25. Ordinary Differential Equations: Systems Of Equations

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25. Ordinary Differential Equations:Systems of EquationsWe now turn to the analysis of systems of equations. As we saw in section 24.14, it ispossible to convert a second order differential equation into a first order system of twoequations.Let’s start with a general first order linear system of m equations relating relatingfunctions y1 , . . . , ym of a variable t. We can write the systemẏ1 a11 y1 a12 y2 · · · a1m ymẏ2 a21 y1 a22 y2 · · · a2m ym. .ẏm am1 y1 am2 y2 · · · amm ymSetting y (y1 , . . . , ym )T , and A [aij ], we can rewrite the equation in vector/matrixform asẏ Ay.Given initial data of the form yi (t0 ) yi , define y0 (y1 , . . . , ym ). We can writethe initial value problem asẏ Ay(25.0.1)y(t0 ) y0We have complete solutions to such initial value problems. The solution has matrixformy(t) e(t t0 )A y0 .This implies that each of the yi (t) can be written as a linear combination of eλi t forλi σ(A) if the eigenvectors of A span Rm , and terms of the form tk eλi t for eigenvalueswith algebraic multiplicity larger than their nullity.

2MATH METHODS25.1 More on Solving First Order Linear SystemsIf the eigenvalues of A are all distinct, or if they all have equal algebraic and geometricmultiplicity, there is a basis P consisting of eigenvectors with basis matrix P wherethe linear transformation defined by A takes the diagonal form D where the diagonalconsists of eigenvalues, λ1 , . . . , λm . The matrices A and D are related by the changeof basis formula P 1 AP D. If an eigenvalue has multiplicity mk , we list it mk times.Then e(t t0 )A P e(t t0 )D P 1 .It follows that eλ1 (t t0 )te(t t0 )A P .eλm (t t0 )tIn fact, we can write y0 as a sum of eigenvectorsy0 mX P 1civii 1wheny(t) mXcieλi (t t0 ) vi .i 1Otherwise, we must use the Jordan canonical form. When unpacking it, this involvesterms of using eλi (t t0 ) , teλi(t t0 ) , . . . , tk eλi (t t0 ) for a Jordan chain of order k belongingto eigenvalue λi .

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS325.2 mth Order Equations as Matrix EquationsAny mth order linear differential equation can written as a first order equation on Rm .Suppose we have the mth order linear ordinary differential equationa0dm 1 ydydm y a · · · am 1 am y 0.1mmdtdtdt(25.2.2)Because it is mth order, we must have a0 6 0. Now define the matrix A by 010 ···0. . 0.01. .A .0 0 0 0······01 a aam0 m 1··· aa10a0Then taking y1 y, we have ẏ ẏ1 y2 , ÿ ẏ2 y3 , . . . , dm 1y/dtm 1 ẏm 1 ym , andẏm dm y/dtmam 1amy1 y2 · · · a0a0amam 1 dy y ··· a0a0 dta1yma0a1 dm 1.a0 dtm 1Rearranging and multiplying by a0 shows that the matrix equation implies equation(25.2.2).For example, if.2 y 5ÿ 6ẏ 16y 0the matrixA puts the equation into matrix form.00 810 301 5/2!

