Chapter 9, Sections 9.1, 9.2, 9.3, 9.4 only" 9.1 Stability and the Phase Plane ==== EXAMPLE 1. x'=14x-2x^2-xy, y'=16y-2y^2-xy PHASE PORTRAITS The actual rules for drawing threaded trajectories don't appear here. A trajectory passes a nearby field vector with visually equal tangent vector. Solutions don't cross. Crossing of field vectors is disallowed, because to graphic resolution they are themselves trajectories. EXAMPLE 2. x'=x-y, y'=1-x^2 EXAMPLE 3. x'=-x, y'=-ky The trajectories DO NOT pass through the critical point and out the other side. They limit (one-sided, if you wish) to the critical point at infinity and/or minus infinity. NODE was used back in chapter 2 to describe "not a funnel and not a spout." Usage in Ch9 is similar, but not a duplicate notion. DEF. node, proper node, improper node, sink, source, saddle point. Keywords funnel, spout from chapter 2 are not used for phase portaits. They have analogs, the spirals. DEF. stable, unstable. EXAMPLE 4. Harmonic oscillator in phase, x'=y, y'=-omega^2 x Computer software will show concentric circles instead of ellipses. Re-emphasize here that solutions don't cross. Similarly, a spiral on a computer monitor may look like an improper node. DEF. center, asymptotically stable, spiral EXAMPLE 5. Spiral x'=y, y'=-2x-2y from a spring-mass system The example has a bad phase portrait. For illustration, I use x'=y, y'=-3x-(1/3)y, whose spiral phase portrait has 7 wraps about (0,0) on window [-1,1]x[-1,1]. EXAMPLE 6. x'=-ky+x(1-x^2-y^2), y'=kx+y(1-x^2-y^2) FOUR POSSIBILITIES A reference should appear to Poincare. 9.1 Application ==== The examples produce stunning pictures. Some faculty and grad students still use PPLANE and DFIELD. The others prefer built-ins in maple and mathematica. Matlab is heavily used in engineering departments. Maple and mathematica packages exist for making phase portraits. The browser-based PPLANE (Polking) has many useful features, better than the CAS packages. ======================================================================== 9.2 Linear and Almost Linear Systems ==== The next example will illustrate substitutions that replace a critical point (x0,y0) by (0,0). EXAMPLE 1. x'=x(3-x-y), y'=y(1-3x-3y) and via u=x-1, v=y-2 the translated equations u'=-(u+1)(u+v), v'=(v+2)(-3u+v). The idea to be communicated is pasting one image over another, critical points superimposed. FIGURE 9.2.1 is created first, then FIGURE 9.2.2 by over-paste. LINEARIZATION near a Critical Point EXAMPLE 1a. x'=x(3-x-y), y'=y(1-3x-3y) linearized at (1,2) Reference should be to equation (7) instead of (4), which is one page back and obscure relative to (8). Page 533 CRITICAL POINTS of Linear Systems The remarks about (0,0) being an isolated critical point of X'=AX iff |A| is nonzero deserves to be a lemma, proposition or theorem. This result is at the heart of the classification scheme in the chapter, and it is not a minor issue at all. Everything that follows depends on |A| nonzero. In chapter 2, there were 3 cases: funnel, spout, node. Here the number of classifications has expanded. We will use the Cayley-Hamilton-Ziebur method (section 7.2) to theoretically solve the system X'=AX. This base requires less theory to decide a classification. We can proceed by example and elimination. C-H-Z solves every system, there is no switch needed to generalized eigenvectors, nor are complex solutions encountered. This approach rquires no tables, all classifications can be done based upon experience with a few examples. Cayley-Hamilton method. It is already in section 7.2. It says for X'=AX that X=(Euler sol 1)v1 + (Euler sol 2)v2 for some vectors v1, v2, where the Euler sols are obtained from the roots