Linear ODEs Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems p. 1 Linear ODEs Types of systems Definition (Linear ODE) A linear ODE is a ifferential equation taking the form x = A(t)x + B(t), t (LNH) where A(t) Mn(R) with continuous entries, B(t) R n with real value, continuous coefficients, an x R n. The associate IVP takes the form x = A(t)x + B(t) t (1) x(t0) = x0. x = A(t)x + B(t) is linear nonautonomous (A(t) epens on t) nonhomogeneous (also calle affine system). x = A(t)x is linear nonautonomous homogeneous. x = Ax + B, that is, A(t) A an B(t) B, is linear autonomous nonhomogeneous (or affine autonomous). x = Ax is linear autonomous homogeneous. If A(t + T ) = A(t) for some T > 0 an all t, then linear perioic. Linear ODEs p. 3 Linear ODEs p. 4
Existence an uniqueness of solutions Linear ODEs Existence of solutions to linear IVPs Resolvent matrix (Existence an Uniqueness) Solutions to (1) exist an are unique on the whole interval over which A an B are continuous. In particular, if A, B are constant, then solutions exist on R. Autonomous linear systems Existence of solutions to linear IVPs p. 6 The vector space of solutions Funamental matrix Consier the homogeneous system t x = A(t)x, with A(t) efine an continuous on an interval J. The set of solutions of (LH) forms an n-imensional vector space. (LH) Definition A set of n linearly inepenent solutions of (LH) on J, {φ1,..., φn}, is calle a funamental set of solutions of (LH) an the matrix Φ = [φ1 φ2... φn] is calle a funamental matrix of (LH). Existence of solutions to linear IVPs p. 7 Existence of solutions to linear IVPs p. 8
Funamental matrix solution Abel s formula Let X Mn(R) with entries [xij]. Define the erivative of X, X (or t X ) as t X (t) = [ t xij(t)]. The system of n 2 equations t X = A(t)X is calle a matrix ifferential equation. A funamental matrix Φ of (LH) satisfies the matrix equation X = A(t)X on the interval J.- If Φ is a solution of the matrix equation X = A(t)X on an interval J an τ J, then ( t ) etφ(t) = etφ(τ) exp tra(s)s τ for all t J. Existence of solutions to linear IVPs p. 9 Existence of solutions to linear IVPs p. 10 The resolvent matrix Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Definition (Resolvent matrix) Let t0 J an Φ(t) be a funamental matrix solution of (LH) on J. Since the columns of Φ are linearly inepenent, it follows that Φ(t0) is invertible. The resolvent (or state transition matrix, or principal funamental matrix) of (LH) is then efine as R(t, t0) = Φ(t)Φ(t0) 1. Autonomous linear systems Resolvent matrix p. 12
Proposition The resolvent matrix satisfies the Chapman-Kolmogorov ientities 1. R(t, t) = I, 2. R(t, s)r(s, u) = R(t, u), as well as the ientities 3. R(t, s) 1 = R(s, t), 4. s R(t, s) = R(t, s)a(s), 5. t R(t, s) = A(t)R(t, s). Proposition R(t, t0) is the only solution in Mn(K) of the initial value problem with M(t) Mn(K). M(t) = A(t)M(t) t M(t0) = I, Resolvent matrix p. 13 Resolvent matrix p. 14 A variation of constants formula The solution to the IVP consisting of the linear homogeneous nonautonomous system (LH) with initial conition x(t0) = x0 is given by φ(t) = R(t, t0)x0. (Variation of constants formula) Consier the IVP x = A(t)x + g(t, x) x(t0) = x0, (2a) (2b) where g : R R n R n a smooth function, an let R(t, t0) be the resolvent associate to the homogeneous system x = A(t)x, with R efine on some interval J t0. Then the solution φ of (2) is given by t φ(t) = R(t, t0)x0 + R(t, s)g(φ(s), s)s, (3) on some subinterval of J. t0 Resolvent matrix p. 15 Resolvent matrix p. 16
