Hello everyone, As promised, the chart mentioned in class about what kind of critical points you get with different types of eigenvalues are included on the following pages (The pages are an ecerpt from Edwards and Penny s Differential equations tet). The chart summarizes the situation for the case where (0, 0) is a critical point. For all of our cases, the system of differential equations is linear, with the eception of the last section. For the non-linear problem, you can linearize and get the equilibrium points and locally (in a small region around the critical point), the phase portrait will resemble the corresponding picture. Also included are some eamples of each kind so you can get an idea of what the homework is asking you to sketch for the phase portrait. On the last page, there are some theorems about stability of the critical points which we will need later, so included that page as well for your reference. hope this will clear up some of the material related to the phase portraits. Best, Josh 1
500 Chapter 7 Nonlinear Systems and Phenomena we plot trajectories of the system 2 FGURE 7.2.21. Phase portrait for the system in Eq. (). 2... 0-1 -2 - - - - 2-1 FGURE 7.2.22. Phase portrait for the system corresponding to Eq. (5). d - = 2-2y, The result is shown in Fig. 7.2.21. 1. 2... s. dy 2 - = 2y - y. Plot similarly some solution curves for the following differential equations. dy - 5y - = d 2 + y dy - 5y = d 2 - y dy d - 2-5y dy 2y d - _ 2 y2 dy 2 + 2y = d y2 + 2y - y Now construct some eamples of your own. Homogeneous functions like those in Problems 1 through 5-rational functions with numerator and denominator of the same degree in and y-work well. The differential equation dy 25 + y(1-2 - y2)( _ 2 _ - d _ -25y + (1-2 - y2)( - 2 of this form generalizes Eample 6 in this section but would be inconvenient to solve eplicitly. ts phase portrait (Fig. 7.2.22) shows two periodic closed trajectoriesthe circles r = 1 and r = 2. Anyone want to try for three circles? y2) y2) () (5) We now discuss the behavior of solutions of the autonomous system d dy =!(, y), - = g(, y) near an isolated critical point (o, Yo) where!(o, Yo) = g(o, Yo) = O. A critical point is called isolated if some neighborhood of it contains no other critical point. We assume throughout that the functions! and g are continuously differentiable in a neighborhood of (o, Yo). We can assume without loss of generality that Xo = Yo = O. Otherwise, we make the substitutions u = - Xo, v = y - Yo. Then d/ = du/ and dy/ = dv/, so (1) is equivalent to the system du - =!(u + o, v + Yo) =!\ (u, v), dv - = g(u + o, v + Yo) = g\ (u, v) that has (0, 0) as an isolated critical point. (1) (2)
7. Linear and Almost Linear Systems 501 Ea m ple 1 The system d 2 - = - - y = ( - - y), dy = y + y 2 _ y = y(1 - + y) has (1, 2) as one of its critical points. We substitute u = -, v = y - 2; that is, = u +, y = v + 2. Then and - - y = - (u + 1) - (v + 2) = -u - v 1 - + y = 1 - (u + 1) + (v + 2) = -u + v, so the system in () takes the form () (1, 2) for the system ' = - X2 y' - y, = y + y2 y _ FGURE 7..1. The saddle point of Eample 1. du 2 - = (u + 1)(-u - v) = -u - v - u - uv, dv = (v + 2) ( -u + v) = -6u + 2v + v2 - uv and has (0, 0) as a critical point. f we can determine the trajectories of the system in () near (0, 0), then their translations under the rigid motion that carries (0, 0) to (1, 2) will be the trajectories near (1, 2) of the original system in (). This equivalence is illustrated by Fig. 7.. 1 (which shows computer-plotted trajectories of the system in () near the critical point (1, 2) in the y-plane) and Fig. 7..2 (which shows computer-plotted trajectories of the system in () near the critical point (0, 0) in the uv-plane). () Figures 7.. 1 and 7..2 illustrate the fact that the solution curves of the ysystem in (1) are simply the images under the translation (u, v) (u + o, v + Yo) of the solution curves of the uv-system in (2). Near the two corresponding critical points-(o, Yo) in the y-plane and (0, 0) in the uv-plane-the two phase portraits therefore look precisely the same. FGURE 7..2. The saddle point (0, 0) for the equivalent system u' = -u - v - u2 - uv, v' = -6u + 2v + v2 - uv. u Linearization Near a Critical Point Taylor's formula for functions of two variables implies that-if the function f (, y) is continuously differentiable near the fied point (o, Yo)-then f(o + u, Yo + v) = f(o, Yo) + f Co, Yo)u + fy (o, Yo)v + r(u, v) where the "remainder term" r(u, v) satisfies the condition r(u, v) lim (U, v)--+ (O,O) + v2 = o. (Note that this condition would not be satisfied if r(u, v) were a sum containing either constants or terms linear in u or v. n this sense, r(u, v) consists of the "nonlinear part" of the function f (o + u, Yo + v) of u and v.)
