Linearization and hyperbolicity of the autonomous nonlinear equations

HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS ————oOo———— NGUYEN THI KIM OANH LINEARIZATION AND HYPERBOLICITY OF THE AUTONOMOUS NONLINEAR EQUATIONS GRADUATION THESIS HANOI, 01/2019 HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS ————oOo———— GRADUATION THESIS LINEARIZATION AND HYPERBOLICITY OF THE AUTONOMOUS NONLINEAR EQUATIONS Supervisor : Dr TRAN VAN BANG Student : NGUYEN THI KIM OANH Class : K41CLC HANOI, 01/2019 Thesis Assurance I assure for this is my research thesis which is completed under the guidance of Dr Tran Van Bang The results presented in the thesis are honest, and have never been published in any other thesis Student Nguyen Thi Kim Oanh Acknowledgment Before presenting the main content of the thesis, I would like to express my gratitude to the mathematics teachers, Hanoi Pedagogical University 2, teachers in the Analysis group as well as the teachers involved Teaching has dedicatedly conveyed valuable knowledge and created favorable conditions for me to successfully complete the course and thesis In particular, I would like to express my deep respect and gratitude to Dr Tran Van Bang, who directly instructed, just told to help me so that I could complete this thesis Due to limited time, capacity and conditions, the discourse cannot avoid errors Therefore, I look forward to receiving valuable comments from teachers and friends Student Nguyen Thi Kim Oanh Linearization and hyperbolicity NGUYEN THI KIM OANH Preface Differential equations are an important discipline of mathematics and have many applications in the fields of science and technology, which are considered as the bridge between theory and application Thus, the differential equation is a subject that is widely taught in universities at home and abroad Stability theory is one of the important qualitative properties in the study of differential equations For linear systems of equations, we have an explicit criterion for studying the stability of trivial solution of the system of equations by examining the sign of the real part of matrix [A] is eigen values However, many differential equations are expressed in non-linear forms, for instance in quasi linear one- the defferential equation system which has the form x = A(t)x + f (t), where [A]n×n is a square matrix A fairly useful method for studying the stability of nonlinear systems is the linearization method, we perform a transformation convert nonlinear defferential equations to linear form Then, stability of nonlinear systems solutions is evaluated through the stability of linearized systems solutions Through this thesis, I focus on the stability of nonlinear systems based on the linearization method, while studying the stability characteristics for solution of differential equations based on the stable manifold of that system Contents PRELIMINARIES 1.1 Differential equation 1.2 Flows 1.3 Limit sets and trajectories 1.4 Stability LINEARIZATION AND HYPERBOLICITY 11 2.1 Poincare’s Linearization theorem 11 2.2 Hyperbolic stationary points and the manifoid theorem 17 2.3 Persistence of hyperbolic stationary points 22 2.4 Structural stability 23 2.5 Nonlinear sink 24 2.6 The proof of the stable manifold theorem 29 References 32 Chapter PRELIMINARIES In this chapter, we discuss about the differential equation including of flows, trajectory and the stability 1.1 Differential equation Consider the differential equation in the form x˙ = f (x, t), x ∈ Rn , f : Rn × R → Rn where the dot denoted by the differentiation with respect to time t A particularly simple example of differential equation is the linear differential equation x˙ = Ax, (1.1) where A is an n × n matrix with constant coefficients With the initial condition at t = is x0 , the equation (1.1) has solutions x = etA x0 , Where etA = ∞ k=0 k (tA) k! = I + tA + (tA) 2! k + + (tA) k! + Theorem 1.1.1 (Local existence and uniqueness ) Suppose x˙ = f (x, t) and f : Rn × R −→ Rn is continuously differentiable Then there exists maximal t1 > 0, t2 > such that a solution x(t) Linearization and hyperbolicity NGUYEN THI KIM OANH with x(t0 ) = x0 exists and is unique for all t ∈ (t0 − t1 , t0 + t1 ) Theorem 1.1.2 (Continuity of solutions ) Suppose that f is C r (r times continuously differentiable) and r ≥ 1, in some neighbourhood of (x0 , t0 ) Then there exists > and δ > such that if |x −x0 | < there is a unique solution x(t) defined on [t0 −δ, t0 +δ] with x(t0 ) = x Solutions depend continuously on x and on t 1.2 Flows In this section, we see that solutions to differential equations can be represented as curves in some appropriate space Consider the Autononoous equation x˙ = f (x), x ∈ Rn (1.2) Definition 1.2.1 The curve (x1 (t), , xn (t)) in Rn is an integral curve of equation (1.2) iff (x˙ (t), , x˙ n (t)) = f (x1 (t), , xn (t)) for all t ∈ I On the other words, (x1 (t), , xn (t)) is solution of (1.2) on I Thus the tangent to the integral curve at (x1 (t0 ), , xn (t0 )) is f (x1 (t0 ), , xn (t0 )) Definition 1.2.2 Consider x˙ = f (x) The solution of this differential equation defines a flow, ϕ(x, t) satisfies ϕ(x, t) is solution of the equation (1.2) with the initial condition x(0) = x Hence d ϕ(x, t) = f (ϕ(x, t)) dt Linearization and hyperbolicity NGUYEN THI KIM OANH for all t and ϕ(x, 0) = x Then the solution x(t) with x(0) = x0 is ϕ(x0 , t) Lemma 1.2.1 (Properties of the Flow) (i) ϕ(x, 0) = x; (ii) ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t) = ϕ(x, s + t) Example 1.2.1 Consider the equation x˙ = Ax with x(0) = x0 The solution of equation is x = x0 etA Then the flow ϕ(x0 , t) = x0 etA Hence the flow ϕ(x, t) = xetA We will go to check properties of the flow in this case,we have: i, ϕ(x, 0) = xe0 = x ii, We have ϕ(x, t) = etA x Then ϕ(x, t + s) = xe(t+s)A ; ϕ(ϕ(x, t), s) = ϕ(x, t)esA = xetA esA = xe(t+s)A ; ϕ(ϕ(x, s), t) = ϕ(x, s)etA = xesA etA = xe(t+s)A Therefore ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t) Definition 1.2.3 A point x is stationary point of the flow iff ϕ(x, t) = x, for all t Thus, at a stationary point f (x) = Linearization and hyperbolicity NGUYEN THI KIM OANH Example 1.2.2 Consider the equation   x˙ = −x  x(0) = x We have x is stationary point iff x = ϕ(x, t) ⇔x = xe−t ∀t ⇔x(e−t − 1) = ∀t ⇔x = Hence, the flow has unique stationary point, that is x = Definition 1.2.4 A point x is periodic of (minimal) period T iff   ϕ(x, t + T ) = ϕ(x, t) ∀t  ϕ(x, t + s) = ϕ(x, t) f or all ≤ s < T The curve Γ = {y|y = ϕ(x, t), ≤ t < T } is called a periodic orbit of the differential equation and is a closed curve in phase space Example 1.2.3 Consider the differential equations   x˙1 = x2 (1.3)  x˙ = −x    and the initial condition is For this equation, the matrix A =  −1   a x(0) =   b We can transform (1.3) into n second order differential equation for x1 xă1 + x1 = Linearization and hyperbolicity NGUYEN THI KIM OANH contain the nonlinear parts of the equation, together with their first derivaties at the origin, (x, y) = (0, 0) Hence