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In this section we discuss another approach, known as Liapunov’s5 second method or direct method. The method is referred to as a direct method because no knowledge of the solution of the system of differential equations is required. Rather, conclusions about the stability or instability of a critical point are obtained by constructing a suitable auxiliary function. The technique is a very powerful one that provides a more global type of information, for example, an estimate of the extent of the basin of attraction of a critical point. In addition, Liapunov’s second method can also be used to study systems of equations that are not almost linear; however, we will not discuss such problems.
Basically, Liapunov’s second method is a generalization of two physical principles for conservative systems, namely, (i) a rest position is stable if the potential energy is a local minimum, otherwise it is unstable, and (ii) the total energy is a constant during any motion. To illustrate these concepts, again consider the undamped pendulum (a conservative mechanical system), which is governed by the equation
^ + g sin 9 = 0. (1)
The corresponding system of first order equations is
dx dy g .
dt = y dt = — L s‘n (2)
5Alexandr M. Liapunov (1857-1918), a student of Chebyshev at St. Petersburg, taught at the University of Kharkov from 1885 to 1901, when he became academician in applied mathematics at the St. Petersburg Academy of Sciences. In 1917 he moved to Odessa because of his wife’s frail health. His research in stability encompassed both theoretical analysis and applications to various physical problems. His second method formed part of his most influential work, General Problem of Stability of Motion, published in 1892.
Chapter 9. Nonlinear Differential Equations and Stability
where x = Q and y = dd/dt. If we omit an arbitrary constant, the potential energy U is the work done in lifting the pendulum above its lowest position, namely,
U(x, y) = mgL(1 — cos x); (3)
see Figure 9.2.2. The critical points of the system (2) are x = ±nn, y = 0, n = 0, 1, 2, 3,..., corresponding to Q = ±nn, dQ/dt = 0. Physically, we expect the points x = 0, y = 0; x = ±2n, y = 0;..., corresponding to Q = 0, ±2n,..., to be stable, since for them the pendulum bob is vertical with the weight down; further, we expect the points x = ±ir, y = 0; x = ±3n, y = 0;..., corresponding to Q = ±ir, ±3n,..., to be unstable, since for them the pendulum bob is vertical with the weight up. This agrees with statement (i), for at the former points U is a minimum equal to zero, and at the latter points U is a maximum equal to 2mgL.
Next consider the total energy V, which is the sum of the potential energy U and the kinetic energy ImL2(dQ/dt)2. In terms of x and y
V(x, y) = mgL(1 — cos x) + 2mL2y2. (4)
On a trajectory corresponding to a solution x = 0(t), y = ty(t) of Eqs. (2), V can be
considered as a function of t. The derivative of V[0(t),ty(t)] with respect to t is called
the rate of change of V following the trajectory. By the chain rule
dV¹(t), f(t)] V [J() ,()] d$(t) , V [J() ,()] df(t)
~z = Vx [<p(t), f(t)]—— + Vy [0(t), f(t)] ——
dt x dt y dt
dx 2 dy
= (mgL sin x)-------1- mL y —, (5)
where it is understood that x = 0(t), y = ty(t). Finally, substituting in Eq. (5) for dx/dt and dy/dt from Eqs. (2) we find that dV/dt = 0. Hence Vis a constant along any trajectory of the system (2), which is statement (ii).
It is important to note that at any point ( x, y) the rate of change of V along the trajectory through that point was computed without actually solving the system (2). It is precisely this fact that allows us to use Liapunov’s second method for systems whose solution we do not know, and is the main reason for its importance.
At the stable critical points, x = ±2nn, y = 0, n = 0, 1, 2,..., the energy V is zero. If the initial state, say, (x1, y^, of the pendulum is sufficiently near a stable critical point, then the energy Vx, yx) is small, and the motion (trajectory) associated with this energy stays close to the critical point. It can be shown that if V(x1, y^ is sufficiently small, then the trajectory is closed and contains the critical point. For example, suppose that x, y^ is near (0, 0) and that V(x1, y^ is very small. The equation of the trajectory with energy V(x1, y^ is
V(x, y) = mgL(1 — cosx) + 2mL2y2 = V(x1, y1).
For x small we have 1 — cosx = 1 — (1 — x2/2! + •••) = x2/2. Thus the equation of the trajectory is approximately
2mgLx2 + 1 mL2y2 = Vx, y1),
2 V(x1, y1)/mgL 2 V(x1, yx)/mL
9.6 Liapunov’s Second Method
This is an ellipse enclosing the critical point (0, 0); the smaller V(xv yx) is, the smaller are the major and minor axes of the ellipse. Physically, the closed trajectory corresponds to a solution that is periodic in time—the motion is a small oscillation about the equilibrium point.
If damping is present, however, it is natural to expect that the amplitude of the motion decays in time and that the stable critical point (center) becomes an asymptotically stable critical point (spiral point). See the phase portrait for the damped pendulum in Figure 9.3.5. This can almost be argued from a consideration of dV/dt. For the damped pendulum, the total energy is still given by Eq. (4), but now from Eqs. (8) of Section9.3 dx/dt = y and dy/dt = — (g/ L) sin x — (c/Lm) y. Substituting for dx/dt and dy/dt in Eq. (5) gives dV/dt = — cLy2 < 0. Thus the energy is nonincreasing along any trajectory and, except for the line y = 0, the motion is such that the energy decreases. Hence each trajectory must approach a point of minimum energy—a stable equilibrium point. If dV/dt < 0 instead of dV/dt < 0, it is reasonable to expect that this would be true for all trajectories that start sufficiently close to the origin.