502 Chapter 7 Nonlinear Systems and Phenomena f we apply Taylor's formula to both f and g in (2) and assume that (o, Yo) is an isolated critical point so f(o, Yo) = g(o, Yo) = 0, the result is du = f (o, yo)u + f y (o, yo)v + r(u, v), dv = g (o, yo)u + g y(o, yo)v + s (u, v) where r (u, v) and the analogous remainder term s (u, v) for g satisfy the condition r(u, v) lim (U, v) -+ (O,O) + v2 s(u, v) lim = O. (u, v)-+ (O,O) + v2 Then, when the values u and v are small, the remainder terms r(u, v) and s(u, v) are very small (being small even in comparison with u and v). f we drop the presumably small nonlinear terms r(u, v) and s(u, v) in (5), the result is the linear system du = f Co, yo)u + f y(o, yo)v, (7) dv = g (o, yo)u + g y(o, yo)v whose constant coefficients (of the variables u and v) are the values f (o, Yo), fy(o, Yo) and g Co, Yo), gy (o, Yo) of the functions f and g at the critical point (o, Yo). Because (5) is equivalent to the original (and generally) nonlinear system u' = f(o + u, Yo + v), v' = g(o + u, Yo + v) in (2), the conditions in (6) suggest that the linearized system in (7) closely approimates the given nonlinear system when (u, v) is close to (0, 0). Assuming that (0, 0) is also an isolated critical point of the linear system, and that the remainder terms in (5) satisfy the condition in (6), the original system ' = f (, y), y' = g (, y) is said to be almost linear at the isolated critical r point (o, Yo). n this case, its linearization at (o, Yo) is the linear system in (7). n short, this linearization is the linear system u ' = Ju (where u = [ u v ) whose coefficient matri is the so-called Jacobian matri of the functions f and g, evaluated at the point (o, Yo). (5) (6) (8) Ea mple 1 Continued J(, y) = [ - 2 - y - ] [ - 1-1 ] -y 1 + 2y - ' so J(1, 2) = -6 2 ' Hence the linearization of the system ' = - 2 - y, y' = y + y 2 - y at its critical point ( 1, 2) is the linear system U = -u - v, v' = -6u + 2v that we get when we drop the nonlinear (quadratic) terms in ().