E s (0, 0) = {(x, y)|x = 0} and E u (0, 0) = {(x, y)|y = 0} Since the stable and unstable manifolds are smooth and are tangential to these manifolds at the origin, so the stable manifold is given by xi = Si (y), i = 1, , nu (2.19) ∂Si (0) = 0, ≤ i ≤ nu , ≤ j ≤ ns ∂yi (2.20) where Similary we can write the unstable manifold as yi = Uj (x), ∂Si (0) = 0, ≤ i ≤ nu , ≤ j ≤ ns ∂yi (2.21) We begin by expanding Uj as a power series in x, so umj xm Uj (x) = (2.22) r≥2 m∈Mr If B is diagonal then with eigenvalues (λi ), i = 1, 2, , ns then y˙ j = −λj yj + g2j (x, y) (2.23) and unstable manifol y = U (x) so y˙ j = −λj Uj (x) + g2j (x, U (x)) (2.24) On the other hand d y˙ j = Uj (x) = dt nu x˙ k k=1 ∂ Uj (x) ∂xk (2.25) Consider (2.24) and (2.25), we find that nu −λi Uj (x) = g2j (x, U (x)) = x˙ k k=1 19 ∂ Uj (x) ∂xk (2.26) Linearization and hyperbolicity NGUYEN THI KIM OANH An example may make this clearer Example: Consider the equations x˙ = x, y˙ = −y + x2 This has a unique stationary point at (x, y) = (0, 0) and the equation in form near the stationary point, the linearized equation is x˙ = x, y˙ = −y Giving a saddle at the origin E s (0, 0) = {(x, y)|x = 0} and E u (0, 0) = {(x, y)|y = 0} By the stable manifold theorem we know that the nonlinear system has a local unstable manifold of the form y = U (x), ∂U (0) = ∂x and uk xk U (x) = k≥2 Now y˙ = −y + x2 = − kuk xk k≥2 on the unstable manifold and also y˙ = x˙ ∂U (x) = ∂x kuk xk k≥2 Equating terms of order x2 , x3 and so on gives −u2 + = 2u2 , and − uk = kuk , k ≥ Hence u2 = 31 , uk = for k ≥ and so u Wloc (0, 0) = {(x, y)|y = x3 } 20 Linearization and hyperbolicity NGUYEN THI KIM OANH Theorem 2.2.1 (Stable manifold theorem) Suppose that the origin is a hyperbolic stationary point for x˙ = f (x) and E s and E u are the stable and unstable manifolds of the linear system x˙ = s Df (0)x Then there exists local stable and unstable manifolds Wloc (0) u and Wloc (0) of the same dimension as E s and E u respectively These manifolds are (respectively) tangential to E s and E u at the origin and as smooth as the original function f Suppose that xo is a hyperbolic stationary point, then there are three s u possibilities Either Wloc (x0 ) = ∅, or Wloc (x0 ) = ∅, or both manifolds are non-empty These three possibilities are given names: x0 is called a source, sink or saddle respectively Theorem 2.2.2 (Hartman’s theorem) If x=0 is a hyperbolic stationary point of x˙ = f (x) then there is a continuous invertible map,h, defined on some neighbourhood of x = which takes orbits of the nonlinear flow to those of the linear flow exp(tDf (0)) This map can be chosen so that the parametrization of orbits by time is preserved Note: The map is only continuous (not nesessary differentiable) and so it does not distinguish between, for example, a logarithmic spiral and the phase portrait obtainned when the Jacobian at the stationary point has real eigenvalues 21 Linearization and hyperbolicity 2.3 NGUYEN THI KIM OANH Persistence of hyperbolic stationary points If the origin is a hyperbolic stationary point of x˙ = f (x) and v is any smooth vector field on Rn then for sufficiently small the equation x˙ = f (x) + v(x) = (2.27) has a hyperbolic stationary point near the origin of the same type as the hyperbolic point of the unperturbed equation Suppose that f (0) = and look for stationary point of the perturbed system They satisfy f (x) + v(x) = (2.28) This equation about x = gives [Df (0) + Dv(0)]x + v(0) + O(|x|2 ) (2.29) x = − [Df (0) + Dv(0)]−1 