Stability of ordinary differential equations

HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS ———o0o——— PHUNG THI HONG LIEN STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS BACHELOR THESIS Hanoi – 2019 HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS ———o0o——— PHUNG THI HONG LIEN STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS BACHELOR THESIS Major: Analysis SUPERVISOR: Dr Tran Van Bang Hanoi – 2019 Bachelor thesis PHUNG THI HONG LIEN Thesis Acknowledgement I would like to express my gratitudes to the teachers of the Department of Mathematics, Hanoi Pedagogical University 2, the teachers in the analytic group as well as the teachers involved The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis In particular, I would like to express my deep respect and gratitude to Dr Tran Van Bang, who has direct guidance, help me complete this thesis Due to time, capacity and conditions are limited, so the thesis can not avoid errors Then, I look forward to receiving valuable comments from teachers and friends Ha Noi, May 7, 2019 Student Phung Thi Hong Lien i Bachelor thesis PHUNG THI HONG LIEN Thesis Assurance I assure that the data and the results of this thesis are true and not identical to other topics I also assure that all the help for this thesis has been acknowledge and that the results presented in the thesis has been identified clearly Ha Noi, May 7, 2019 Student Phung Thi Hong Lien ii Contents Notation iv Preface 1 PRELIMINARIES 1.1 Introduction 1.2 Existence and uniqueness theorems 1.3 Flows 1.4 Limit sets and trajectories 1.5 Example 1.6 Definitions of stability 10 STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS 16 2.1 Lyapunov functions 16 2.2 Strong linear stability 24 2.3 Orbital stability 32 2.4 Bounding functions 33 2.5 Non-autonomous equations 35 Conclusion 37 References 38 iii Bachelor thesis PHUNG THI HONG LIEN Notation Rn n-dimensional space cl(A) The closure of A ∂A Boundary A B(x, y) The open ball of radius y centred at x x˙ Derivates with respect to time γ(x) The trajectory through x γ + (x) The positive semi-trajectory through x γ − (x) The negative semi-trajectory through x Λ(x) The w−limit set of x A(x) The α-limit set of x Γ Simple closed curve (or periodic orbit) T Period iv Bachelor thesis PHUNG THI HONG LIEN Preface Differential equations are one of the most important areas of mathematics and many applications in the fields of science and technology This is considered to be the bridge between theory and application In our undergraduate program, we learned about the basic concepts, theorems of differential equations and the orbits of motion systems In systems of dynamics, a trajectory called Lyapunov stability if the trajectory ahead of any point in a neighbourhood is small enough or it is in a small neighbourhood (but perhaps, larger) Various standards have been developed to demonstrate the stability or instability of an orbit A more general method is the Lyapunov function In fact, any one of the numbers of different stabilization standards is used Therefore, we will study the stability of ordinary differential equations Chapter PRELIMINARIES 1.1 Introduction In this thesis, we explore the stability of ordinary differential equations And we will consider differential equations in the form x˙ = f (x, t), x ∈ Rn , f : Rn × R −→ Rn , where the dot denotes differentiation with