Outline. Summary of topics relevant for the final. Outline of this part. Scalar difference equations. General theory of ODEs. Linear ODEs.

Outline Scalar ifference equations Summary of topics relevant for the final General theory of ODEs Linear ODEs Linear maps Analysis near fixe points (linearization) Bifurcations How to analyze a system p. 1 Some matrix properties p. 2 Outline of this part Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Scalar ifference equations p. 3 Scalar ifference equations p. 4

First-orer ifference equation A ifference equation takes the form x(n + 1) = f (x(n)), which is also enote xn+1 = f (xn). Starting from an initial point x0, we have x1 = f (x0) x2 = f (x1) = f (f (x0)) = f 2 (x0) x3 = f (x2) = f (f (f (x0))) = f 3 (x0)... Scalar ifference equations p. 5 Definition 1 (Iterates) f (x0) is the first iterate of x0 uner f ; f 2 (x0) is the secon iterate of x0 uner f. More generally, f n (x0) is the nth iterate of x0 uner f. By convention, f 0 (x0) = x0. Definition 2 (Orbits) The set {f n (x0) : n 0} is calle the forwar orbit of x0 an is enote O + (x0). The backwar orbit O (x0) is efine, if f is invertible, by the negative iterates of f. Lastly, the (whole) orbit of x0 is {f k (x0) : < k < }. The forwar orbit is also calle the positive orbit. The function f is always assume to be continuous. If its erivative or secon erivative is use in a result, then the assumption is mae that f C 1 or f C 2.. Scalar ifference equations p. 6 Perioic points Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Definition 3 (Perioic point) A point p is a perioic point of (least) perio n if f n (p) = p an f j (p) p for 0 < j < n. Definition 4 (Fixe point) A perioic point with perio n = 1 is calle a fixe point. Definition 5 (Eventually perioic point) A point p is an eventually perioic point of perio n if there exists m > 0 such that f m+n (p) = f m (p), so that f j+n (p) = f j (p) for all j m an f m (p) is a perioic point. Scalar ifference equations p. 7 Scalar ifference equations p. 8

Fining fixe points an perioic points A fixe point is such that f (x) = x, so it lies at the intersection of the first bisectrix y = x with the graph of f (x). A perioic point is such that f n (x) = x, it is thus a fixe point of the nth iterate of f, an so lies at the intersection of the first bisectrix y = x with the graph of f n (x). Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Scalar ifference equations p. 9 Stable set Scalar ifference equations p. 10 Unstable set Definition 6 (Forwar asymptotic point) q is forwar asymptotic to p if f j (q) f j (p) 0 as j. If p is n-perioic, then q is asymptotic to p if Definition 7 (Stable set) The stable set of p is f jn (q) p 0 as j. Definition 8 (Backwar asymptotic point) If f is invertible, then q is backwar asymptotic to p if Definition 9 (Unstable set) The unstable set of p is f j (q) f j (p) 0 as j. W u (p) = {q : q backwar asymptotic to p}. W s (p) = {q : q forwar asymptotic to p}. Scalar ifference equations p. 11 Scalar ifference equations p. 12

Stability Definition 10 (Stable fixe point) A fixe point p is stable (or Lyapunov stable) if, for every ε > 0, there exists δ > 0 such that x0 p < δ implies f n (x0) p < ε for all n > 0. If a fixe point p is not stable, then it is unstable. Definition 11 (Attracting fixe point) A fixe point p is attracting if there exists η > 0 such that x(0) p < η implies lim x(n) = p. n If η =, then p is a global attractor (or is globally attracting). Definition 12 (Asymptotically stable point) A fixe point p is asymptotically stable if it is stable an attracting. It is globally asymptotically stable if η =. The point oes not have to be a fixe point to be stable. Definition 13 A point p is stable if for every ε > 0, there exists δ > 0 such that if x p < δ, then f k (x) f k (p) < ε for all k 0. Another characterization of asymptotic stability: Definition 14 A point p is asymptotically stable if it is stable an W s (p) contains a neighborhoo of p. Can be use with perioic point, in which case we talk of attracting perioic point (or perioic sink). A perioic point p for which W u (p) is a neighborhoo of p is a repelling perioic point (or perioic source). Scalar ifference equations p. 13 Scalar ifference equations p. 14 Conition for stability/instability Theorem 15 Let f : R R be C 1. 1. If p is a n-perioic point of f such that (f n ) (p) < 1, then p is an attracting perioic point. 2. If p is a n-perioic point of f such that (f n ) (p) > 1, then p is repelling. Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Scalar ifference equations p. 15 Scalar ifference equations p. 16