v(0) + O( ) (2.30) With solution provided [Df (0) + Dv(0)] is invertible If x = is a hyperbolic stationary point, the eigenvalues of Df (0) are bounded away from zero and hence the eigenvalues of [Df (0) + Dv(0)] are bounded away from zero for suffciently small So det [Df (0) + Dv(0)] = for sufficiently small trix is invertible By continuity in and hence this ma- , there is a neighbourhood of =0 for which the real parts of the eigenvalues of [Df (0) + Dv(0)] are all non-zero for sufficiently small x In particular, no eigenvalue can cross the imaginary axis and on the left of the imaginary axis is the same for all x sufficiently small in this neighbourhood of = and the stationary 22 Linearization and hyperbolicity NGUYEN THI KIM OANH point of the perturbed equation has sufficiently small |x| Then for all values of sufficiently small the stationary point of the perturbed equa- tion is hyperbolic To ensure existence of a stationary point for small enough perturbations of the original equations it is enough that no eigenvalue of Df (0) is zero 2.4 Structural stability In the previous section we established that sufficiently small perturbations of a flow with a hyperbolic stationary point have a hyperbolic stationary point of the same type The idea of the equivalence for the flows is basically the same as for Hartman’s theorem, except that the parametrization of solutions by time is not necessary preserved although the sense of time is preserved Definition 2.4.1 Two smooth vector fields f and g are flow equivalent iff there exists a homeomorphism, h, ( so both h and its inverse exists and are continuous)which takes trajectories under f , ϕf (x, t) to trajectories of g,ϕg (x, t), an which preserves the sense of parametrization by time A symple consequence of this definition is that for any x and t1 there exists t2 such that h(ϕt (x, t1 )) = ϕg (h(x), t2 ) Definition 2.4.2 A vector field f : Rn → Rn is structurally stable if for all twice differentiable vector fields v : Rn → Rn there exists such that f is flow equivalent to f + for all ∈ (0, ) 23 >0 Linearization and hyperbolicity NGUYEN THI KIM OANH Definition 2.4.3 Suppose u(t) is aperiodic orbit of the system x˙ = f (x) with least period T Then u(t) is hyperbolic if all the Floquet multipliers of the T -periodic equation v˙ = Df (u(t))v lie off the unit circle except one, which must equal unity Theorem 2.4.1 (Peixoto’s theorem.) Let f : Rn → Rn be a twice differentiable vector field and let D be a compact, connected subset of Rn bounded by the simple closed curve ∂D with outward normal n Suppose that f.n = on ∂D Then f is structurally stable on D iff i) all stationary point are hyperbolic; ii) all periodic orbits are hyperbolic; iii) if x and y are hyperbolic saddles (with x=y possibly) then W s (x) ∩ W u (y) = Ø Furthermore, the set of structurally stable vector fields is open and dense in the set of twice conditinuously diffrentiable vector fields satisfying the conditions of the theorem 2.5 Nonlinear sink Theorem 2.5.1 Suppose x0 is a sink of the differential equation x˙ = f (x), where x ∈ Rn and f is a smooth function Then x0 is a asymptotically stable Proof: We begin by making a number of obvious coordinate transformations Without loss of generality we can assume that x0 is the origin 24 Linearization and hyperbolicity NGUYEN THI KIM OANH and that the linear of the differential equation is in normal form Hence we consider x˙ = Ax + g(x) (2.31) where A is a n × n matrix with eigenvaluees all having strictly