4MATH METHODS25.3 First Order Systems vs. mth Order EquationsAre systems of m equations the same as mth order equations of one variable? Notexactly. Although every mth order linear equation defines an equivalent first ordersystem in Rm , the converse is false. It is possible to write uncoupled equations inmatrix form, but not as a single equation. Example 25.3.1: Uncoupled System. For example, consider the uncoupled system.ẏ1 2y1ẏ2 y2The general solution is y1 c1 e2t and y2 c2 et , and there is no relation betweenwhat happens with y1 and y2 . In particular, we cannot use information about y1 andẏ1 to find out anything about y2 . This may seem a bit odd, as our solution method involves changing coordinates to theeigenvector system, where the system will usually take this form. However, this changeof basis often means that returning to the standard basis will mix the coordinates. Inthe standard basis, one of the variables may contain information about all of the others.This is always the case when the matrix form is derived from a single equation. Example 25.3.2: Reducible System. For comparison, here is a system in Rm that canbe reduced to an equation of order m (or lower). For example, takeẏ1 ay2ẏ2 by1 ,where a, b 0. Notice that ẏ1 depends on y2 and ẏ2 depends on y1 .We can write this as a single equation. Begin by setting y y1 . Now ẏ ay2 . Wetake the derivative, yielding ÿ aẏ2 . The second equation is ẏ2 by, allowing usto combine them into a single second order equationÿ aẏ2 a( by) aby.The solution is y1 (t) y(t) c1 cos (ab)1/2 t c2 sin (ab)1/2 t .Here knowledge of y1 (0) c1 and ẏ1 (0) c2 (ab)1/2 entirely determine y2 (1/a)ẏ1 .

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS525.4 Vector FieldsA vector field on Rm is a mapping F : Rm Rm that assigns a vector in Rm to anypoint in Rm .If A is an m m matrix, we can define a vector field on Rm by F(x) Ax. Manyother vector fields are possible, such asF(x) x21 sin x2 !22x1 x3 ex1 x2x2 x3Another way to define a vector field is to take the gradient of any differentiablefunction φ. Just set F(x) φ(x). This method is often used in physics where φ ispotential energy and ( φ) is the corresponding force field. Under Newtonian gravity,the potential energy from a mass M located at the origin is φ(x) GM/kxk2 whereG is the gravitational constant. Then ( φ) is the gravitational field, indicating that thegravitational force on a mass m at x is m φ(x) GMm xGMmx̂ 2kxk2 kxk2kxk22where x̂ is the unit vector in the x direction. As Isaac Newton discovered, the gravitational force points toward the mass M at the origin and has magnitude proportional tothe inverse square of the distance from the origin.More generally, vector fields make sense on any differentiable manifold. The onlychange is that they map points x M to the tangent space of M at x. We define thetangent bundle of a C1 manifold M by T M {(x, y) : y Tx M}.Vector Field. A vector field on a C1 manifold M is a mapping F : M T M such thatF(x) Tx M for all x M.When M Rm , the tangent space at x, Tx (Rm ) Rm . As a result, the manifolddefinition of vector field generalizes the previous definition because when M Rm ,Tx M Rm for all x Rm . It follows that a vector field on Rm (as a manifold) is just amapping F : Rm Rm , which was our original definition.

6MATH METHODS25.5 Differential Systems and Vector FieldsVector fields are of interest to us because any first order differential system ẏ F(y) onRm defines an associated vector field F. The point is that since F : Rm Rm , F canbe interpreted as a vector field.If y1F(y) y22 /2we can represent the vector field by the following diagram.bFigure 25.5.1: The vector field F is plotted by plotting the vector from x to x F(x) forvarious x. At the origin, the vector field is zero. We indicate that with a dot at the origin.The length of the arrows increases away from the origin, indicating that motion acceleratesas you move away from the origin.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS725.6 Phase PortraitsAn integral curve or trajectory of F is the set of points {y(t)} solving ẏ F(y) for someinitial values y0 .The Peano Existence Theorem gives conditions showing that integral curves exist overa small time interval. If the Picard-Lindelöf Theorem holds, they will be locally unique.In many cases of interest, we are able to write integral curves over a substantial domain.In the case of a linear system, that domain is often (t0 , ) or even ( , ). A plotof sample trajectories is called a phase portrait.Of course, the derivative of an integral curve defines the vector field F.Suppose we take the equation ẏ 2y. The solutions are y(t) (c1 e 2t , c2e 2t ).If c1 6 0, we can write y2 (t)/y1 (t) c2 /c1 , which shows that the trajectories are raysabout the origin. This yields the following phase portrait.y2by1Figure 25.6.1: The phase portrait plots trajectories of the form y(t) (c1 e t , c2 e t ).

