In principle, bifurcation diagrams with a varying parameter, such as those presented earlier in this chapter, may be obtained by methodically solv- ing for all the steady states at one value of the parameter, then stepping over to the next value of parameter and repeating the process, and so on.
However, a far more efficient way of constructing a bifurcation diagram is to start at a steady state and “continue” along the branch. Thus, for exam- ple, in case of the saddle-node bifurcation diagram in Figure 3.13, instead of finding the two steady states at one value of parameter and doing so for every successive parameter value, one would start with a steady state and continue along that branch with increasing parameter, go around the fold (critical point), and track the other set of steady states with decreasing parameter. In this process, whenever bifurcations are encountered, those critical points are noted and the type of bifurcation identified. One then needs to go back to these critical points and further continue along bifur- cated branches that originate from these points, if any. Algorithms that perform this kind of computation are called continuation algorithms.
While it is beyond the scope of this book to investigate continuation algorithms in depth, since we will be using them for our work and expect the reader to get familiar with one as well, some hints about how they operate may be worthwhile.
We would like the continuation algorithm to solve for steady states and periodic states of dynamical systems of the form Equation 3.1. Additionally, we would like it to compute the eigenvalues of the Jacobian matrix as in Equation 3.15 and Floquet multipliers of the Poincaré map as in Equation 3.21. And we would want it to identify the bifurcations in Figures 3.13 and 3.20 and correctly pick the newly bifurcated branches to be tracked.
Imagine that the continuation algorithm needs to track a branch of steady states x* = x*(u) as sketched in Figure 3.22 starting from an equi- librium point (x*, u*). The steady states are solutions of the set of algebraic equations:f x u( , )=0. In general, these are a set of n equations in (n + 1) unknowns—the n states and the single parameter. An obvious, though simplistic approach, would be to implement a predictor-corrector scheme.
As pictured in Figure 3.22, for a fixed step Δu, one can find the “predicted”
values of the states as follows:
f x u f
x x f
u u x f
x
f ( , )= ⇒∂ u u
∂ +∂
∂ = ⇒ = − ∂
∂
⋅ ∂
∂
−
0 0
1
∆ ∆ ∆ ∆ (3.23)
In practice though, there is no need to always select the parameter u as the stepping variable; any one of the (n + 1) variables can be used for this purpose, even variably at different points along a branch depending on convenience. Another option is to use an altogether different variable, such as arc-length along a branch, as the stepping variable. Several variants of this approach are possible; however, the predictor step basically uses “local”
information at an equilibrium point (x*, u*) to advance in one direction along a branch, as depicted in Figure 3.22. Note that the Jacobian matrix
x
u (x1, u1)
(x1, u1)
Δu
f (x, u)
x (x1*, u1)
(x*, u*)
f (x1*, u1) = 0
(a) (b)
x* = x*(u)
FIGURE 3.22 (a) Predictor step and (b) corrector step in a continuation.
(∂ ∂f x/ )( *, *)x u used in Equation 3.23 is the same one that is needed to evalu- ate the stability of the equilibrium state (x*, u*) anyway. At critical points where an eigenvalue may be zero, one needs to be careful in dealing with the Jacobian matrix. Here, one of the alternative formulations suggested above may be helpful. The predictor step yields an approximate state (x1, u1), which must then be corrected to the “nearest” steady state on that branch. This may be done by a “corrector” step that may well be a standard Newton–
Raphson algorithm. For further details about the continuation algorithm, one may refer to Reference 2.
