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dx/dt = f(x), x(t0) = x0, (3)
where x = xi + yj, f(x) = F(x, y)i + G(x, y)j, and x0 = x0i + y0j. In this case the solution is expressed as x = ^(t), where ^(t) = 0(t)i + ^(t)j. As usual, we interpret a solution x = ^(t) as a curve traced by a moving point in the xy-plane, the phase plane.
Chapter 9. Nonlinear Differential Equations and Stability
Observe that the functions F and G in Eqs. (1) do not depend on the independent variable t, but only on the dependent variables x and /. A system with this property is said to be autonomous. The system
x' = Ax, (4)
where A is a constant matrix, is a simple example of a two-dimensional autonomous system. On the other hand, if one or more of the elements of the coefficient matrix A is a function of the independent variable t, then the system is nonautonomous. The distinction between autonomous and nonautonomous systems is important because the geometric qualitative analysis in Section 9.1 can be effectively extended to two-dimensional autonomous systems in general, but is not nearly as useful for nonautonomous systems.
In particular, the autonomous system (1) has an associated direction field that is independent of time. Consequently, there is only one trajectory passing through each point (X0, /0) in the phase plane. In other words, all solutions that satisfy an initial condition of the form (2) lie on the same trajectory, regardless of the time /0 at which they pass through X, /0). Thus, just as for the constant coefficient linear system (4), a single phase portrait simultaneously displays important qualitative information about all solutions of the system (1). We will see this fact confirmed repeatedly in this chapter.
Autonomous systems occur frequently in applications. Physically, an autonomous system is one whose configuration, including physical parameters and external forces or effects, is independent of time. The response of the system to given initial conditions is then independent of the time at which the conditions are imposed.
Stability and Instability. The concepts of stability, asymptotic stability, and instability have already been mentioned several times in this book. It is now time to give a precise mathematical definition of these concepts, at least for autonomous systems of the form
x' = f(x). (5)
In the following definitions, and elsewhere, we use the notation ||x|| to designate the length, or magnitude, of the vector x.
The points, if any, where f(x) = 0 are called critical points of the autonomous system (5). At such points x' = 0 also, so critical points correspond to constant, or equilibrium, solutions of the system of differential equations. A critical point x0 of the system (5) is said to be stable if, given any ˆ > 0, there is a S > 0 such that every solution x = $(t) of the system (1), which at t = 0 satisfies
IIW0) - x°|| <S, (6)
exists for all positive t and satisfies
HWO - x°|| < ˆ (7)
for all t > 0. This is illustrated geometrically in Figures 9.2.1a and 9.2.1b. These mathematical statements say that all solutions that start “sufficiently close” (that is, within the distance S) to x0 stay “close” (within the distance ˆ) to x0. Note that in Figure
9.2.1a the trajectory is within the circle ||x - x°|| = S at t = 0 and, while it soon passes outside of this circle, it remains within the circle ||x - x01| = ˆ for all t > 0. However, the trajectory of the solution does not have to approach the critical point x0 as t ^ to, as illustrated in Figure 9.2.1b. A critical point that is not stable is said to be unstable.
9.2 Autonomous Systems and Stability
FIGURE 9.2.1 (a) Asymptotic stability. (b) Stability.
A critical point x0 is said to be asymptotically stable if it is stable and if there exists a 5°, with 0 < 5° <5, such that if a solution x = $(t) satisfies
||^(0) - x°|| <V (8)
lim $(t) = x°. (9)
Thus trajectories that start “sufficiently close” to x° must not only stay “close” but must eventually approach x° as t ^ to. This is the case for the trajectory in Figure
9.2.1a but not for the one in Figure 9.2.1b. Note that asymptotic stability is a stronger property than stability, since a critical point must be stable before we can even talk about whether it might be asymptotically stable. On the other hand, the limit condition (9), which is an essential feature of asymptotic stability, does not by itself imply even ordinary stability. Indeed, examples can be constructed in which all of the trajectories approach x° as t ^ to, but for which x° is not a stable critical point. Geometrically,
all that is needed is a family of trajectories having members that start arbitrarily close
to x°, then depart an arbitrarily large distance before eventually approaching x° as t ^ TO.
While we specified originally that the system (5) is of second order, the definitions just given are independent of the order of the system. If you interpret the vectors in Eqs.
(5) through (9) as n-dimensional, then the definitions of stability, asymptotic stability, and instability apply also to nth order systems. These definitions can be made more concrete by interpreting them in terms of a specific physical problem.