7. Linear and Almost Linear Systems 50 t turns out that in most (though not all) cases, the phase portrait of an almost linear system near an isolated critical point (o, Yo) strongly resemblesqualitatively-the phase portrait near the origin of its linearization. Consequently, the first step toward understanding general autonomous systems is to characterize the critical points of linear systems. Critical Points of Linear Systems We can use the eigenvalue-eigenvector method of Section 5. to investigate the critical point (0, 0) of a linear system (9) with constant-coefficient matri A. Recall that the eigenvalues A and A2 of A are the solutions of the characteristic equation det(a - A) = a - A b c d _ A = (a - A)(d - A) - bc = O. We assume that (0, 0) is an isolated critical point of the system in (9), so it follows that the coefficient determinant ad - bc of the system a + by = 0, c + dy = 0 is nonzero. This implies that A = 0 is not a solution of (9), and hence that both eigenvalues of the matri A are nonzero. The nature of the isolated critical point (0, 0) then depends on whether the two nonzero eigenvalues A and A2 of A are real and unequal with the same sign; real and unequal with opposite signs; real and equal; comple conjugates with nonzero real part; or pure imaginary numbers. u - - - (u, v) _ - -1 V FGURE 7... The oblique u v-coordinate system determined by the eigenvectors V and V2. These five cases are discussed separately. n each case the critical point (0, 0) resembles one of those we saw in the eamples of Section 7.2-a node (proper or improper), a saddle point, a spiral point, or a center. UNEQUAL REAL EGENVALUES r WTH THE SAME SGN : n this case the matri A has linearly independent eigenvectors V and V2, and the general solution (t) = [ (t) yet) of (9) takes the form This solution is most simply described in the oblique uv-coordinate system indicated in Fig. 7.., in which the u- and v-aes are determined by the eigenvectors V and V2. Then the uv-coordinate functions u (t) and vet) of the moving point (t) are simply its distances from the origin measured in the directions parallel to the vectors V and V2, so it follows from Eq. (10) that a trajectory of the system is described by where Uo = u (O) and Vo = v(o). f Vo = 0, then this trajectory lies on the u ais, whereas if Uo = 0, then it lies on the v-ais. Otherwise-if Uo and Vo are (10) (1 1)
50 Chapter 7 Nonlinear Systems and Phenomena both nonzero-the parametric curve in (1 1) takes the eplicit form v = Cu k where k = A2/)"1 > O. These solution curves are tangent at (0, 0) to the u-ais if k > 1, to the v-ais if 0 < k < 1. Thus we have in this case an improper node as in Eample of Section 7.2. f A and A2 are both positive, then we see from (10) and (1 1) that these solution curves "depart from the origin" as t increases, so (0, 0) is a nodal source. But if Al and A2 are both negative, then these solution curves approach the origin as t increases, so (0, 0) is a nodal sink. 2 1 - - 2 - Ea mple 2 FGURE 7... The improper nodal source of Eample 2. 5 (a) The matri A = g [ 7-1 ] has eigenvalues Al = 1 and A2 = 2 with associated eigenvectors V = [ 1 f and V 2 = [ 1 f. Figure 7.. shows a direction field and typical trajectories of the corresponding linear system ' = A. Note that the two eigenvectors point in the directions of the linear trajectories. As is typical of an improper node, all other trajectories are tangent to one of the oblique aes through the origin. n this eample the two unequal real eigenvalues are both positive, so the critical point (0, 0) is an improper nodal source. (b) The matri B = -A = g [ -7 -] -17 has eigenvalues Al = - 1 and A2 = -2 with the same associated eigenvectors V = [ 1 f and V2 = [ 1 f. The new linear system ' = B has the same direction field and trajectories as in Fig. 7.. ecept with the direction field arrows now all reversed, so (0, 0) is now an improper nodal sink. -2 - - FGURE 7..5. The saddle point of Eample. Ea m ple 5 UNEQUAL REAL EGENVALUES WTH OPPOSTE SGNS : Here the situation is the same as in the previous case, ecept that A2 < 0 < Al in (1 1). The trajectories with Uo = 0 or Vo = 0 lie on the u- and v-aes through the critical point (0, 0). Those with Uo and Vo both nonzero are curves of the eplicit form v = Cu k, where k = A2/A < O. As