8MATH METHODS25.7 Drawing Phase Portraits ISupposed x 2 1dt y1 x .y2We could easily solve this equation, which has eigenvalues σ(A) {1, 3}. It’s easy tosee that limt ky(t)k unless we start at the steady state, (0, 0).We will use the phase portrait to study the long-run behavior of this system. We seethat ẋ 0 when 2x y 0 and ẏ 0 when x 2y 0. These lines intersect at thesteady state, (0, 0). Above the ẋ 0 line, ẋ is positive. Below it, ẋ is negative. Similarly,above the ẏ 0 line, ẏ is positive, and below it, ẏ is negative.Those lines are drawn on the figure below, breaking it into four “quadrants”. The SEand NW “quadrants” are shaded. The arrows indicate the direction of motion withineach of the four “quadrants”.In this case, the motion is away from the steady state at the origin, regardless of which“quadrant” you start in. The steady state is unstable. Starting near the steady state willlead to a trajectory that moves away from the origin.yẋ 0xẏ 0Figure 25.7.1: This diagram indicates the lines where ẋ 0 and ẏ 0 as when as thedirection of motion (outward!) within each region.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS925.8 Drawing Phase Portraits IIIt’s obvious that any trajectories starting in the NE or SW “quadrants” will move furtherNE or SW, off toward infinity. What happens in the gray zones is a little unclear, butyou might imagine that one could start left of the ẏ 0 axis in the SE and move towardit, before crossing it. The trajectory must be horizontal (ẏ 0) at the crossing. It thenmoves off the the NE. Or perhaps one could start a little above the ẋ 0 line, moveSE to cross it vertically (ẋ 0), then head SW.yẋ 0xẏ 0Figure 25.7.1: This diagram indicates the lines where ẋ 0 and ẏ 0 as when as thedirection of motion (outward!) within each region.

10MATH METHODS25.9 Drawing Phase Portraits IIITo sort the details out it helps to look at the eigenvectors. Here v1 (1, 1) is aneigenvector for λ 3 and v2 (1, 1) is an eigenvector for λ 1. If we start on anymultiple of the eigenvectors, we end up with solutions of the the form c1 e3t (1, 1) orc2 et (1, 1). Everywhere else, the trajectories are a blend of these types of solutions,meaning that the e3t term will eventually dominate.For example, if we start in the SE cone at (1, 0.8), then c1 c2 1 and c1 c2 0.8, implying c1 0.1 and c2 0.9. The trajectory starting at (1, 0.8)is y(t) (.1e3t 0.9et, .1e3t 0.9et). This shows that limt y1 /y2 1. Thetrajectory is asymptotic to a line parallel to the 45 line, even though it starts in the graycone.yẋ 0v1xv2ẏ 0Figure 25.9.1: This diagram indicates the lines where ẋ 0 and ẏ 0 as when as thedirection of motion (outward!) within each region. It also shows the eigenvectors (the heavyarrows) and their trajectories.Keep in mind that the eigenvector lines cannot be crossed and that movement in the (1, 1)direction is faster than in the (1, 1) direction.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS1125.10 Stability of Steady StatesOne question of interest is whether the steady states of a differential system are stable.A homogeneous system always has a steady state at 0. That is, y(t) 0 always solvesthe systemẏ Ay(25.0.1)y(t0 ) y0when the initial data is y0 0.For general autonomous systems ẏ F(y), a vector ȳ is a steady state if F(ȳ) 0.It implies that y(t) ȳ solves the differential system with initial value ȳ becauseẏ F(ȳ) 0. For linear systems, the steady state condition becomes Aȳ 0.It follows that ker A is the set of steady states for equation (25.0.1). The system hasa unique steady state if A is invertible. If A is not invertible, any vector in ker A is asteady state, meaning that the set of steady states is a vector subspace of Rm .Steady states need not be at zero for an inhomogeneous linear system. Let y0 be theinitial data andẏ A(y y0 )Now ẏ 0 if y y0 , so y0 is a steady state, and y(t) y0 solves the system.We will have to make precise what we mean by “stable”. There are many, many,different definitions in the literature. One author claimed there are over 100!. We willcontent ourselves with four.The basic idea of stability is that trajectories starting near a steady state remain near it.In the case of asymptotic stability, those trajectories y(t) converge to that steady state.

