Two Dimensional Linear Systems of ODEs

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2 CHAPTER 3 Two Dimensional Linear Sstems of ODEs A first-der, autonomous, homogeneous linear sstem of two ODEs has the fm x t ax + b, t cx + d where a, b, c, d are real constants The matrix fm is 31 x t A x where x 32 x The initial condition f 31 is a b, A c d 33 x x where x and x, R are arbitrar constants x 31 The exponential of a linear map The solution of the linear, autonomous, scalar IVP x t ax, x x ma be written as xt e ta x An analogous fmula holds f sstems with a suitable definition of the exponential of a linear map, matrix The definition does not depend on the dimension of the sstem, so we initiall consider the d- dimensional case Definition 31 If A : R d R d is a linear map, its cresponding matrix, and t R, then the exponential e ta : R d R d is defined b 34 e ta I + ta + 1 2! t2 A n! tn A n + The series 34 is a natural generalization of the Tal series of the scalar exponential function It makes sense if A is a d d matrix, so that its powers are well-defined, and then the series defines a d d matrix e ta We remark that it is useful conceptuall to distinguish between a linear map, which is a geometrical object like a rotation of vects, and the matrix that represents it with respect to some basis The matrix of a given linear map depends on the basis, not onl on the map We consider linear maps acting on R d with its 35

3 36 3 TWO DIMENSIONAL LINEAR SYSTEMS OF ODES standard basis, so usuall we will not distinguish between linear maps and matrices, but one should still be able to view the results we discuss from a geometrical perspective We summarize some properties of the matrix exponential in the following result Theem 32 F ever linear map A : R d R d, its cresponding matrix, the series 34 converges absolutel f all t R It is a differentiable function of t, and 35 Meover, e A I and d dt eta Ae ta e ta A e sa e ta e s+ta f all s, t R, and e ta is invertible with e ta 1 e ta The proof of these results is similar to the proof of the analogous results f the usual scalar exponential function F example, term b term differentiation of the series which one can show is justified gives d dt eta A + ta A Ae ta 1 n 1! tn 1 A n + I + ta + 1 2! t2 A n 1! tn 1 A n 1 + Similarl, if A, B commute meaning that AB BA then multiplication and rearrangement of the series f e A, e B implies that e A e B e A+B Note, however, that this relation does not hold f non-commuting maps matrices; this is a significant difference from the scalar case It follows from 35 that the solution of the IVP 31, 33 is given b 36 xt e ta x To verif this, note that if xt is given b 36 then d dt xt AetA x A xt, and x x since e A I Thus, 36 is the unique solution of 31, and Φ t e ta is the flow map associated with the linear vect field A x Note that the flow map is also linear, as follows from the superposition propert of linear equations Let us consider some simple 2 2 examples with Example 33 The sstem x t x, 1 A 1 t

4 31 THE EXPONENTIAL OF A LINEAR MAP 37 has a vect field pointing radiall inward toward the igin The solutions are xt x e t, t e t, x e t e t x In this case, A I and A n 1 n I, so that e ta I ti + 1 2! t2 I n t n I + n! 1 t + 12! t n t n + I n! e t I [ 1 exp t 1 ] e t e t in agreement with the above solution The trajecties consist of radial lines that approach as t, and the equilibrium point The flow map is a contraction b a fact e t with is Example 34 The solution of In this case, and Thus, x t x, A xt x e t, x A n e ta t t n t n e t e t 1 1 t t e t e t e t x 1 n tn 1 n n! 1 n! t + + tn n! + [ 1 exp t 1 ] e t e t, + in agreement with the above solution The trajecties consists of branches of the hperbolas x constant, the positive and negative x and axes, and the equilibrium point The x-values approach as t while the -values go off

5 38 3 TWO DIMENSIONAL LINEAR SYSTEMS OF ODES to infinit as t unless The flow map is a contraction b e t in the x-direction and an expansion b e t in the -direction Example 35 The sstem x t, t x with 1 A 1 has a vect field that is thogonal to the radial vect field Eliminating, we get x tt + x, and the solutions are In this case, xt x cos t sin t, t x sin t + cos t, x cos t sin t x sin t cos t A 2n 1 n I, A 2n+1 1 n A, and using the Tal series f cos t, sin t we get Thus, e ta I + ta 1 2! t2 I 1 3! t3 A + 1 4! t4 I ! t ! t4 I + cos ti + sin ta [ 1 exp t 1 t 1 3! t ! t5 ] cos t sin t sin t cos t, A in agreement with the solution above The trajecties consist of circles centered at the igin, and the equilibrium All solutions are 2π-periodic functions of t, and the flow map is a counter-clockwise rotation about the igin through an angle t with Example 36 The sstem has the solution x t, t A 1 xt x + t, t, x 1 t 1 x In this case, A n f n 2 a matrix whose powers are eventuall zero is said to be nilpotent and 34 shows, in agreement with this solution, that [ ] 1 1 t exp t 1 The x-axis consists of equilibrium, and the other trajecties are lines with The flow map consists of a shear b t in the x-direction