ω-limit points an sets α-limit points an sets Definition 16 A point y is an ω-limit point of x for f is there exists a sequence {nk} going to infinity as k such that lim k (f nk (x), y) = 0. The set of all ω-limit points of x is the ω-limit set of x an is enote ω(x). Definition 17 Suppose that f is invertible. A point y is an α-limit point of x for f is there exists a sequence {nk} going to minus infinity as k such that lim k (f nk (x), y) = 0. The set of all α-limit points of x is the α-limit set of x an is enote α(x). Scalar ifference equations p. 17 Scalar ifference equations p. 18 Invariant sets Theorem 19 Let f : X X be continuous on a complete metric space X. Then Definition 18 Let S X be a set. S is positively invariant (uner the flow of f ) if f (x) S for all x S, i.e., f (S) S. S is negatively invariant if f 1 (S) S. S is invariant if f (S) = S. 1. If f j (x) = y for some j, then ω(x) = ω(y). 2. For any x, ω(x) is close an positively invariant. 3. If O + (x) is containe in some compact subset of X, then ω(x) is nonempty an compact an (f n (x), ω(x)) 0 as n. 4. If D X is close an positively invariant, an x D, then ω(x) D. 5. If y ω(x), then ω(y) ω(x). Scalar ifference equations p. 19 Scalar ifference equations p. 20

Parametrize families of functions Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Consier the logistic map xt+1 = µxt(1 xt), (1) where µ is a parameter in R+, an x will typically be taken in [0, 1]. Let fµ(x) = µx(1 x). (2) The function fµ is calle a parametrize family of functions. Scalar ifference equations p. 21 Bifurcations Scalar ifference equations p. 22 Topological conjugacy Definition 20 (Bifurcation) Let fµ be a parametrize family of functions. Then there is a bifurcation at µ = µ0 (or µ0 is a bifurcation point) if there exists ε > 0 such that, if µ0 ε < a < µ0 an µ0 < b < µ0 + ε, then the ynamics of fa(x) are ifferent from the ynamics of fb(x). An example of ifferent woul be that fa has a fixe point (that is, a 1-perioic point) an fb has a 2-perioic point. Definition 21 (Topological conjugacy) Let f : D D an g : E E be functions. Then f topologically conjugate to g if there exists a homeomorphism τ : D E, calle a topological conjugacy, such that τ f = g τ. Formally, fa an fb are topologically conjugate to two ifferent functions. Scalar ifference equations p. 23 Scalar ifference equations p. 24

Theorem 22 Let D an E be subsets of R, f : D D, g : E E, an τ : D E be a topological conjugacy of f an g. Then 1. τ 1 : E D is a topological conjugacy. 2. τ f n = g n τ for all n N. 3. p is a perioic point of f with least perio n iff τ(p) is a perioic point of g with least perio n. 4. If p is a perioic point of f with stable set W s (p), then the stable set of τ(p) is τ (W s (p)). 5. The perioic points of f are ense in D iff the perioic points of g are ense in E. 6. f is topologically transitive on D iff g is topologically transitive on E. 7. f is chaotic on D iff g is chaotic on E. Scalar ifference equations First-orer ifference equations Perioic points Stability Limit sets The cascae of bifurcations to chaos Chaos Devaney s efinition Scalar ifference equations p. 25 Topologically transitive function Scalar ifference equations p. 26 Sensitive epenence on initial conitions Definition 23 The function f : D D is topologically transitive on D if for any open sets U an V that interset D, there exists z U D an n N such that f n (z) D. Equivalently, f is topologically transitive on D if for any two points x, y D an any ε > 0, there exists z D such that z x < ε an f n (x) y < ε for some n. Definition 24 The function f : D D exhibits sensitive epenence on initial conitions if there exists δ > 0 such that for any x D an any ε > 0, there exists y D an n N such that x y < ε an f n (x) f n (y) > δ. Scalar ifference equations p. 27 Scalar ifference equations p. 28

