameter conditions. Increasing parameter finds the system in one sta- ble state and decreasing values of parameter in the other. Note that this multivalued behavior is not because the dynamical system function f(x) is multivalued—it is not! For a contrast, Figure 3.14a shows a jump-like dynamics where the single stable branch of steady states falls steeply with changing parameter but does not fold over at a saddle-node bifurcation.
Homework Exercise: Jump phenomenon and hysteresis are known to occur in airplane flight dynamics as well, for example, in rolling maneu- vers. Look up cases of jump in the roll rate with aileron deflection. We shall be studying this later in this text.
Small forcing Small forcing
S2
S1
(a) (b)
FIGURE 3.14 (a) Jump-like condition and (b) jump and hysteresis with a pair of saddle-node bifurcations.
Homework Exercise: Consider the plot of steady states in Figure 3.15.
Here, the branch of equilibrium states is unstable even though it folds over like the saddle-node bifurcation in Figure 3.13. Is this mathematically pos- sible? How would the movement of the eigenvalues look as one traversed along this branch of steady states and went round the curve?
3.3.1.2 Transcritical Bifurcation
The second type of stationary bifurcation of an equilibrium state in Figure 3.13 is called a transcritical bifurcation. In this case, a pair of steady states, one stable and the other unstable, intersect. Obviously, at the point of intersection, the steady state must have a zero eigenvalue—hence, it is a critical point. After the critical point, there is an exchange of stability with the previously stable branch becoming unstable and vice versa. If one were to follow a branch from its stable states to the unstable states, then a single real eigenvalue would cross over from the left to the right half com- plex plane at the critical point. At the same time, on the other intersecting branch, going from unstable to stable states, a real eigenvalue would cross over in the reverse sense from right to left half-plane.
Homework Exercise: In both the saddle-node and transcritical cases, as one follows a branch of steady states through the critical point, we see a change in stability due to the movement of a single real eigenvalue. Then what distinguishes one bifurcation from the other? Or what decides if the bifurcation type corresponding to a single zero eigenvalue is a saddle- node or a transcritical?
x
μ FIGURE 3.15 Plot of equilibrium states with varying parameter.
Dynamically, in case of a transcritical bifurcation, there is no “jump.”
As the parameter is slowly varied, the steady state changes smoothly from one stable branch pre-bifurcation to the other post-bifurcation, except for the kink in the curves at the critical point.
3.3.1.3 Pitchfork Bifurcation
The third type of bifurcation in Figure 3.13 is called a pitchfork bifurca- tion. Here again, there is a branch of steady states that loses stability as a parameter is varied. We shall call this the primary steady-state branch. At the critical point, once again, there is a single zero eigenvalue. However, in this case, a symmetric pair of secondary steady states is created at the critical point. There are two possibilities: Either a pair of stable steady- state branches is created about the unstable segment of the primary steady state—this is called a supercritical pitchfork bifurcation, or an unstable secondary branch pair coexists with the stable primary segment—this case is a subcritical pitchfork bifurcation. The dynamic behavior as one traverses past the critical point is vastly different in the two cases.
For the supercritical bifurcation, as one traverses along the stable pri- mary branch and crosses the critical point with the parameter being slowly varied, one steps on to the unstable branch of steady states. Here, the slightest perturbation will see the system diverging from the unstable primary steady state and approaching one of the two stable bifurcated sec- ondary branches—which one, depends on the nature of the perturbation.
This is indicated by the arrows in Figure 3.13. Similarly, traversing from stable to unstable primary branch through the critical point in case of a subcritical bifurcation, perturbations from an unstable steady state seem to carry the system away to infinity. In real-life systems, there will usu- ally be another steady state for that value of the parameter and the system will most probably tend to it. Thus, stepping across a subcritical pitchfork bifurcation usually results in a jump phenomenon, somewhat like what was seen for the saddle-node case. This is shown in Figure 3.16 where the unstable secondary steady states created at a subcritical pitchfork bifurca- tion fold over at a saddle-node point to form large-amplitude stable steady states. As the parameter is varied past the subcritical pitchfork bifurcation point, a perturbation off the now unstable primary steady state carries the system to one of these stable large-amplitude stable states. Hysteresis also accompanies the jump phenomenon in Figure 3.16, as was the case in Figure 3.14b, as the parameter is varied in a quasi-steady manner back and forth across the subcritical pitchfork point. The hysteresis is indicated
by the arrows in Figure 3.16. Note that in this case too there is a range of values of the parameter where the system may find itself in one of three possible stable steady states. It is not possible to definitely predict which of these three steady states will be observed—it depends on the history and perturbation in the states, and changes in the parameter.
3.3.1.4 Perturbation to Transcritical and Pitchfork Bifurcations
Since all the stationary bifurcations in Figure 3.13 correspond to the same criterion—passage of a single real eigenvalue through the origin of the complex plane—one may wonder if one of them is more fundamen- tal, in some sense, than the others. The answer is, yes. The saddle-node bifurcation is the most general form of stationary bifurcation in case of a zero real eigenvalue. The transcritical and pitchfork bifurcations are special cases that occur only when the system satisfies some additional constraint. Imagine a second parameter, in addition to the parameter μ in Figure 3.13. Only for a particular value of this second parameter would a transcritical or pitchfork bifurcation be formed. If the second parameter were perturbed (imagine an axis into/out of the plane of the paper in Figure 3.13), then the structure of the transcritical or pitchfork bifurcation may not be maintained. The typical forms of the transcriti- cal and pitchfork bifurcations under perturbation of a second parameter are as sketched in Figure 3.17. In each case, the branches intersecting at the bifurcation point break apart to form either a saddle-node bifurca- tion or an unbifurcated branch of steady states, as indicated in Figure 3.17. Thus, for a general (or well-modeled) dynamical system, one would not expect to see a transcritical or pitchfork bifurcation unless it had an inherent constraint or symmetry that forced the second parameter to
x
μ
FIGURE 3.16 Jump phenomenon and hysteresis at a subcritical pitchfork bifur- cation point.
have precisely that particular value for which one of these bifurcations would occur.
Homework Exercise: Review the bifurcation diagram in Figure 3.4 for the rotating hoop with a bead, which shows a supercritical pitchfork bifur- cation. What is the constraint or symmetry that causes this particular bifurcation? Physically, how can this constraint be “broken” or the system perturbed to yield a perturbed bifurcation diagram as in Figure 3.17?
Homework Exercise: Sketch the perturbed bifurcation diagram for the subcritical pitchfork bifurcation in Figure 3.16. Identify the jump phenom- enon and hysteresis loops. How is it different from the form of the bifurca- tion diagram for the jump in Figure 3.14b?
3.3.2 Hopf Bifurcation of Equilibrium States
As seen in Figure 3.12, a Hopf bifurcation occurs when a pair of com- plex conjugate eigenvalues cross the imaginary axis from left to right half- plane. At a Hopf bifurcation, the equilibrium state changes its stability and a family of periodic orbits emerges. Just as in the case of pitchfork bifur- cations, Hopf bifurcation may be supercritical or subcritical. The super- critical Hopf bifurcation case is pictured in greater detail in Figure 3.18.
When the eigenvalue pair is in the left half-plane, the equilibrium state is
Perturbed transcritical bifurcation
Perturbed supercritical pitchfork bifurcation
x x
x
μ μ
μ μ
x
FIGURE 3.17 Form of transcritical and pitchfork bifurcation under perturbation of a second parameter.