EXAMPLE 3.1: NONLINEAR PHUGOID DYNAMICS
To illustrate the use of the continuation procedure on a dynamical system with one parameter, let us consider a simplified model of the airplane dynamics representing a nonlinear phugoid motion. Refer back to the equations of motion in Chapter 1 (Table 1.5). Assuming longitudinal motion and setting all lateral-directional variables to zero, from Equation 1.59, we have
mV T D mg
m q V T L mg
= − −
− = + −
cos sin
( ) sin cos
α γ
α α γ (3.24)
From Equation 1.47,
q− =α qw (3.25)
And further using Equation 1.28 we have
qw =γ (3.26)
So, Equation 3.24 may be written as
Vg T
mg D
mg Vg T
mg L
mg
= − −
= + −
cos sin
sin cos
α γ
γ α γ (3.27)
Converting the velocity V to a nondimensional variable as y = (V/V0) where V0= 2mg SC/ρ L , and transforming to a
nondimensional time, τ = t/(V0/g), Equation 3.27 may be repre- sented as
dyd b ay f y
dd y y f y
τ γ γ
γ
τ γ γ
= − − =
= − =
2 1
2
sin ( , )
cos ( , ) (3.28)
where small α has been assumed (sin α ≈ 0, cos α ≈ 1). The param- eters in Equation 3.28 are: b = (T/mg); a = (CD/CL). Clearly, the dynamics in Equation 3.28 is nonlinear in y and γ. Note that the approximation in Equation 3.28 is used here only to illustrate the use of the continuation and bifurcation procedure. In general, in this text, we do not advocate the use of simplified or approximate models of the aircraft flight dynamics equations instead preferring to deal with the complete form of the equations as presented in Chapter 1.
Of the two parameters, let us select b as the continuation param- eter. The first step of the analysis is to determine the equilibrium states. The equilibrium states of Equation 3.28 are given by
dyd b ay d
d y y
τ γ γ
τ
= = −0 * sin *;2− = =0 *−cos *γ
* (3.29) where the * represents a steady state. In terms of the parameters (a, b), one can arrive at the following analytical solution for equilibrium states:
y ab a b
*2 cos * ( a )
2 2
2 1
= = ± 1 − +
γ + (3.30)
Of course the analytical solution is not required since the con- tinuation algorithm will determine the steady states numerically.
But Equation 3.30 can be used to suggest a starting equilibrium state needed for the continuation method. For instance, setting a = b in Equation 3.30 yields the steady state γ* = 0, y* = 1, which can be used to start the continuation algorithm.
The second step is to determine the stability of each of the com- puted equilibrium states. This is obtained by calculating the eigen- values of the Jacobian matrix of the system in Equation 3.28. The Jacobian may be written as
J
fy f fy f
ay
y
=
∂
∂ ∂
∂
∂
∂ ∂
∂
=
− −
+
1 1
2 2
2 1 γ
γ
γ
( *, *)γ
* cos *
cosγγ* γ
*
sin * y 2 y*
(3.31)
Note that the Jacobian matrix does not directly depend on the parameter b. However, with varying b, the steady state (y*, γ*) changes, and the entries in the matrix of Equation 3.31 do depend on the values of (y*, γ*). Hence, one can expect a change in the steady states and their stability with varying parameter b. For instance, at the steady state γ* = 0, y* = 1,
J a
=− −
2 1
2 0 (3.32)
Whose eigenvalues are − ±a a2−2 . Since a is usually of the order of 0.1, we may safely assume a2 << 2; hence the eigenvalues are approximately − ±a i 2 . In other words, the damping of the phu- goid motion depends primarily on the parameter a, which is the ratio of drag to lift, and the damped natural frequency is, to a first approximation, equal to 2, independent of any parameter.
Then again, we do not really need to make any approximation or even calculate the eigenvalues analytically—all the computations can be managed by the continuation algorithm. However, it is good practice to have an estimate of values that may be expected in the computation and to understand the nature of the changes in steady states and their stability as the continuation run is carried out.
We shall hold a = 0.2 for these computations and vary b between 0.2 and 0.8. Remember, b = a = 0.2 corresponds to a level, steady- state flight, and with increasing b, the thrust is being increased.