6 31 THE EXPONENTIAL OF A LINEAR MAP 39 It is not eas to use the series definition to compute the exponential of a matrix directl except in special cases We can however use a similarit transfmation to reduce a general matrix to a canonical fm whose matrix exponential can be easil computed Proposition 37 Suppose that A P 1 BP is similar to a matrix B b a nonsingular matrix P Then the matrix exponential of A is similar to the matrix exponential of B Proof We have e ta P 1 e tb P A n P 1 B n P because of the cancelation in the inner facts P 1 P I Thus, e ta I + ta n! tn A n + I + tp 1 BP n! tn P 1 B n P + P I 1 + tb n! tn B n + P P 1 e tb P A matrix is said to be diagonalizable if it is similar to a diagonal matrix Λ In the 2 2 case, this means that A P 1 λ1 ΛP, Λ λ 2 where λ 1, λ 2 are the not necessaril distinct eigenvalues of A We compute from the series definition that Λ n λ n 1 λ n 2 and e tλ e λ 1t e λ2t Thus, the exponential of a diagonalizable 2 2 matrix is given b e ta P 1 e λ 1t e λ2t P Most matrices are diagonalizable Sufficient conditions f the diagonalizabilit of a matrix are: a it has simple eigenvalues; b it is smmetric, in which case its eigenvalues are real and there is an thonmal basis of eigenvects There exist, however, nondiagonalizable matrices In the case of 2 2 matrices, it follows from the Jdan canonical fm that ever nondiagonalizable matrix A is similar to a 2 2 Jdan block A P 1 λ 1 JP, J λ

7 4 3 TWO DIMENSIONAL LINEAR SYSTEMS OF ODES where λ is the necessaril repeated eigenvalue of A To compute the exponential of J, note that 1 J Λ + N, Λ λi, N Since Λ is a multiple of the identit, it commutes with N and therefe e tj e tλ e tn The series definition implies that e tλ e λt I and, from Example 36, e tn 1 t 1 Thus, the exponential of a nondiagonalizable 2 2 matrix is given b e ta e λt P 1 1 t P 1 32 Eigenvalues and eigenvects We can represent the solution of a general linear sstem 31 in terms of the eigenvalues and generalized eigenvects of A The result is equivalent to the representation of the matrix exponential in terms of a similarit transfmation 1 F simplicit, we consider the 2 2 case where A is given b 32, although the ideas generalize in a fairl straightfward wa to sstems of an dimension A scalar λ C is an eigenvalue of A if there exists a nonzero eigenvect r C 2 such that A r λ r The eigenvalues are solutions of the characteristic equation of A, deta λi In that case, a solution of 31 is given b xt e λt r If the 2 2 matrix A is diagonalizable, then it has has possibl repeated eigenvalues {λ 1, λ 2 } with linearl independent eigenvects { r 1, r 2 } The general solution of 31 is given b xt c 1 e λ1t r 1 + c 2 e λ2t r 2 To satisf the initial condition 33, we choose the scalar constants c 1, c 2 C so that x c 1 r 1 + c 2 r 2 There exist unique such constants because { r 1, r 2 } are linearl independent and hence fm a basis of C 2 It is convenient here to allow real complex eigenvalues even though the sstem is real An complex eigenvalues come in complex-conjugate pairs, and we can take their eigenvects to be complex conjugates also In that case the constants c 1, c 2 are complex, but f real-valued initial data c 1, c 2 are complex conjugates and the solution xt is also real-valued 1 The columns of the similarit matrix P 1 consist of generalized eigenvects of A

8 33 CLASSIFICATION OF 2 2 LINEAR SYSTEMS 41 If A is nondiagonalizable with a repeated eigenvalue λ, then it has linearl independent generalized eigenvects { r, s} such that the solution of the ODE is A λi r, A λi r s xt c 1 e λt r + c 2 e λt t r + s where the scalar constants c 1, c 2 are chosen so that x c 1 r + c 2 s 33 Classification of 2 2 linear sstems The linear sstem 31 has the equilibrium x, and this is the onl equilibrium if A is nonsingular We classif the equilibrium of the 2 2 linear sstem 31 in terms of the possibl repeated eigenvalues λ 1, λ 2 of A as follows Saddle point: If λ 1, λ 2 are real and of opposite signs Node: If λ 1, λ 2 are real and of the same sign Stable if λ 1, λ 2 < and unstable if λ 1, λ 2 > Spiral point: If λ 1, λ 2 τ ± iω are a complex conjugate pair with nonzero real part Stable if τ < and unstable if τ > Center: If λ 1, λ 2 ±iω are a pure imaginar, complex conjugate pair Singular: If λ 1 λ 2 is zero, when there is a one two dimensional subspace of equilibria F instance, Example 33 is a stable node, Example 34 is a saddle point, Example 35 is a center, and Example 36 is singular A stable node stable spiral is asmptoticall stable A center is stable but not asmptoticall stable A saddle point, unstable node unstable spiral is unstable The eigenvalues λ λ 1, λ 2 of the 2 2 matrix A in 32 satisf λ 2 τλ + D where τ is the trace of A and D is the determinant of A, τ tr A a + d λ 1 + λ 2, D det A ad bc λ 1 λ 2 The eigenvalues are λ 1,2 1 τ ± τ 2 2 4D We therefe get the following criteria f the tpe of the equilibrium in terms of the trace and determinant of A Saddle point: If D < Node: If < 4D τ 2 Stable if τ < and unstable if τ > Spiral point: If τ and 4D > τ 2 Stable if τ < and unstable if τ > Center: If τ and D > Singular: If D We distinguish between hperbolic and non-hperbolic equilibria accding to the following definition Definition 38 The equilibrium x of 31 is hperbolic if no eigenvalue of A has zero real part