r=r1,r2 of the char eq |A-rI|=0. This result works for all 2x2 real matrices A. No exceptions. No eigenanalysis. No complex solutions. Ditto for nxn. Page 534 EXAMPLE 2a. A:=(1/8)*Matrix([[7,3],[-3,17]]) The C-H method gives solution X=exp(t)v1+exp(2t)v2. Do an experiment with v1, v2 equal to cols of the identity matrix. Then decide it is an unstable node (in less than 30 seconds). To decide if it is improper or proper, do extra work. The sub-classification is needed, if the phase portrait is indecisive. Page 534 EXAMPLE 2b. A:=(-1/8)*Matrix([[7,3],[-3,17]]) Roots of |A-rI|=0 are r=-1,-2. The C-H solution is still a node (in 30 seconds), and stable. With extra work, it is an improper node. Page 534 Unequal Eigenvalues with opposite signs: Saddle point We repeat the reasoning of the previous case and obtain again (10), (11) and the equation v=C u^k, but this time k=lambda[2]/lambda[1] is negative. EXAMPLE 3. Saddle point A:=(1/4)*Matrix([[5,-3],[3,-5]]) Eigenpairs (1,v1), (-1,v2), v1=<3,1>, v2=<1,3>. Edit the single occurrence of "k = -1": Change "k = -1" to "k = lambda[2]/lambda[1] = -1". Page 535 EQUAL REAL ROOTS This derivation requires generalized eigenanalysis. That topic was removed from the course, so the discourse on page 535 has a hurdle for the reader. On reading page 535. First, (r,v1) is an eigenpair. If no eigenvector v2 exists, then v2 comes from solving (A-r I)v2=v1. Putzer's formula for exp(At) is recalled from chapter 8 topics. exp(At) = exp(r t)[I+ t (A-r I)], r=double eigenvalue Last detail: Multiply the Putzer formula by v1 and v2, resp., to give solutions x1(t)=exp(r t)v1 and x2(t)=exp(r t)(v2+ t v1). Page 535 EXAMPLE 4. Improper nodal sink A:=(1/8)*Matrix([[-11,9],[-1,-5]]) Eigenpair (-1,v1), v1=<3,1>, no second eigenpair, solve (A+I)v2=v1 for v2=<1,3>. EXERCISE. In Example 4, solve the equation (A+I)v2=<3,1> for v2=<3,1>. Page 535 COMPLEX CONJUGATE EIGENVALUES PURE IMAGINARY EIGENVALUES DEF. spiral point, center EXAMPLE 5. Spiral point A:=(1/4)*Matrix([[-10,15],[-15,8]]) Eigenvalues -1/4+3i, -1/4-3i. Text should say "we don't compute eigenvectors." EXAMPLE 6. Center A:=(1/4)*Matrix([[-9,15],[-15,9]]) Eigenvalues 3i, -3i. FIGURE 9.2.9 Table of geometric portraits based on eigenvalues What is missing is the classification advice: Find the eigenvalues. Write down Euler's solution basis based on these roots (chapter 5). If there are sines and cosines, then the portrait has rotation and it must be a spiral or a center. Otherwise, there is no rotation and it must be a saddle or node. This advice cuts in half the possible number of phase portraits to be considered. A one-step binary search for the matching portrait. The search can be continued by looking more deeply at the Euler solutions, finally selecting one of the four types. Page 537 THEOREM 1. Stability of Linear Systems. Stability test for X'=AX with |A| nonzero, based upon the roots of the characteristic equation. THEOREM 2. Stability of Almost Linear Systems. The phase portrait of X'=AX and almost linear system X'=AX+G(x,y) at common isolated critical point (0,0) are of the same type and stability, with two exceptions. Exception 1. Real equal eigenvaues.... Exception 2. Purely imaginary eigenvalues.... FIGURE 9.2.12. Classification of critical points of an almost linear system. What is the organization of the table? According to the ordering of Theorem 2, lines 2,4,8 should appear first, in that order, followed by the others. EXAMPLE 7. x'=4x+2y+2x^2-3y^2, y'=4x-3y+7xy, saddle point at (0,0) A:=Matrix([[4,2],[4,-3]]) has eigenvalues -4,5 It is unfortunate that this example has such complicated critical points. I like the 2 spirals and a node, however. The portrait is