Chaos Outline of this part The following in ue to Devaney. There are other efinitions. Definition 25 The function f : D D is chaotic if 1. the perioic points of f are ense in D, 2. f is topologically transitive, 3. an f exhibits sensitive epenence on initial conitions. General theory of ODEs ODEs Existence of solutions to IVPs Scalar ifference equations p. 29 General theory of ODEs ODEs Existence of solutions to IVPs General theory of ODEs p. 31 General theory of ODEs p. 30 Orinary ifferential equations Definition 26 (ODE) An orinary ifferential equation (ODE) is an equation involving one inepenent variable (often calle time), t, an a epenent variable, x(t), with x R n, n 1, an taking the form x = f (t, x), t where f : R R n R n is a function, calle the vector fiel. Definition 27 (IVP) An initial value problem (IVP) consists in an ODE an an initial conition, x = f (t, x) t (3) x(t0) = x0, where t0 R an x0 R n is the initial conition. General theory of ODEs p. 32

General theory of ODEs ODEs Existence of solutions to IVPs Flow Consier an autonomous IVP, x = f (t, x) t x(0) = x0, that is, where f oes not epen explicitly on t. Let φ t (x0) (the notations φt(x0) an φ(t, x0) are also use) be the solution of (4) with given initial conition. We have (4) φ 0 (x0) = x0 an t φt (x0) = f (φ t (x0)) for all t for which it is efine. φ t (x0) is the flow of the ODE. General theory of ODEs p. 33 Lipschitz function General theory of ODEs p. 34 Existence an uniqueness Definition 28 (Lipschitz function) Let f : U R n R n. If there exists K > 0 such that f (x) f (y) K x y for all x, y U, then f is calle a Lipschitz function with Lipschitz constant K. The smallest K for which the property hols is enote Lip(f ). Theorem 29 (Existence an Uniqueness) Let U R n be an open set, an f : U R n be a Lipschitz function. Let x0 U an t0 R. Then there exists α > 0, an a unique solution x(t) to the ifferential equation x = f (x) efine on t0 α t t0 + α, such that x(t0) = x0. Remark: f C 1 f is Lipschitz. General theory of ODEs p. 35 General theory of ODEs p. 36

Continuous epenence on IC Interval of existence of solutions Theorem 30 (Continuous epenence on initial conitions) Let U R n be an open set, an f : U R n be a Lipschitz function. Then the solution φ t (x0) epens continuously on the initial conition x0. Theorem 31 Let U be an open subset of R n, anf f : U R n be C 1. Given x U, let (t, t+) be the maximal interval of efinition for φ t (x). If t+ <, then given any compact subset C U, there exists tc with 0 tc < t+ such that φ tc (x) C. Similarly, if t >, then there exists tc with t < tc 0 such that φ tc (x) C. In particular, if f : R n R n is efine on all of R n an f (x) is boune, then the solutions exist for all t. General theory of ODEs p. 37 Outline of this part General theory of ODEs p. 38 Linear ODEs Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Nonautonomous nonhomogeneous linear equations Definition 32 (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 (5) x(t0) = x0. Linear ODEs p. 39 Linear ODEs p. 40

Types of systems 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. Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Nonautonomous nonhomogeneous linear equations If A(t + T ) = A(t) for some T > 0 an all t, then linear perioic. Linear ODEs p. 41 Existence an uniqueness of solutions Linear ODEs p. 42 The vector space of solutions Theorem 33 (Existence an Uniqueness) Solutions to (5) 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. Theorem 34 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) Linear ODEs p. 43 Linear ODEs p. 44

Funamental matrix Funamental matrix solution Definition 35 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). 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. Theorem 36 A funamental matrix Φ of (LH) satisfies the matrix equation X = A(t)X on the interval J.- Linear ODEs p. 45 Linear ODEs p. 46 Abel s formula Theorem 37 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. Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Nonautonomous nonhomogeneous linear equations Linear ODEs p. 47 Linear ODEs p. 48

