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9.2 Autonomous Systems and Stability
FIGURE 9.1.9 Stability diagram.
9.2 Autonomous Systems and Stability
In this section we begin to draw together and to expand on the geometrical ideas introduced in Section 2.5 for certain first order equations and in Section 9.1 for second order linear homogeneous systems with constant coefficients. These ideas concern the qualitative study of differential equations and the concept of stability, an idea that will be defined precisely later in this section.
Autonomous Systems. We are concerned with systems of two simultaneous differential equations of the form
dx/dt = F(x, y), dy/dt = G(x, y). (1)
We assume that the functions F and G are continuous and have continuous partial derivatives in some domain D of the xy-plane. If (x0, y0) is a point in this domain, then
by Theorem 7.1.1 there exists a unique solution x = ˘(│), y = ty(t) of the system (1)
satisfying the initial conditions
x (t0) = x0> y(t0) = y0 Ž (2)
The solution is defined in some time interval I that contains the point t0.
Frequently, we will write the initial value problem (1), (2) in the vector form
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 + y0 j. In this case the solution is expressed as x = ˘() where ˘(0 = ˘(│)│ + ty(t)j. As usual, we interpret a solution x = ˘^) 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 y. 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, y0) 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 t0 at which they pass through (x0, y0). 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 e > 0, there is a S > 0 such that every solution x = ˘(^ of the system (1), which at t = 0 satisfies
||˘(0) - x░|| <S, (6)
exists for all positive t and satisfies
▓▓ď(0 - x0|| < e (7)
for all t > 0. This is illustrated geometrically in Figures 9.2.1a and 9.2.16. These
mathematical statements say that all solutions that start ôsufficiently closeö (that is,