negative real parts and g is a smooth function which vanishes together with its first derivatives at the origin Since A is in normal form A = diag(A1 , , Ak ) (2.32) where the Ai are Jordan blocks We need to make one further change of coordinates Suppose that Aq  λ   0      o  is the r × r block  ···   λ · · · 0      · · · λ 1  ··· ··· λ involing the variables xm to xm+r−1 Let ym+s = (2.33) −s xm+s for ≤ s ≤ r − Then the linear part of the equation is transformed form x˙ m = λxm + xm+1 to λym + ym+1 and so on Hence the block is  λ   0      o  transformed to  ···   λ · · · 0       ··· λ  ··· ··· λ 25 (2.34) Linearization and hyperbolicity NGUYEN THI KIM OANH in the new coordinates (ym , , ym+r−1 ) Similary, for a 2r × 2r block  C   0      o  of the form I ···    C I ··· 0      ··· C I  ··· ··· C (2.35) where     λ −ω  and I =   C= ω λ the transformation of the corresponding coordinates (xp , , xp+2r−1 ) to (yp , , yp+2r−1 ) where s yp+2s = xp+2s , s yp+2s+1 = xp+2s+1 for ≤ s ≤ r − gives the new block  C I   0 C I      o ···  0 ··· ···    ··· 0      C I  ··· C (2.36) With this preparation the differential equation (2.31) becomes y˙ = By + h(y) Where B is the normal form matrix with the (2.37) modiffications described above and h(y) is smooth and vanished together with it’s frist derivatives 26 Linearization and hyperbolicity NGUYEN THI KIM OANH at the origin Consider the standard first guess for a Liapounov function: n V (y) = yi2 (2.38) yi y˙ i (2.39) i=1 with n ˙ =2 V (y) i=1 if yi corresponds to a single real block in the matrix B the yi y˙ i = λi yi2 + O(|y|3 ) (2.40) where the eigenvalues λi is strictly negative and the cubicorder terms come from h(y) Similary, if (yi , yy+1 ) corresponds to a complex conjugate pair of eigenvalues in a single block,   λi −ωi  C= ωi λ i then yi y˙ i + yi+1 y˙ i+1 = λi (yi2 + yi+1 ) + O(|y|3 ) (2.41) Furthermore, if (yi , , yi+r−1 ) correspond to a block like (2.34) then r−1 r−1 yi+s + (yi yi+1 + + yi+r−2 yi+r−1 ) + O(|y|3 ) yi+s y˙ i+s = λ s=0 s=0 Nothing that 2 − yi+s+1 ) yi+s yi+s+1 = ((yi+s+yi+s+1 )2 − yi+s equation (2.5) can be tidied up to give r−1 (yi+s y˙ i+s ) = (λ − s=0 )(yi2 + yi+r+1 ) r−2 (λ − + )yi+s s=1 + r−2 (yi+s yi+s+1 )2 + O(|y|3 ) s=0 (2.42) 27 Linearization and hyperbolicity NGUYEN THI KIM OANH Example If (yi , , yi+2r−1 ) is associated with a block like (2.36) show that 2r−1 yi+s y˙ i+s =(λ − s=0 2 )(yi2 + yi+1 + yi+2r−2 + yy+2r−1 ) 2r−3 yi+s + (λ − ) s=2 + 2r−2 (yi+s + yi+s+1 )2 s=0 + O(|y|3 ) (2.43) Now, all the eigenvalues of B have negative real parts, so choosing < such that (Re(λi ) − ) < µ < (2.44) For all eigenvalues λi of B we fine that ( using (2.40), (2.41), (2.42), (2.43) n yi2 + O(|y|3 ) V˙ (y) ≤ 2µ (2.45) i=1 since h and its derivatives re smooth, there exists K > such that O(|y|3 ) ≤ K(|y|3 ) (2.46) on some neighbourhood of y = So, choosing |y| ≤ K −1 |µ| we fine that K |y|3 ≤ |µ || y|2 (2.47) from (2.45) we have n V˙ (y) ≤ µ yi2 (2.48) i=1 This cpmpletes the proof: V is a Liapounov function on some neighnourhood of y = and V˙ < for y = Hence, by Liapounov’s second stability theorm, y = is asymptotically stable 28 Linearization and hyperbolicity 2.6 NGUYEN THI KIM OANH The proof of the stable manifold theorem In particular, we will restrict attention to analytic differential equation The idea of this sketch is simple: given the differential equation x˙ = Ax + g1 (x, y) (2.49) y˙ = By + g2 (x, y) (2.50) where the eigenvalues of both A and B are all positive and contain only nonlinear