The resolvent matrix Definition 38 (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. Proposition 1 The resolvent matrix satisfies the ientities 1. R(t, t) = I, 2. R(t, s)r(s, u) = R(t, u), 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). Linear ODEs p. 49 Proposition 2 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, Theorem 39 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. Linear ODEs p. 50 A variation of constants formula Theorem 40 (Variation of constants formula) Consier the IVP x = A(t)x + g(t, x) x(t0) = x0, (6a) (6b) 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 (6) is given by t φ(t) = R(t, t0)x0 + R(t, s)g(φ(s), s)s, (7) t0 on some subinterval of J. Linear ODEs p. 51 Linear ODEs p. 52

Autonomous linear systems Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Nonautonomous nonhomogeneous linear equations Consier the autonomous affine system x = Ax + B, t (A) an the associate homogeneous autonomous system t x = Ax. (L) Linear ODEs p. 53 Exponential of a matrix Linear ODEs p. 54 Properties of the matrix exponential Definition 41 (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. Linear ODEs p. 55 Linear ODEs p. 56

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. Linear ODEs p. 57 Linear ODEs p. 58 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 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 Linear ODEs p. 59 Linear ODEs p. 60

Noniagonalizable case Therefore, e λ1t 0 e P 1AP =... 0 e λnt 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 Linear ODEs p. 61 Linear ODEs p. 62 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 Linear ODEs p. 63 Linear ODEs p. 64

Fixe points (equilibria) 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 Definition 42 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. Linear ODEs p. 65 Linear ODEs p. 66 Orbits, limit sets Orbits an limit sets are efine as for maps. 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 3 A point q is in ω(x0) iff there exists a sequence {tk} such that limk tk = an limk x(tk, x0) = q V. Definition 43 (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 44 (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}. Linear ODEs p. 67 Linear ODEs p. 68

Contracting linear equation Theorem 45 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, Hyperbolic linear equation Definition 46 The linear ifferential equation (L) is hyperbolic if A has no eigenvalue with zero real part. 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). Linear ODEs p. 69 Definition 47 (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 48 (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 49 (Unstable eigenspace) The unstable eigenspace of A Mn(R) is E u = span{v : v generalize eigenvector for eigenvalue λ, with R(λ) > 0} Linear ODEs p. 71 Linear ODEs p. 70 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 50 (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. Linear ODEs p. 72

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 ± }. Theorem 51 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 ±. Linear ODEs p. 73 Topologically conjugate linear ODEs Definition 52 (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 53 (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 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. Linear ODEs p. 74 Theorem 54 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. Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Nonautonomous nonhomogeneous linear equations Theorem 55 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. Linear ODEs p. 75 Linear ODEs p. 76

Outline of this part Theorem 56 Consier x = A(t)x + g(t) (LNH) an x = A(t)x (LH) Linear maps 1. If x1 an x2 are two solutions of (LNH), then x1 x2 is a solution to (LH). 2. If xn is a solution to (LNH) an xh is a solution to (LH), then xn + xh is a solution to (LNH). 3. If xn is a solution to (LNH) an M is a funamental matrix solution of (LH), then any solution of (LNH) can be written as xn + M(t)v. Linear ODEs p. 77 Similarities between ODEs an maps Let A Mn(R). Let v be an eigenvector associate to the eigenvalue λ. Then By inuction, A 2 v = A(Av) = A(λv) = λav = λ 2 v A n v = λ n v, i.e., v is an eigenvector of the matrix A n, associate to the eigenvalue λ n. Thus, if λ < 1, then A n v = λ n v goes to zero as n. Linear maps p. 79 Linear maps p. 78 Linear map corresponing to a matrix with all eigenvalues of moulus less than 1 is a linear contraction, with the origin a linear sink or attracting fixe point. If all eigenvalues have moulus larger than 1, then the map inuce by A is a linear expansion, an the origin is a linear source or repelling fixe point. The map Ax is a hyperbolic linear map if all eigenvalues of A have moulus ifferent of 1. Linear maps p. 80