in the case k < 0 of Eample in Section 7.2, the nonlinear trajectories resemble hyperbolas, and the critical point (0, 0) is therefore an unstable saddle point. _.. The matri -] has eigenvalues Al = 1 and A2 = - 1 with associated eigenvectors V = [ 1 f and V 2 = [ 1 f. Figure 7..5 shows a direction field and typical trajectories of the corresponding linear system ' = A. Note that the two eigenvectors again point in the directions of the linear trajectories. Here k = - 1 and the nonlinear trajectories are (true) hyperbolas in the oblique uv-coordinate system, so we have the saddle point indicated in the figure. Note that the two eigenvectors point in the directions of the asymptotes to these hyperbolas. -5
Ea m ple 7. Linear and Almost Linear Systems 505 EQUAL REAL ROOTS : n this case, with A = A = A2 i= 0, the character of the critical point (0, 0) depends on whether or not the coefficient matri A has two linearly independent eigenvectors V and V2. f so, then we have oblique uvcoordinates as in Fig. 7.., and the trajectories are described by u (t) = uoe A t, vet) = voeat (12) as in (1 1). But now k = A2/A =, so the trajectories with Uo i= 0 are all of the form v = Cu and hence lie on straight lines through the origin. Therefore, (0, 0) is a proper node (or star) as illustrated in Fig. 7.2., and is a source if A > 0, a sink if A < O. f the multiple eigenvalue A i= 0 has only a single associated eigenvector V, then (as we saw in Section 5.6) there nevertheless eists a generalized eigenvector V2 such that (A - AJ)V2 = V, and the linear system ' = A has the two linearly independent solutions We can still use the two vectors V and V2 to introduce oblique uv-coordinates as in Fig. 7... Then it follows from (1) that the coordinate functions u(t) and vet) of the moving point (t) on a trajectory are given by (1) u (t) = (uo + vot)e A t, vet) = voe A t, (1) where Uo = u(o) and Vo = v(o). f Vo = 0 then this trajectory lies on the u-ais. Otherwise we have a nonlinear trajectory with dv dv/ AvOeAt AVO = = du du/ voeat + A(UO + vot)eat Vo + A(UO + vot) We see that dv/du 0 as t ±oo, so it follows that each trajectory is tangent to the u-ais. Therefore, (0, 0) is an improper node. f A < 0, then we see from (1) that this node is a sink, but it is a source if A > O. - - - _._-. _- _. _. _- --_.. _- - -._----_._.._.._. _. ---_. _ --------- -- - -_......_.._ - _.... - ----- - _.. -_. The matri 9J A _.! [ - l 1-8 - 1-5 '" 0-1 - 2 - - FGURE 7..6. The improper nodal sink of Eample. has the multiple eigenvalue A = - 1 with the single associated eigenvector V = [ 1 r. t happens that V2 = [ 1 r is a generalized eigenvector based on VJ, but only the actual eigenvector shows up in a phase portrait for the linear system ' = A. As indicated in Fig. 7..6, the eigenvector V determines the u-ais through the improper nodal sink (0, 0), this ais being tangent to each of the nonlinear trajectories. COMPLEX CONJUGATE EGENVALUES : Suppose that the matri A has eigenvalues A = P + qi and = p - qi (with p and q both nonzero) having associated comple conjugate eigenvectors V = a + bi and v = a - bi. Then we saw in Section 5.-see Eq. (22) there-that the linear system ' = A has the two independent real-valued solutions (t) = ept (a cos qt - b sin qt) and 2(t) = ept (b cos qt + a sin qt). (15)
506 Chapter 7 Nonlinear Systems and Phenomena Ea m ple 5 Thus the components (t) and yet) of any solution (t) = CX (t) +C2X2(t) oscillate between positive and negative values as t increases, so the critical point (0, 0) is a spiral point as in Eample 5 of Section 7.2. f the real part p of the eigenvalues is negative, then it is clear from (15) that (t) 0 as t +00, so the origin is a spiral sink. But if p is positive, then the critical point is a spiral source. The matri 1 [ - 10 A = - 15 ] 1... 0-1 - 2 - has the comple conjugate eigenvalues A = - ± i with negative real part, so (0, 0) is a spiral sink. Figure 7..7 shows a direction field and a typical spiral trajectory approaching the origin as t +00. PURE MAGNARY EGENVALUES : f the matri A has conjugate imaginary eigenvalues A = qi and = -qi with associated comple conjugate eigenvectors v = a + bi and v = a - bi, then (15) with p = 0 gives the independent solutions Xl (t) = a cos qt - b sin qt and X2(t) = b cos qt + a sin qt (16) FGURE 7..7. The spiral sink of Eample 5.... 