9 42 3 TWO DIMENSIONAL LINEAR SYSTEMS OF ODES Thus, a saddle point, node spiral point is hperbolic, but a center singular point is not Hperbolic equilibria are robust under small perturbations such as the inclusion of non-linear terms, but non-hperbolic equilibria are not The stable respectivel, unstable subspace E s respectivel, E u of is the subspace spanned b the generalized eigenvects associated with eigenvects whose eigenvalues have negative respectivel, positive real parts Thus, x E s if the solution approaches the equilibrium fward in time, e ta x as t, while x E u if the solution approaches the equilibrium backward in time, e ta x as t Note that this is not equivalent to the condition that e ta x as t The center subspace E c of is the subspace spanned b generalized eigenvects with zero real part The cresponding solutions ma remain bounded grow algebraicall in time F example, a stable node spiral point of a 2 2 linear sstem has E s R 2 and E u E c {}; a center has E c R 2 and E u E s {}; and a saddle point has one-dimensional stable and unstable subspaces E s, E u and E c {} The center subspace of an hperbolic equilibrium is {} 34 The linear oscillat Consider a mass m > on a Hookean spring with spring constant k > subject to linear damping with coefficient δ The displacement xt of the mass from equilibrium satisfies 37 mx tt + δx t + kx Introducing the momentum mx t, we ma write this as first der sstem x x t 1 m, The eigenvalues of this sstem are [ t t kx δ 1/m x k δ λ 1 2 δ ± δ 2 4k m Depending on the strength of the damping, we have three possibilities f the equilibrium: : Undamped When δ, x, is a center All solutions are periodic with angular frequenc ω k/m : Underdamped When < δ < 2 km, x, is a stable spiral point All solutions deca to zero in an oscillat fashion as t : Overdamped When δ > 2 km, x, is a stable node All solutions eventuall deca monotonicall to zero with the displacement x passing through zero at most once ]

10 35 FLOW MAPS AND AREAS 43 The damping δ 2 km at which the oscillat switches from underdamped to overdamped is called critical damping Note that this equation is dimensionall consistent: both δ and km have the dimension of Mass/Time Exactl the same analsis applies to linear electrical circuits Suppose a circuit consists of an induct with inductance L, a resist with resistance R, and a capacit with capacitance C, placed in series If It is the current in the circuit and Qt is the charge on the capacit, then the change in voltage across the induct is V L LI t, the change in voltage across the resist is given b Ohm s law V R RI, and the change of voltage across the capacit is V C 1 C Q Accding to Kirchoff s law the total change in voltage around the circuit is zero, meaning that V L + V L + V L Meover, since charge is conserved, we have I Q t It follows that Qt satisfies LQ tt + RQ t + 1 C Q This equation is identical to 37 with L plaing the role of mass, R the role of the damping constant, and 1/C the role of the spring constant 35 Flow maps and areas An imptant aspect of flows is how the affect areas volumes in phase space F example, do areas contract increase under the flow do the remain the same? We consider this briefl here in the special case of 2 2 linear sstems The iented area P of the parallelogram P spanned b vects x, R 2 is given b P det [ x, ] If A : R 2 R 2 is a linear map, then the iented area of the parallelogram AP spanned b the vects A x, A R 2 is given b AP det [A x, A ] det A [ x, ] det Adet [ x, ] det A P Thus, det A is the fact b which A multiplies iented areas The following result related the determinant of a matrix exponential to the trace of the matrix Exactl the same result is true f a linear map A : R d R d in an number of dimensions d, as can be shown using the general Jdan canonical fm, but we state and prove the theem onl f the case d 2 Theem 39 If A : R 2 R 2 is a linear map, then det e ta e t tr A

11 44 3 TWO DIMENSIONAL LINEAR SYSTEMS OF ODES and Hence Proof If A is diagonalizable, then A P 1 λ1 P, λ 2 e ta P 1 e λ 1t e λ2t det e ta det P 1 e λ 1t det e λ2t If A is non-diagonalizable, then and A P 1 λ 1 λ det e ta det P 1 e λt te det λt e λt P det P e λ1+λ2t e t tr A P, det P e 2λt e t tr A 36 References F a classification of equilibria based on real canonical fms, see [9]