Download (direct link):
-8/3 - ê
= -(8/3 + ê)[ê2 + 11ê - 10(r - 1)] = 0. (6)
534 Chapter 9. Nonlinear Differential Equations and Stability
8 —11 — V81 + 40r —11 + V81 + 40r _
ß, — —, X2 —-----------------------, k3 —-------------------------. (7)
1 3 2 2 3 2 w
Note that all three eigenvalues are negative for r < 1; for example, when r = 1/2, the eigenvalues are Ë1 = —8/3, X2 = —10.52494, X3 = —0.47506. Hence the origin is asymptotically stable for this range of r both for the linear approximation (5) and for the original system (1). However, X3 changes sign when r = 1 and is positive for r > 1. The value r = 1 corresponds to the initiation of convective flow in the physical problem described earlier. The origin is unstable for r > 1; all solutions starting near the origin tend to grow except for those lying precisely in the plane determined by the eigenvectors associated with Ë1 and X2 [or, for the nonlinear system (1), in a certain surface tangent to this plane at the origin].
Next let us consider the neighborhood of the critical point P2(V8(r — 1)/3, V8(r — 1)/3, r — 1) for r > 1. If u, v, and w are the perturbations from the critical point in the x, y, and z directions, respectively, then the approximating linear system is
u ó ( —10 10 0
v ² = | 1 —1 —V8(r — 1)/3 I I v I . (8)
w \V8(r — 1)/3 V8(r — 1)/3 —8/3
The eigenvalues of the coefficient matrix of Eq. (8) are determined from the equation
3A3 + 4U2 + 8(r + 10)A + 160(r — 1) = 0, (9)
which is obtained by straightforward algebraic steps that are omitted here. The solutions of Eq. (9) depend on r in the following way:
For 1 < r < r1 = 1.3456 there are three negative real eigenvalues.
For r1 < r < r2 = 24.737 there are one negative real eigenvalue and two complex eigenvalues with negative real part.
For r2 < r there are one negative real eigenvalue and two complex eigenvalues with positive real part.
The same results are obtained for the critical point P3. Thus there are several different situations.
For0 < r < 1 the only critical point is P1 and it is asymptotically stable. All solutions approach this point (the origin) as t — to.
For 1 < r < r1 the critical points P2 and P3 are asymptotically stable and P1 is unstable. All nearby solutions approach one or the other of the points P2 and P3 exponentially.
For r1 < r < r2 the critical points P2 and P3 are asymptotically stable and P1 is unstable. All nearby solutions approach one or the other of the points P2 and P3; most of them spiral inward to the critical point.
For r2 < r all three critical points are unstable. Most solutions near P2 or P3 spiral away from the critical point.
However, this is by no means the end of the story. Let us consider solutions for r somewhat greater than r2. In this case P1 has one positive eigenvalue and each of P2 and P3 has a pair of complex eigenvalues with positive real part. A trajectory can approach any one of the critical points only on certain highly restricted paths. The
9.8 Chaos and Strange Attractors: The Lorenz Equations
slightest deviation from these paths causes the trajectory to depart from the critical point. Since none of the critical points is stable, one might expect that most trajectories will approach infinity for large t. However, it can be shown that all solutions remain bounded as t ^to; see Problem 5. In fact, it can be shown that all solutions ultimately approach a certain limiting set of points that has zero volume. Indeed, this is true not only for r > r2 but for all positive values of r.
A plot of computed values of x versus t for a typical solution with r > r2 is shown in Figure 9.8.2. Note that the solution oscillates back and forth between positive and negative values in a rather erratic manner. Indeed, the graph of x versus t resembles a random vibration, although the Lorenz equations are entirely deterministic and the solution is completely determined by the initial conditions. Nevertheless, the solution also exhibits a certain regularity in that the frequency and amplitude of the oscillations are essentially constant in time.
The solutions of the Lorenz equations are also extremely sensitive to perturbations in the initial conditions. Figure 9.8.3 shows the graphs of computed values of x versus t
FIGURE 9.8.2 A plot of x versus t for the Lorenz equations (1) with r = 28; initial point is (5, 5, 5).
FIGURE 9.8.3 Plots of x versus t for two initially nearby solutions of Lorenz equations with r = 28; initial point is (5, 5, 5) for dashed curve and (5.01, 5, 5) for solid curve.