respect to time That is the first-order differential equation We can also write a differential equation in general with the n-order form dx dn x dn−1 x = F (t, x, , , n−1 ) dtn dt dt In the particular case, we consider the autonomous equation x˙ = Ax (1.1) where A is a square matrix with constant coefficient If the initial condition at t = is x0 then this equation has solutions x = etA x0 tA with e (tA)k (tA)2 (tA)k = = I + tA + + + + k! 2! k! k=0 ∞ Bachelor thesis 1.2 PHUNG THI HONG LIEN Existence and uniqueness theorems The existence of the differential equation solution gives us the following theorems Theorem 1.2.1 (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) with x(t0 ) = x0 exists and is unique ∀t ∈ (t0 − t1 , t0 + t1 ) Theorem 1.2.2 (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 ε > and δ > such that if |x −x0 | < ε there is a unique solution x(t) defined on [t0 −δ, t0 +δ] with x(t0 ) = x Solutions are depended continuously on x and on t 1.3 Flows We have x˙ = f (x), x ∈ Rn then x˙ i = fi (x1 , x2 , , xn ), i = 1, , n Definition 1.3.1 The curve (x1 (t), , xn (t)) in Rn is an integral curve of equation x˙ = f (x) iff (x˙ (t), , x˙ n (t)) = f (x1 (t), , xn (t)), ∀t Thus the tangent to the integral curve at (x1 (t0 ), , xn (t0 )) is f (x1 (t0 ), , xn (t0 )) Example 1.3.2 The differential equation x˙ = −x has solution x = c.e−t is an integral curve Definition 1.3.3 Suppose that x˙ = f (x) The solution of this differential equation define a flow, ϕ(x, t), such that ϕ(x, t) is solution of the differential equation at time t with initial value, (at t = 0) x Hence d ϕ(x, t) = f (ϕ(x, t)) dt Bachelor thesis PHUNG THI HONG LIEN for all t and ϕ(x, 0) = x The solution x(t) with x(0) = x0 is ϕ(x0 , t) Lemma 1.3.4 (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(ϕ(x, s + t)) Example 1.3.5 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 We infer the flow ϕ(x, t) = xetA We will go to check properties of the flow, we have: i, ϕ(x, 0) = xe0 = x ii, We have ϕ(x, t) = etA x We infer that 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 and ϕ(x, s + t) = xe(s+t)A Hence, ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t) = ϕ(x, s + t) Definition 1.3.6 A point x is stationary point of the flow if and only if ϕ(x, t) = x, ∀t Thus, at a stationary point f (x) = Example 1.3.7 Consider the equation  x˙ = −x x(0) = x x is stationary point ⇔ x = ϕ(x, t) ⇔ x = xe−t , ∀t ⇔ x(e−t − 1) = 0, ∀t ⇔ x = Bachelor thesis PHUNG THI HONG LIEN stable We will try a function V (x, y) = x6 + 3y We see that V (0, 0) = 0, V (x, y) > 0, ∀(x, y) = (0, 0) V˙ (x, y) = 6x5 x˙ + 6y y˙ = 6x5 (−y − x3 ) + 6yx5 = −6x8 < Hence, V (x, y) = x6 + 3y is a Lyapunov function and the origin is asymptotically stable 2.2 Strong linear stability In this part, it easily for us to find Lyapunov function Suppose that, x˙ = f (x) and x = a is a stationary point, so f (a) = We can expand f locally as a Taylor expansion about x = a to get, in component form with ξ = x − a We have x = (x1 , x2 , , xn ) then x˙ = (x˙ , x˙ , , x˙ n ) f = (f1 (x), f (x), , f (xn ))  x˙ = f1 (x)      x˙ = f (x) 2 x˙ = f (x) ⇔       x˙ = f (x) n n Where x = a, a ∈ Rn is a stationary point of f ⇔ f (a) = Then we have   f1 (a) =      f (a) =       f (a) = n Let ξ = x − a, then 24 Bachelor thesis PHUNG THI HONG LIEN   ξ1 = x1 − a1      ξ =x −a 2 ⇔       ξ =x −a n n n or        ξ˙1 = x˙ = f1 (x) ξ˙2 = x˙ = f2 (x)       ξ˙ = x˙ = f (x) n n n Taylor expansion on neighbourhood of x = a of function f can be given as T a(f ) T a(f ) T k−1 a(f ) T k a(f ) f (x) = f (a) + + + + + 1! 