Definition 57 (Stable eigenspace) The stable eigenspace of A Mn(R) is E s = span{v : v generalize eigenvector for eigenvalue λ, with λ < 1} Definition 58 (Center eigenspace) The center eigenspace of A Mn(R) is E c = span{v : v generalize eigenvector for eigenvalue λ, with λ = 1} Outline of this part Analysis near fixe points (linearization) Stable manifol theorem Hartman-Grobman theorem Lyapunov functions Perioic orbits Definition 59 (Unstable eigenspace) The unstable eigenspace of A Mn(R) is E u = span{v : v generalize eigenvector for eigenvalue λ, with λ > 1} Linear maps p. 81 Analysis near fixe points (linearization) p. 82 Objective Consier the autonomous nonlinear system in R n x = f (x) (8) The object here is to show two results which link the behavior of (8) near a hyperbolic equilibrium point x to the behavior of the linearize system x = Df (x )(x x ) (9) about that same equilibrium. Analysis near fixe points (linearization) Stable manifol theorem Hartman-Grobman theorem Lyapunov functions Perioic orbits Analysis near fixe points (linearization) p. 83 Analysis near fixe points (linearization) p. 84

Stable manifol theorem Theorem 60 (Stable manifol theorem) Let f C 1 (E), E be an open subset of R n containing a point x such that f (x ) = 0, an let φt be the flow of the nonlinear system (8). Suppose that Df (x ) has k eigenvalues with negative real part an n k eigenvalues with positive real part. Then there exists a k-imensional ifferentiable manifol S tangent to the stable subspace E s of the linear system (9) at x such that for all t 0, φt(s) S an for all x0 S, Analysis near fixe points (linearization) Stable manifol theorem Hartman-Grobman theorem Lyapunov functions Perioic orbits lim φt(x0) = x t an there exists an (n k)-imensional ifferentiable manifol U tangent to the unstable subspace E u of (9) at x such that for all t 0, φt(u) U an for all x0 U, lim φt(x0) = x t Analysis near fixe points (linearization) p. 85 HG theorem Formulation 1 Theorem 61 (Hartman-Grobman) Suppose that x is an equilibrium point of the nonlinear system (8). Let ϕt be the flow of (8), an ψt be the flow of the linearize system x = Df (x )(x x ). If x is a hyperbolic equilibrium, then there exists an open subset D of R n containing x, an a homeomorphism G with omain in D such that G(ϕt(x)) = ψt(g(x)) whenever x D an both sies of the equation are efine. Analysis near fixe points (linearization) p. 86 HG theorem Formulation 2 Theorem 62 (Hartman-Grobman) Let f C 1 (E), E an open subset of R n containing x where f (x ) = 0, an let φt be the flow of the nonlinear system (8). Suppose that the matrix A = Df (x ) has no eigenvalue with zero real part. Then there exists a homeomorphism H of an open set U containing x onto an open set V containing the origin such that for each x0 U, there is an open interval I0 R containing x such that for all x0 U an t I0, H φt(x0) = e At H(x0); i.e., H maps trajectories of (8) near the origin onto trajectories of x = Df (x )(x x ) near the origin an preserves the parametrization by time. Analysis near fixe points (linearization) p. 87 Analysis near fixe points (linearization) p. 88

Lyapunov function Analysis near fixe points (linearization) Stable manifol theorem Hartman-Grobman theorem Lyapunov functions Perioic orbits We consier x = f (x), x R n, with flow φt(x). Let p be a fixe point. Definition 63 (Weak Lyapunov function) The function V C 1 (U, R) is a weak Lyapunov function for φt on the open neighborhoo U p if V (x) > V (p) an t V (φt(x)) 0 for all x U \ {p}. Definition 64 (Lyapunov function) The function V C 1 (U, R) is a (strong) Lyapunov function for φt on the open neighborhoo U p if V (x) > V (p) an t V (φt(x)) < 0 for all x U \ {p}. Analysis near fixe points (linearization) p. 89 Analysis near fixe points (linearization) p. 90 Theorem 65 Suppose that p is a fixe point of x = f (x), U is a neighborhoo of p, an V : U R. 1. If V is a weak Lyapunov function for φt on U, then p is Liapunov stable. 2. If V is a Lyapunov function for φt on U, then p is asymptotically stable. Analysis near fixe points (linearization) Stable manifol theorem Hartman-Grobman theorem Lyapunov functions Perioic orbits Analysis near fixe points (linearization) p. 91 Analysis near fixe points (linearization) p. 92