5 2-1 - 2 - Ea mple 6 of the linear system ' = A. Just as in Eample of Section 7.2, it follows that any solution (t) = CX (t) + C2X2(t) describes an ellipse centered at the origin in the y-plane. Hence (0, 0) is a stable center in this case. The matri 1 [ -9 A = - 15 has the pure imaginary conjugate eigenvalues A = ±i, and therefore (0, 0) is a stable center. Figure 7..8 shows a direction field and typical elliptical trajectories enclosing the critical point. For the two-dimensional linear system ' = A with det A ==- 0, the table in Fig. 7..9 lists the type of critical point at (0, 0) found in the five cases discussed here, according to the nature of the eigenvalues A 1 and A2 of the coefficient matri A. Our discussion of the various cases shows that the stability of the critical point (0, 0) is determined by the signs of the real parts of these eigenvalues, as summarized in Theorem. Note that if Al and A2 are real, then they are themselves their real parts. ] FGURE 7..8. The stable center of Eample 6. Real, unequal, same sign Real, unequal, opposite sign Real and equal Comple conjugate Pure imaginary mproper node Saddle point Proper or improper node Spiral point Center FGURE 7..9. Classification of the critical point (0, 0) of the two-dimensional system ' = A.
7. Linear and Almost Linear Systems 507 TH EOREM 1 Stability of Linear Systems Let A 1 and A2 be the eigenvalues of the coefficient matri A of the twodimensional linear system with ad - d = a + by, dy = e + dy be : O. Then the critical point (0, 0) is 1. Asymptotically stable if the real parts of Al and A2 are both negative; 2. Stable but not asymptotically stable if the real parts of Al and A2 are both zero (so that A, A2 = ±qi);. Unstable if either A 1 or A2 has a positive real part. (17) 1-11 = r + si FGURE 7..10. The effects of perturbation of pure imaginary roots. FGURE 7..11. The effects of perturbation of real equal roots. t is worthwhile to consider the effect of small perturbations in the coefficients a, b, e, and d of the linear system in (17), which result in small perturbations of the eigenvalues A 1 and A2. f these perturbations are sufficiently small, then positive real parts (of A 1 and A2) remain positive and negative real parts remain negative. Hence an asymptotically stable critical point remains asymptotically stable and an unstable critical point remains unstable. Part 2 of Theorem is therefore the only case in which arbitrarily small perturbations can affect the stability of the critical point (0, 0). n this case pure imaginary roots At. A2 = ±qi of the characteristic equation can be changed to nearby comple roots Lt. L2 = r ± si, with r either positive or negative (see Fig. 7.. 10). Consequently, a small perturbation of the coefficients of the linear system in (7) can change a stable center to a spiral point that is either unstable or asymptotically stable. There is one other eceptional case in which the type, though not the stability, of the critical point (0, 0) can be altered by a small perturbation of its coefficients. This is the case with Al = A2, equal roots that (under a small perturbation of the coefficients) can split into two roots L 1 and L2, which are either comple conjugates or unequal real roots (see Fig. 7..1 1). n either case, the sign of the real parts of the roots is preserved, so the stability of the critical point is unaltered. ts nature may change, however; the table in Fig. 7..9 shows that a node with Al = A2 can either remain a node (if Ll and L2 are real) or change to a spiral point (if Ll and L2 are comple conjugates). Suppose that the linear system in (17) is used to model a physical situation. t is unlikely that the coefficients in (17) can be measured with total accuracy, so let the unknown precise linear model be d - = a * + b * y, dy - = e * + d * (17 * ) y. f the coefficients in (17) are sufficiently close to those in (17*), it then follows from the discussion in the preceding paragraph that the origin (0, 0) is an asymptotically stable critical point for (17) if it is an asymptotically stable critical point for (17*), and is an unstable critical point for (17) if it is an unstable critical point for (17*). Thus in this case the approimate model in (17) and the precise model in (17 * )