rich and interesting. The method for solving for the critical points ended in the maple code below, as an example of how to solve for (x,y). plots[implicitplot]({4*x+2*y+2*x^2-3*y^2=0,4*x-3*y+7*x*y=0},x=-3..3,y=-2..2); The plot predicted x=-2.39, y=-0.49 as an approximate intersection. The Example 7 answer x = -2.354243070, y = -.4834248761 is verified by fsolve({4*x+2*y+2*x^2-3*y^2=0, 4*x-3*y+7*x*y=0},{x=-2.39,y=-0.49}); EXAMPLE 8. Asymptotically stable spiral at (4,3) for the system x'=33-10x-3y+x^2, y'=-18+6x+2y-xy, the Jacobian A=J(4,3) equals Matrix([[-2,-3],[3,-2]]) with roots -2+3i, -2-3i. The portraits are rich and interesting. The saddle is at a messy point x = .3944487245, y = 9.737034184 while the spiral is at x=4, y=3 (not messy). Page 541 EXERCISES We don't have much time at the end of the course to assign and collect problems. Problems 1-32 are possible to do. I like problem 38, but no option to assign it. 9.2 APPLICATION: Phase portraits of almost linear systems ==== We have been using the phase portrait tool in maple to do the computer plots of examples and problems. Polking's PPLANE has more features that are genuinely useful. Page 545 The maple command evalf(Eigenvals(A)) should be evalf(linalg[eigenvals](A)), a case error at least, but the package should be specified. 9.3 Ecological Models ==== We have one day for 9.3 and one day for 9.4. Often they are done in 1.5 days, in a hurry. Page 545 MODELING The model for predator-prey. x'=(a-py)x, y'=(-b+qx)y The text discusses identifying which is the predator, and which the prey. EXAMPLE 1. Critical points of the predator-prey equations. (0,0) is a saddle (b/q,a/p) is a center or spiral PHASE PLANE PORTRAIT FIGURE 9.3.1 We work on the board Exercise 2, which verifies the closed curves in this figure. Page 547 EXAMPLE 2. Oscillating Populations x'=0.005(40-y)x, y'=0.01(-50+x)y The critical point x=50, y=40 is studied on the board and a computer projection is done for the figure in 9.3.2. FIGURE 9.3.2 Predator-Prey of Example 2 phase portrait FIGURE 9.3.3 Predator-Prey of Example 2 component functions Page 548 Competing Species We do enough of the theory to explain inhibition and competition. Then go to examples. EXAMPLE 3. Survival of a single species x'=(14-x/2-y)x, y'=(16-y/2-x)y with c1 c2 > b1 b2 Critical points (0,0), (0,32), (28,0), (12,8). This is a main example worked in great detail, then on to discuss the impact of Theorem 2 on the phase portrait. Page 551 Separatrix No definition in the book. EXAMPLE 4. Peaceful coexistence of two species x'=(14-2x-y)x, y'=(16-2y-x)y with c1 c2 < b1 b2 Critical points (0,0), (0,8), (7,0), (4,6). This is the second main example of 9.3, which is worked completely in class, with the aid of the textbook details to speed the presentation. Page 554 EXERCISES They are offered as extra credit. Student have worked on problems 4 to 10. 9.3 APPLICATION ==== Page 558 9.4 Nonlinear Mechanical Systems ==== One day or 1/2 day lecture. We do hard and soft springs. EXAMPLE 0. Hard spring 2x'' = - 2x - 4x^3 Conservation law y^2+x^2+x^4=E, center at (0,0) FIGURE 9.4.2 shows the phase portrait Example done in detail in the lecture. However, it is not a formal example in the textbook. EXAMPLE 1. Soft spring x'' + 4x - x^3 = 0 Conservation law y^2/2 + 2x^2 - x^4/4 = E I do this example in detail. It is a favorite. Separatrix again. Definition: "a separatrix is a union of critical points and trajectories that join them at t=infinity or t=-infinity." DAMPED NONLINEAR VIBRATIONS EXAMPLE 2. Not done for lack of time. NONLINEAR PENDULUM We do the basic results and reproduce the graphics in FIGURE 9.4.8 and FIGURE 9.4.10. 9.4 APPLICATION ==== No time to do it.