Perioic orbits for flows Definition 66 (Perioic point) Let x = f (x), an φt(x) be the associate flow. p is a perioic point with (least) perio T, or T -perioic point, if φt (p) = p an φt(p) p for 0 < t < T. Definition 67 (Perioic orbit) If p is a T -perioic point, then O(p) = {φt(p) : 0 t T } is the orbit of p, calle a perioic orbit or a close orbit. Definition 68 (Stable perioic orbit) A perioic orbit γ is stable if for each ε > 0, there exists a neighborhoo U of γ such that for all x U, (γ + x, γ) < ε, i.e., if for all x U an t 0, (φt(x), γ) < ε. Definition 69 (Unstable perioic orbit) A perioic orbit that is not stable is unstable. Definition 70 (Asymptotically stable perioic orbit) A perioic orbit γ is asymptotically stable if it is stable an for all x in some neighborhoo U of γ, lim (φt(x), γ) = 0. t Analysis near fixe points (linearization) p. 93 Hyperbolic perioic orbits Definition 71 (Characteristic multipliers) If γ is a perioic orbit of perio T, with p γ, then the eigenvalues of the Poincaré map DφT (p) are 1, λ1,..., λn 1. The eigenvalues λ1,..., λn 1 are calle the characteristic multipliers of the perioic orbit. Definition 72 (Hyperbolic perioic orbit) A perioic orbit is hyperbolic if none of the characteristic multipliers has moulus 1. Definition 73 (Perioic sink) A perioic orbit which has all characteristic multipliers λ such that λ < 1. Definition 74 (Perioic source) A perioic orbit which has all characteristic multipliers λ such that λ > 1. Analysis near fixe points (linearization) p. 95 Analysis near fixe points (linearization) p. 94 Theorem 75 1. If φt(x) is a solution of x = f (x), γ is a perioic orbit of perio T, an p γ, then DφT (p) has 1 as an eigenvalue with eigenvector f (p). 2. If p an q belong to the same T -perioic orbit γ, then DφT (p) an DφT (q) are linearly conjugate an thus have the same eigenvalues. Analysis near fixe points (linearization) p. 96

Outline of this part Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf Bifurcations p. 97 Bifurcations p. 98 The general context of bifurcations Consier the iscrete time system or the continuous time system xt+1 = f (xt, µ) = fµ(xt) (10) x = f (x, µ) = fµ(x) (11) for µ R. We start with a function f : R 2 R, C r when a map is consiere, C 1 when continuous time is consiere. Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf In both cases, the function f can epen on some parameters. We are intereste in the ifferences of qualitative behavior, as one of these parameters, which we call µ, varies. Bifurcations p. 99 Bifurcations p. 100

Types of bifurcations (iscrete time) Types of bifurcations (continuous time) Sale-noe (or tangent): xt+1 = µ + xt + xt 2 Transcritical: xt+1 = (µ + 1)xt + xt 2 Pitchfork: xt+1 = (µ + 1)xt xt 3 Sale-noe Transcritical Pitchfork supercritical subcritical x = µ x 2 x = µx x 2 x = µx x 3 x = µx + x 3 Bifurcations p. 101 Bifurcations p. 102 Sale-noe for maps Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf Theorem 76 Assume f C r with r 2, for both x an µ. Suppose that 1. f (x0, µ0) = x0, 2. f (x0) = 1, µ0 3. f (x0) 0 an µ0 f 4. (x0, µ0) 0. µ Then I x0 an N µ0, an m C r (I, N), such that 1. f m(x) (x) = x, 2. m(x0) = µ0, 3. the graph of m gives all the fixe points in I N. Bifurcations p. 103 Bifurcations p. 104