Autonomous linear systems Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Consier the autonomous affine system x = Ax + B, t (A) an the associate homogeneous autonomous system t x = Ax. (L) Autonomous linear systems p. 18 Exponential of a matrix Properties of the matrix exponential Definition (Matrix exponential) Let A Mn(K) with K = R or C. The exponential of A, enote e At, is a matrix in Mn(K), efine by e At t k = I + k! Ak, k=1 where I is the ientity matrix in Mn(K). Φ(t) = e At is a funamental matrix for (L) for t R. The resolvent for (L) is given for t J by R(t, t0) = e A(t t0) = Φ(t t0). e At1 e At2 = e A(t1+t2) for all t1, t2 R. 1 Ae At = e At A for all t R. (e At ) 1 = e At for all t R. The unique solution φ of (L) with φ(t0) = x0 is given by φ(t) = e A(t t0) x0. Autonomous linear systems p. 19 Autonomous linear systems p. 20
Computing the matrix exponential Let P be a nonsingular matrix in Mn(R). We transform the IVP t x = Ax x(t0) = x0 using the transformation x = Py or y = P 1 x. The ynamics of y is y = (P 1 x) = P 1 x = P 1 Ax = P 1 APy (L IVP) We have thus transforme IVP (L IVP) into t y = P 1 APy y(t0) = P 1 x0 From the earlier result, we then know that the solution of (L IVP y) is given by ψ(t) = e P 1 AP(t t0) P 1 x0, an since x = Py, the solution to (L IVP) is given by φ(t) = Pe P 1 AP(t t0) P 1 x0. So everything epens on P 1 AP. (L IVP y) The initial conition is y0 = P 1 x0. Autonomous linear systems p. 21 Autonomous linear systems p. 22 Diagonalizable case Assume P nonsingular in Mn(R) such that λ1 0 P 1 AP = 0 λn with all eigenvalues λ1,..., λn ifferent. We have k e P 1AP t k λ1 0 = I + k! k=1 0 λn Autonomous linear systems p. 23 Autonomous linear systems p. 24
For a (block) iagonal matrix M of the form m11 0 M = 0 mnn there hols m k 11 0 M k = 0 mnn k Therefore, e P 1AP t k λ k 1 0 = I + k! k=1 0 λ k n k=0 tk k! λk 1 0 = 0 e λ1t 0 = 0 e λnt k=0 tk k! λk n Autonomous linear systems p. 25 Autonomous linear systems p. 26 Noniagonalizable case An so the solution to (L IVP) is given by e λ1t 0 φ(t) = P P 1 x0. 0 e λnt The Joran canonical form is J0 0 P 1 AP = 0 Js so we use the same property as before (but with block matrices now), an e J0t 0 e P 1APt = 0 e Jst Autonomous linear systems p. 27 Autonomous linear systems p. 28
The first block in the Joran canonical form takes the form λ0 0 J0 = 0 λk an thus, as before, e λ0t 0 e J0t = 0 e λkt Other blocks Ji are written as Ji = λk+ii + Ni with I the ni ni ientity an Ni the ni ni nilpotent matrix 0 1 0 0 Ni = 1 0 0 λk+ii an Ni commute, an thus e Ji t = e λk+i t e Ni t Autonomous linear systems p. 29 Autonomous linear systems p. 30 Since Ni is nilpotent, Ni k = 0 for all k ni, an the series e Ni t terminates, an t 1 t n i 1 (ni 1)! e Ji t = e λk+i t t 0 1 n i 2 (ni 2)! 0 1 For all (t0, x0) R R n, there is a unique solution x(t) to (L IVP) efine for all t R. Each coorinate function of x(t) is a linear combination of functions of the form t k e αt cos(βt) an t k e αt sin(βt) where α + iβ is an eigenvalue of A an k is less than the algebraic multiplicity of the eigenvalue. Autonomous linear systems p. 31 Autonomous linear systems p. 32
Fixe points (equilibria) Orbits, limit sets Orbits an limit sets are efine as for maps. Definition A fixe point (or equilibrium point, or critical point) of an autonomous ifferential equation x = f (x) is a point p such that f (p) = 0. For a nonautonomous ifferential equation x = f (t, x), a fixe point satisfies f (t, p) = 0 for all t. A fixe point is a solution. For the equation x = f (x), the subset {x(t), t I }, where I is the maximal interval of existence of the solution, is an orbit. If the maximal solution x(t, x0) of x = f (x) is efine for all t 0, where f is Lipschitz on an open subset V of R n, then the omega limit set of x0 is the subset of V efine by ( ) ω(x0) = {x(t, x0) : t τ} V }. τ=0 Proposition A point q is in ω(x0) iff there exists a sequence {tk} such that limk tk = an limk x(tk, x0) = q V. Autonomous linear systems p. 33 Autonomous linear systems p. 34 Contracting linear equation Definition (Liapunov stable orbit) The orbit of a