terms, we see immediately that for the linearzation of (2.50) x˙ = Ax, y˙ = −By (2.51) the stable and unstable manifolds of (x, y) = (0, 0) are given by E s (0, 0) = {(x, y)|x = 0} and E u (0, 0) = {(x, y)|x = 0} Hence we expect the stable and unstable manifolds of the nonlinear problem (2.50) to be perturbations of these axes if they exist So, for example, we might look for a stable manifold of (2.50) of the form x = S(y) for some function S When ξ = x − S(y) , the axis |xi = should be invariant, such that after the change of variable ξ = x − S(y) (2.52) ξ˙ = Aξ + ξF1 (ξ, y) (2.53) y˙ = −By + F2 (ξ, y) (2.54) it’s transformed into 29 Linearization and hyperbolicity NGUYEN THI KIM OANH From (ref19) , the axis ξ = is invariant and the motion on ξ = is given by y˙ = −By + F2 (0, y) (2.55) We begin by nothing that for real analytic g1 (x, y) (2.50) can be written as x˙ = Ax + xG1 (x, y) + vr (y) (2.56) r≥2 If y ∈ Rs , x ∈ Ru , m = (m1 , , ms ) and y m = y1m1 y2m2 ysms then am y m vr (y) = (2.57) m∈Mr where am ∈ Ru , as in Section 4.1, s Mr = (m1 , , ms )|mi ≥ 0, mi + r Suppose that we have used a sequence of transformations ξ i+1 = ξ i − Si (y), y = y (2.58) with ξ (2) = x , such that at the k th step the function S2 (y) to Sk−1 (y) have been chose such that in the new coordinate ξ (k) equation (2.56) is ξ˙(k) = Aξ (k) + ξ (k) Gk (ξ (k) , y) + Vk (y) + Vr (y) r>k 30 (2.59) Linearization and hyperbolicity NGUYEN THI KIM OANH where the Vj (y) are sums of monomials of order j as the vj (y) were We have change the coordinate ξ (k+1) = ξ (k) − Sk (y) (2.60) bm y m (2.61) With Sk (y) = m∈Mk and so, in coordinate form with ≤ i ≤ u, (k+1) ξ (k) = ξi bmi y m + (2.62) m∈Mk Then, from (2.8) we have that for m ∈ Mk s d m y =( mp βp )y m + O(|y|k+1 ) dt p=1 (2.63) where the βj are the eigenvalues of −B, so βj < Then recall the convention that s mp βp = (m.β) p=1 Differentiating (2.60) in component from and using (2.59) and (2.62) we fine that (k+1) )k+1 ξ˙(k+1) =αj ξi + ξi Gk+1 (ξ (k+1) , y) (ami + (αi − (m, β))bmi )y m + O(|y|k+1 ) + (2.64) m∈Mk so, choosing ami (m, β) − αi bmi = (k+1) The new equation satisfiled by ξi (2.65) is ξ˙(k+1) = Aξ (k+1) + ξ (k+1) Gk+1 (ξ (k+1) , y) + Vr (y) (2.66) r>k y˙ = −By + Fk+1 (ξ (k+1) , y) (2.67) 31 Linearization and hyperbolicity NGUYEN THI KIM OANH where Vr are modified sums of monomials of order r Hence there is a sequence of coordinate changes (2.62) such that we obtain, in the limit, a coordinate change ξ = x − S(y) (2.68) where S is defined as a formal power series, sk (y) S(y) = (2.69) k≥2 such that in the coordinate (ξ, y) the equation (2.49) becomes (2.53) 32 Bibliography [1] Paul Glendinning, Stability and Instability and Chaos, USA, 1994 [2] Cung The Anh, Co so li thuyet phuong trinh vi phan, NXB Dai hoc Su Pham Ha Noi, 2015 [3] Nguyen The Hoan - Pham Phu, Co so phuong trinh vi phan va li thuyet on dinh, NXB Giao duc Viet Nam, 2010 33 ... The quantity |m| = n mk is called the order of the resonance The problem associated with resonance is one of the convergence of the power series expansion of the new coordinates in terms of the. .. x˙ = f (x) and E s and E u are the stable and unstable manifolds of the linear system x˙ = s Df (0)x Then there exists local stable and unstable manifolds Wloc (0) u and Wloc (0) of the same dimension... eigenvalue can cross the imaginary axis and on the left of the imaginary axis is the same for all x sufficiently small in this neighbourhood of = and the stationary 22 Linearization and hyperbolicity