Figure 3.23a shows the branch of steady states with varying param- eter b. It can be noticed that beginning from b = 0.2, the equilibrium
HB
HB
b = 0.2 b = 0.557 b = 0.8
–1.5 –0.2 –1 –0.5 0 0.5 1 1.5
Imaginary axis
–0.15 –0.1 –0.05 0 0.05 0.1 0.15
Real axis 0.80.2
0.9 1 1.1 1.2 y (= V/V0)
1.3 1.4 1.5 1.5 (a)
(b) 1
0.5
0
0.3 0.4 0.5 0.6 0.7 0.8
0.2
γ (rad)
0.3 0.4 0.5 0.6 0.7 0.8
b (= T/mg)
0.2
FIGURE 3.23 (a) Bifurcation diagrams of γ and y as function of parameter b (solid lines: stable states; dashed lines: unstable states; darker solid lines: unstable limit cycles; HB: Hopf bifurcation). (b) Plot of eigenvalues as function of parameter b.
states are stable and the steady-state climb angle increases with increasing thrust (parameter b), the velocity remaining nearly con- stant. The eigenvalues are a complex conjugate pair in the left half complex plane but are seen to move toward the imaginary axis with increasing b. At b = 0.557, the pair of complex eigenvalues cross the imaginary axis, indicating a Hopf bifurcation (HB)—the same is noted by the continuation algorithm and marked on the bifurcation diagram. As we have seen, a branch of limit cycles is expected to be created at the Hopf bifurcation point—either stable ones at a super- critical bifurcation or an unstable branch at a subcritical one. In this instance, the HB is subcritical and unstable limit cycles are formed—
these are also computed and plotted by the continuation algorithm.
Beyond the HB point, the equilibrium branch has unstable states—the eigenvalue pair has moved into the right half complex plane as seen in Figure 3.23b. In this regime, according to the model (Equation 3.28), small perturbations off the steady state will diverge to infinity. In practice, alternative steady states and bifurcations may be expected when nonlinearities in the aerodynamic model- ing, presently absent, are included. Nevertheless, the simple model in Equation 3.28 is of use to illustrate the use of the continuation and bifurcation method to a problem in flight dynamics.
To complement the bifurcation analysis, one can carry out time simulations for certain fixed values of the parameter b. Figure 3.24 shows the phase plane trajectories for four different values of b. It can be seen that for b < 0.557, trajectories within a certain bound spiral in to the stable steady state at γ* = 0, y* = 1. The boundary is deter- mined by the unstable limit cycle seen in Figure 3.23a. As b tends to the critical value of 0.577, the extent of the limit cycle shrinks and the domain within the limit cycle wherefrom trajectories spiral in to the steady state reduces as well. For the case of b = 0.7 in Figure 3.24, the trajectories spiral out the unstable steady state and tend toward increasing values of γ. However, γ being a periodic variable with bounds (−π,+π), one can imagine the trajectories to fold back along the X axis in Figure 3.24 and repeatedly pass from −π to +π. Effectively, the aircraft appears to be pitching nose-over and looping repeatedly—an unrealistic solution.
Homework Exercise: Carry out a bifurcation analysis with param- eter b held fixed and parameter a varied. Analyze your results.
0–3 0.5
1
y 1.5
2 2.5 3
–2 a = 0.2; b = 0.2
–1 0 1 2 3
γ
0 0.5 1
y 1.5
2 2.5 3
0 0.5 1
y 1.5
2 2.5 3
0 0.5 1
y 1.5
2 2.5 3
–3 –2 –1 0 1 2 3
–3 –2 –1 0 1 2 3
γ (rad)
γ (rad)
–3 –2 –1 0 1 2 3
γ (rad) a = 0.2; b = 0.41
a = 0.2; b = 0.5
a = 0.2; b = 0.7
FIGURE 3.24 Phase portrait results for different sets of parameter values (fixed a = 0.2; different b).
3.5 CONTINUATION FRAMEWORK FOR