point p is Liapunov stable for a flow φt if, given ε > 0, there exists δ > 0 such that (x, p) < δ implies that (φt(x), φt(p)) < ε for all t 0. If p is a fixe point, then this is written (φt(x), p) < ε. Definition (Asymptotically stable orbit) The orbit of a point p is asymptotically stable (or attracting) for a flow φt if it is Liapunov stable, an there exists δ1 > 0 such that (x, p) < δ1 implies that limt (φt(x), φt(p)) = 0. If p is a fixe point, then it is asymptotically stable if it is Liapunov stable an there exists δ1 > 0 such that (x, p) < δ1 implies that ω(x) = {p}. Let A Mn(R), an consier the equation (L). Then the following conitions are equivalent. 1. There is a norm A on R n an a constant a > 0 such that for any x0 R n an all t 0, e At x0 A e at x0 A. 2. There is a norm B on R n an constants a > 0 an C 1 such that for any x0 R n an all t 0, e At x0 B Ce at x0 B. 3. All eigenvalues of A have negative real parts. In that case, the origin is a sink or attracting, the flow is a contraction (antonyms source, repelling an expansion). Autonomous linear systems p. 35 Autonomous linear systems p. 36
Hyperbolic linear equation Definition (Stable eigenspace) The stable eigenspace of A Mn(R) is E s = span{v : v generalize eigenvector for eigenvalue λ, with R(λ) < 0} Definition The linear ifferential equation (L) is hyperbolic if A has no eigenvalue with zero real part. Definition (Center eigenspace) The center eigenspace of A Mn(R) is E c = span{v : v generalize eigenvector for eigenvalue λ, with R(λ) = 0} Definition (Unstable eigenspace) The unstable eigenspace of A Mn(R) is Autonomous linear systems p. 37 We can write R n = E s E u +E c, an in the case that E c =, then R n = E s E u is calle a hyperbolic splitting. The symbol stans for irect sum. Definition (Direct sum) Let U, V be two subspaces of a vector space X. Then the span of U an V is efine by u + v for u U an v V. If U an V are isjoint except for 0, then the span of U an V is calle the irect sum of U an V, an is enote U V. Autonomous linear systems p. 39 E u = span{v : v generalize eigenvector for eigenvalue λ, with R(λ) > 0} Autonomous linear systems p. 38 Trichotomy Define V s = {v : there exists a > 0 an C 1 such that e At v Ce at v for t 0}. V u = {v : there exists a > 0 an C 1 such that e At v Ce a t v for t 0}. V c = {v : for all a > 0, e At v e a t 0 as t ± }. The following are true. 1. The subspaces E s, E u an E c are invariant uner the flow e At. 2. There hols that E s = V s, E u = V u an E c = V c, an thus e At E u is an exponential expansion, eat E s is an exponential contraction, an e At E c grows subexponentially as t ±. Autonomous linear systems p. 40
Topologically conjugate linear ODEs Definition (Topologically conjugate flows) Let φt an ψt be two flows on a space M. φt an ψt are topologically conjugate if there exists an homeomorphism h : M M such that for all x M an all t R. h φt(x) = ψt h(x), Definition (Topologically equivalent flows) Let φt an ψt be two flows on a space M. φt an ψt are topologically equivalent if there exists an homeomorphism h : M M an a function α : R M R such that Let A, B Mn(R). 1. If all eigenvalues of A an B have negative real parts, then the linear flows e At an e Bt are topologically conjugate. 2. Assume that the system is hyperbolic, an that the imension of the stable eigenspace of A is equal to the imension of the eigenspace of B. Then the linear flows e At an e Bt are topologically conjugate. h φ α(t+s,x) (x) = ψt h(x), for all x M an all t R, an where α(t, x) is monotonically increasing in t for each x an onto all of R. Autonomous linear systems p. 41 Autonomous linear systems p. 42 Let A, B Mn(R). Assume that e At an e Bt are linearly conjugate, i.e., there exists M with e Bt = Me At M 1. Then A an B have the same eigenvalues. Autonomous linear systems p. 43