Sale-noe for continuous equations Theorem 77 (cont.) Moreover, m (x0) = 0 an 2 f (x0, µ0) m (x0) = x 2 0. f µ (x0, µ0) These fixe points are attracting on one sie of x0 an repelling on the other. Consier the system x = f (x, µ), x R. Suppose that f (x0, µ0) = 0. Further, assume that the following nonegeneracy conitions hol: 1. a0 = 1 2 f 2 x2 (x0, µ0) 0, f 2. µ (x0, µ0) 0. Then, in a neighborhoo of (x0, µ0), the equation x = f (x, µ) is topologically equivalent to the normal form x = γ + sign(a0)x 2 Bifurcations p. 105 Sale-noe for continuous systems Theorem 78 Consier the system x = f (x, µ), x R n. Suppose that f (x, 0) = x0 = 0. Further, assume that 1. The Jacobian matrix A0 = Df (0, 0) has a simple zero eigenvalue, 2. a0 0, where a0 = 1 2 p, B(q, q) = 1 2 p, f (τq, 0) 2 τ 2 τ=0 3. fµ(0, 0) 0. B is the bilinear function with components n 2 fj(ξ, 0) Bj(x, y) = xkyl, j = 1,..., n ξk ξl k,l=1 ξ=0 an p, q = p T q the stanar inner prouct. Bifurcations p. 107 Bifurcations p. 106 Theorem 79 (cont.) Then, in a neighborhoo of the origin, the system x = f (x, µ) is topologically equivalent to the suspension of the normal form by the stanar sale, y = γ + sign(a0)y 2 y S = ys y U = yu with y R, ys R ns an yu R nu, where ns + nu + 1 = n an ns is number of eigenvalues of A0 with negative real parts. Bifurcations p. 108

Pitchfork bifurcation The ODE x = f (x, µ), with the function f (x, µ) satisfying Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf (f is o), f (x, µ) = f ( x, µ) f x (0, µ0) = 0, 2 f x 2 (0, µ0) = 0, 3 f (0, µ0) 0, x 3 f r (0, µ0) = 0, 2 f (0, µ0) 0. r x has a pitchfork bifurcation at (x, µ) = (0, µ0). The form of the pitchfork is etermine by the sign of the thir erivative: 3 f (0, µ0) x 3 { < 0, supercritical > 0, subcritical Bifurcations p. 109 Bifurcations p. 110 Canonical example Bifurcations General context A few types of bifurcations Sale-noe Pitchfork Hopf Consier the system x = y + x(µ x 2 y 2 ) y = x + y(µ x 2 y 2 ) Transform to polar coorinates: r = r(µ r 2 ) θ = 1 Bifurcations p. 111 Bifurcations p. 112

Hopf bifurcation Supercritical or subcritical Hopf? Theorem 80 (Hopf bifurcation theorem) Let x = A(µ)x + F (µ, x) be a C k planar vector fiel, with k 0, epening on the scalar parameter µ such that F (µ, 0) = 0 an DxF (µ, 0) = 0 for all µ sufficiently close enough to the origin. Assume that the linear part A(µ) at the origin has the eigenvalue α(µ) ± iβ(µ), with α(0) = 0 an β(0) 0. Furthermore, assume the eigenvalues cross the imaginary axis with nonzero spee, i.e., µ α(µ) 0. µ=0 Then, in any neighborhoo U (0, 0) in R 2 an any given µ0 > 0, there exists a µ with µ < µ0 such that the ifferential equation x = A( µ)x + F ( µ, x) has a nontrivial perioic orbit in U. Transform the system into ( ) ( ) ( x α(µ) β(µ) x = t y β(µ) α(µ) y ) + ( ) f1(x, y, µ) = g1(x, y, µ) The Jacobian at the origin is ( ) α(µ) β(µ) J(µ) = β(µ) α(µ) ( ) f (x, y, µ) g(x, y, µ) an thus eigenvalues are α(µ) ± iβ(µ), an α(0) = 0 an β(0) > 0. Bifurcations p. 113 Supercritical or subcritical Hopf? (cont.) Bifurcations p. 114 Outline of this part Define C = fxxx + fxyy + gxxy + gyyy + 1 ( fxy (fxx + fyy) + gxy (gxx + gyy) + fxxgxx fyygyy), β(0) evaluate at (0, 0) an for µ = 0. Then, if α(0)/µ > 0, 1. If C < 0, then for µ < 0, the origin is a stable spiral, an for µ > 0, there exists a stable perioic solution an the origin is unstable (supercritical Hopf). 2. If C > 0, then for µ < 0, there exists an unstable perioic solution an the origin is unstable, an for µ > 0, the origin is unstable (subcritical Hopf). 3. If C = 0, the test is inconclusive. How to analyze a system Bifurcations p. 115 How to analyze a system p. 116