12MATH METHODS25.11 Example: Purely Imaginary RootsBefore defining any type of stability, we first examine behavior near a steady state in R2 .In these exercises, we will work in the eigenvector basis. If all of the eigenvalues aredistinct, this is an orthonormal basis that is a rotation/reflection of the standard basis.We can also choose an orthonormal basis if the algebraic multiplicity of each eigenvalueis equal to its geometric multiplicity. Example 25.11.1: Circular Motion. Consider the system 0 1 yẏ 1 0The characteristic equation is λ2 1 0, so the eigenvalues are σ(A) { i, i}. Itfollows that the general solution is sin t cos t c2y c1 cos tsin tGiven initial data y0 (x0 , y0 ), we solve the problem by setting c1 x0 , c2 y0 .Nowky(t)k2 x20 cos2 t 2x0 y0 sin t cos t y20 sin2 t y20 cos2 t 2x0 y0 sin t cos t x20 sin2 t (x20 y20 )(cos2 t sin2 t) x20 y20 .This shows that the trajectory is a circle. If we start at a point at distance ε from thesteady state at 0, we remain at distance ε. The trajectory does not approach the steadystate, but doesn’t move away either. This is illustrated in Figure 25.16.2.ybxFigure 25.11.2: The phase portrait can be drawn either in y-space or in (yi , ẏi )-space. Herey2 ẏ1 , so y-space is (y1 , ẏ1 )-space. Here we illustrate several trajectories. Each is an circleabout the origin, which is our steady state. Each is also counter-clockwise about the steadystate, as the arrows indicate.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS1325.12 Example: Elliptical MotionElliptical motion can occur in second order linear systems in R2 , which can be writtenas first order systems in R4 . Example 25.12.1: Elliptical Motion. Consider the system ÿ y with initial datay(0) (x0 , 0) and ẏ(0) (0, y0 ). It has solutiony(t) (x0 cos t)e1 (y0 sin t)e2 .It follows thatky(t)k2 x20 cos2 t y20 sin2 tand the trajectory traces out an ellipse: (y1 (t)/x0 )2 (y2 (t)/y0 )2 1.This will be a circle if x20 y20 . The direction of motion is either clockwise orcounter-clockwise, depending on the initial data. When x0 y0 1 we getcounter-clockwise motion, while if x0 1, y0 1, the motion is clockwise.ybxFigure 25.12.2: Here the phase portrait is drawn in y-space. We illustrate three trajectories,with the direction of motion indicated. Each is an ellipse of the form y21 4y22 c2 . Thesteady state is at 0.

14MATH METHODS25.13 Lyapunov StabilityThe circular and elliptical examples examined cases with a unique steady state. Theyhave shown us that trajectories will sometimes orbit the steady state. Moreover thoseorbits can be eccentric—they can get nearer and farther from the steady state as theyorbit around it.Non-linear equations can lead to even more complex behavior. Lyapunov’s notionof stability is intended to allow this.Lyapunov Stable. The system (25.0.1) is Lyapunov stable at 0 if for every ε 0 there isa δ 0 so that ky(t)k ε for all t t0 whenever ky0 k δ.Stability means that a trajectory y(t) stays within a distance ε of the steady state ifit starts within some distance δ of the steady state. The elliptical trajectories of Figure25.12.2 are an example of Lyapunov stability. These trajectories are twice as far awayat their farthest point than at their closest, so we would need to choose δ ε/2

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS1525.14 Asymptotic Stability of Steady StatesWhile Lyapunov stability only requires that trajectories remain near a steady state, thetwo types of asymptotic stability require convergence to a steady state.Locally Asymptotically Stable. A steady state 0 is locally asymptotically stable if it isLyapunov stable and there exists an ε 0 so that limt y(t) 0 whenever ky0 k ε.Local asymptotic stability takes Lyapunov stability and adds convergence to a steadystate for trajectories that start nearby. Global asymptotic stability requires convergenceto a steady state from any starting point. For this, the steady state must be unique.Globally Asymptotically Stable. A steady state 0 is globally asymptotically stable if everysolution to (25.0.1) converges to 0.

