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0(a) = x0, 0(a) = y0. Then use the fact that x = x0, y = y0 is a solution of the given
system satisfying the initial condition x = x0, y = y0 at t = a.
26. Assuming that the trajectory corresponding to a solution x = 0(t), y = 0 (t), m < t < to, of an autonomous system is closed, show that the solution is periodic.
9.3 Almost Linear Systems
Hint: Since the trajectory is closed, there exists at least one point (x0, y0) such that 0 (t0) x0, 0(t0) y0 and a number T > 0 such that 0 (t0 + T) x0, x^(t0 + T) y0. Show that x ฎ(t) 0(t + T) and y ^(t) 0(t + T) is a solution and then use the existence and uniqueness theorem to show that ฎ (t) 0(t) and ^(t) 0 (t) for all t.
9.3 Almost Linear Systems
In Section 9.1 we gave an informal description of the stability properties of the equilibrium solution x 0 of the two-dimensional linear system
X Ax. (1)
ฎThe results are summarized in Table 9.1.1. Recall that we required that det A 0, so that x 0 is the only critical point of the system (1). Now that we have defined the concepts of asymptotic stability, stability, and instability more precisely, we can restate these results in the following theorem.
Theorem 9.3.1 The critical point x 0 of the linear system (1) is asymptotically stable if the eigenvalues rx, /2 are real and negative or have negative real part; stable, but not asymptotically stable, if /1 and /2 are pure imaginary; unstable if /1 and /2 are real and either is positive, or if they have positive real part.
It is apparent from this theorem or from Table 9.1.1 that the eigenvalues /1, /2 of the coefficient matrix A determine the type of critical point at x 0 and its stability characteristics. In turn, the values of /1 and /2 depend on the coefficients in the system
(1). When such a system arises in some applied field, the coefficients usually result from the measurements of certain physical quantities. Such measurements are often subject to small uncertainties, so it is of interest to investigate whether small changes (perturbations) in the coefficients can affect the stability or instability of a critical point and/or significantly alter the pattern of trajectories.
Recall that the eigenvalues /1, /2 are the roots of the polynomial equation
det(A /I) 0. (2)
It is possible to show that small perturbations in some or all the coefficients are reflected in small perturbations in the eigenvalues. The most sensitive situation occurs when /1 ip and /2 iix, that is, when the critical point is a center and the trajectories are closed curves surrounding it. If a slight change is made in the coefficients, then the eigenvalues /1 and /2 will take on new values /1 X' + ip' and /2 X' ip!, where X' is small in magnitude and p' = p (see Figure 9.3.1). If X' 0, which almost always occurs, then the trajectories of the perturbed system are spirals, rather than closed curves. The system is asymptotically stable if X' < 0, but unstable if X' > 0. Thus, in the case of a center, small perturbations in the coefficients may well change a stable system into an unstable one, and in any case may be expected to alter radically the pattern of trajectories in the phase plane (see Problem 25).
Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.3.1 Schematic perturbation of r1 = ip, r2 = ip.
Another slightly less sensitive case occurs if the eigenvalues rx and r2 are equal; in this case the critical point is a node. Small perturbations in the coefficients will normally cause the two equal roots to separate (bifurcate). If the separated roots are real, then the critical point of the perturbed system remains a node, but if the separated roots are complex conjugates, then the critical point becomes a spiral point. These two possibilities are shown schematically in Figure 9.3.2. In this case the stability or instability of the system is not affected by small perturbations in the coefficients, but the trajectories may be altered considerably (see Problem 26).
In all other cases the stability or instability of the system is not changed, nor is the type of critical point altered, by sufficiently small perturbations in the coefficients of the system. For example, if rx and /2 are real, negative, and unequal, then a smaJJ change in the coefficients will not change the sign of r1 and r2 nor allow them to coalesce. Thus the critical point remains an asymptotically stable node.
Now let us consider a nonlinear two-dimensional autonomous system
X = f(x). (3)
Our main object is to investigate the behavior of trajectories of the system (3) near a critical point x0. We will seek to do this by approximating the nonlinear system (3) by an appropriate linear system, whose trajectories are easy to describe. The crucial question is whether the trajectories of the linear system are good approximations to those of the nonlinear system. Of course, we also need to know how to find the approximating linear system.
It is convenient to choose the critical point to be the origin. This involves no loss of generality, since if x0 = 0, it is always possible to make the substitution u = x x0 in Eq. (3). Then u will satisfy an autonomous system with a critical point at the origin.