Position of the problem Battle plan You are given an autonomous system, whether in iscrete time or in continuous time xt+1 = f (xt) (12) x = f (x) (13) with x R n an f : R n R n a C k function (k 2 for (12) or k 1 for (13)). 1. If you can solve explicitly (12) or (13), solve it explicitly. 2. If not (99% of the time, in real life), plan B: 2.1 Determine invariants. 2.2 Determine equilibria. 2.3 Stuy (local) stability of the equilibria. 2.4 Seek Lyapunov functions for global stability. 2.5 Stuy bifurcations that occur equilibria lose stability. 2.6 Use numerical techniques (not relevant for the final, though). What o you o now? How to analyze a system p. 117 Explicit solutions How to analyze a system p. 118 Look for invariants It happens.. so infrequently with nonlinear systems that most of the times, you will overlook the possibility. If a nonlinear system is integrable explicitly, it is often linke to the presence of invariants, that allow to reuce the imension (typically, 2 to 1). In case of linear systems, solutions can be foun explicitly (they can be complicate, or can be in an implicit form). If the system lives on a hyperplane, which is characterize by xi(t) C R or i x i = 0 i then its imension can be reuce, since one of the variables, say xi, can be expresse as C j i xi. The same can be true with subparts of the system, if for example some variables always appear as sums in the remaining equations. How to analyze a system p. 119 How to analyze a system p. 120

Stuy local stability Seek Lyapunov functions Compute the Jacobian matrix, an evaluate it at the equilibria (fixe points). If DTE, the fixe point is locally asymptotically stable if all eigenvalues have moulus less than 1, repelling (unstable) otherwise. In your case, if you nee to use a Lyapunov function, it will be provie.. Be sure to know how to ifferentiate the function, it is not always simple.. If ODE, the fixe point is locally asymptotically stable if all eigenvalues have negative real parts, unstable otherwise. How to analyze a system p. 121 Stuy bifurcations How to analyze a system p. 122 Outline of this part It can be a goo way to figure out what is happening.. Also, sometimes checking for a bifurcation can give you information about the stability of the equilibrium, without having to o the stability analysis. Some matrix properties Perron-Frobenius theorem Routh-Hurwitz criterion How to analyze a system p. 123 Some matrix properties p. 124

Some matrix properties Perron-Frobenius theorem Routh-Hurwitz criterion Nonnegative matrices Definition 81 (Nonnegative matrix) Let A = (aij) Mn(R). A is nonnegative iff i, j, aij 0. Definition 82 (Positive matrix) A is positive iff aij > 0 for all i, j = 1,..., n. Definition 83 (Irreucible matrix) A is irreucible iff for all i, j, there exists q N such that a q ij > 0. If A is not irreucible, it is reucible, an there is a permutation matrix P such that A is written in block triangular form, ( ) P 1 A11 0 AP = A21 A22 Definition 84 (Primitive matrix) A is primitive iff there exists q N such that i, j, a q ij > 0. Some matrix properties p. 125 Some matrix properties p. 126 Perron-Frobenius theorem Theorem 85 Let A Mn(R) be primitive. 1. There exists an eigenvalue λ1, real an positive, that is a simple, an such that any other eigenvalue λ verifies λ < λ1. To this eigenvalue, there correspons a strongly positive eigenvector, i.e., with all entries positive, an all other (left an right) eigenvectors of A have components of both signs. Some matrix properties Perron-Frobenius theorem Routh-Hurwitz criterion Some matrix properties p. 127 Some matrix properties p. 128

Properties of 2 2 matrices Consier the matrix ( ) a b M = c The characteristic polynomial of M is P(λ) = (a λ)( λ) bc = λ 2 (a + )λ + (a bc) = λ 2 tr(m)λ + et(m) Theorem 86 The matrix M has eigenvalues with negative real parts if, an only if, et(m) > 0 an tr(m) < 0. Some matrix properties p. 129