16MATH METHODS25.15 Stability in Linear SystemsIn the case of a linear system, the stability is controlled by the eigenvalues.Theorem 25.15.1. Consider a homogeneous linear first order differential system. If allof the eigenvalues λ σ(A) satisfy Re λ 0, then the system is both globally and locallyasymptotically stable.Proof. We know the solution has the formy(t) e(t t0 )A y0 .We can write it asy(t) Pe(t t0 )J P 1 y0where J is in Jordan canonical form.Nowky(t)k kPkke(t t0 )J kkP 1 kky0 kwhere we use the ℓ matrix norm. Now each term of e(t t0 )J is a most a constanttimes a polynomial in t and the exponential of Re λt. The last goes to zero fast enoughto swamp the polynomial, so all terms (and the norm) converge to zero. Since this isbounded, we have Lyapunov stability. Since it converges to zero, we get both local andglobal asymptotic stability.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS1725.16 Saddlepoint StabilityThere is one other type of stability that we will be interested in, saddlepoint stability.With saddlepoint stability, there is a manifold (vector subspace for linear equations) sothat trajectories starting on the manifold converge to the steady state, and trajectoriesstarting elsewhere do not converge to the steady state.An example will help clarify how this works. Example 25.16.1: Saddlepoint Stability. Let 2 0 A 0 1The eigenvalues are σ(A) {2, 1} and the general solution is y(t) c1 e2t e1 c2 e t e2 .If we start on the vertical axis, with y0 αe2 , the coefficients are c1 0 and c2 α.That yields solution y(t) αe t e2 . It follows that limt y(t) 0, showing the steadystate is stable in this direction.But it is unstable in any other direction. If y0 has any other form, c1 6 0, andlimt y(t) (sgn c1 ) . This system is saddlepoint stable.y2by1Figure 25.16.2: We illustrate several trajectories, with the arrows indicating the direction ofmotion. Notice how solutions starting on the vertical axis converge to the steady state at theorigin, while trajectories starting anywhere else are asymptotic to the horizontal axis.

18MATH METHODS25.17 The m 2 CaseWhen studying stability, it will be useful to examine the m 2 case in detail. Theintuition gained here will tell use much about how stability works for arbitrary m.Consider the system a b yẏ c dwhere a, b, c, and d are real numbers.The characteristic equation is λ2 (a d)λ (ad bc) 0. This can be written interms of the trace, tr A (a d) and determinant det A (ad bc) asλ2 (tr A)λ det A 0.This has solutionsλi (tr A) (tr A)2 4(det A)2for i 1, 2.The expression (tr A)2 4(det A)is called the discriminant. It tells us whether the roots are complex (if and only if 0),identical ( 0), or real and distinct ( 0).We found earlier that tr A λ1 λ2 and det A λ1 λ2 where the λi are the twoeigenvalues of A. We can use this fact to characterize the types of solutions in terms ofthe trace and determinant.Applied to the discriminant, we obtain (tr A)2 4(det A) (λ1 λ2 )2 4λ1 λ2 (λ1 λ2 )2 .The discriminant is zero if and only if λ1 λ2 . Provided that any complex eigenvaluesare conjugates, the discriminant is positive if and only if the eigenvalues are real anddistinct, and negative if and only if they are complex conjugates.

25. ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQ

ORDINARY DIFFERENTIAL EQUATIONS: SYSTEMS OF EQUATIONS 5 25.4 Vector Fields A vector field on Rm is a mapping F: Rm Rm that assigns a vector in Rm to any point in Rm. If A is an m mmatrix, we can define a vector field on Rm by F(x) Ax. Many other vector fields are possible, such as F(x) x2

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