2! (k − 1)! k! n (xi − ) with T a(f ) = i=1 ∂f (a) In particular, we consider k = 2, i.e ∂xi T a(f ) T θ(f ) f (x) = f (a) + + , (θ ∈ [a, x]) 1! 2! n (xi − ) = f (a) + i=1  ∂f (a) + o(|θ|) ∂xi  x − a1    x −a    Note: ξ˙ = f (x), ξ = x − a =       x n − an   ∂f1 (a)  ∂xi  ∂f2   ∂f (a) ∂fj  ∂xi (a)  =  (a) , j = 1, n =   ∂xi ∂xi   ∂fn ∂xi (a) 25 Bachelor thesis PHUNG THI HONG LIEN ξ˙ = f (x) = f (a) + n (xi − ) i=1 ∂f (a) + o(|θ|) ∂xi ∂fj ξ˙ = fi (x) = fi (a) + ξi + o(|θ|) ∂xi ∂fj Since f (a) = 0, so ξ˙ = (a)ξi + o(|ξ|) ∂xi ∂fj (a) then ξ˙ = Aξi + o(|ξ|) A is the matrix of partial Put Aij = ∂xi derivatives of f evaluated at x = a The linear system ξ˙ = Aξ is called the linearization of x˙ = f (x) at x = a In this thesis, we concentrate on these result by explicitly constructing a Lyapunov function for x = a When A has n distinct eigenvalues, all of which have strictly negative real part So, any stationary point whose linearization has this spectral property is asymptotically stable Example 2.2.1 We consider the differential equations x˙ = x(y − 1) (2.4) y˙ = 3x − 2y + x2 − 2y (2.5) Firstly, we need to find stationary points We have x˙ = y˙ = ⇔ x(y − 1) = 3x − 2y + x2 − 2y = ⇔     x=0 y=1    3x − 2y + x2 − 2y = Case 1: With x = 0, substituting x = into equation (2.5) and y˙ = y = −1 then 3.0 − 2.y + − 2y = ⇔ y=0 We have two points (0, 0), (0, −1) Case 2: With y = 1, substituting into equation (2.5) and y˙ = We x = −4 have 3.x − + x2 − 2.1 = ⇔ x2 + 3.x − = ⇔ We have x=1 26 Bachelor thesis PHUNG THI HONG LIEN two stationary points (−4, 1), (1, 1) Hence, there are four stationary points (x, y) = (0, 0), (0, −1), (−4, 1), (1, 1) The Jacobian matrix is ∂ x˙  ∂x  ∂ y˙ ∂x   ∂ x˙ ∂y  = ∂ y˙  ∂y y−1 x + 2x −2 − 4y Next, we have to find the eigenvalues of this matrix at points At (0, 0) the matrix becomes A − λI = −1 −2 −1 − λ −2 − λ , det(A − λI) = ⇔ −1 − λ −2 − λ ⇔ + λ + 2λ + λ2 = ⇔ λ2 + 3λ + = ⇔ = λ = −1 λ = −2 Then, λ = −1, −2 < 0, so we think that the origin to be asymptoti- cally stable At the rest cases, then we found one eigenvalue is positive and the other is negative so these points are not asymptotically stable From exercise in this chapter illustrate how to show that such points can not be Lyapunov stable Theorem 2.2.2 Suppose x˙ = f (x) has linearization x˙ = Ax at x = If A has n distinct eigenvalues, each of which has strictly negative real part, then x = is asymptotically stable 27 Bachelor thesis PHUNG THI HONG LIEN Proof To prove theorem, we need to make clearly a few results from linear algebra about the orthogonality of eigenvalues in order to prove this result We concerned on real matrices, eigenvalues and eigenvectors are either real or complex conjugate pairs Given two (possibly complex) vectors x and y We define an inner product < y, x > = y ∗T x with ∗ denotes complex conjugation and T denotes the tranpose matrix Given a matrix A (real, n × n), we define the adjoint of A, B to be the n × n matrix such that < y, Ax > = < By, x > i.e y ∗T Ax = y ∗T B ∗T x Therefore, A = B ∗T or B = A∗T Since A is real so B = AT We know that, the eigenvalues of A and AT are the same Suppose that, they are all distinct, denoted by (λi ), i = 1, 2, n Leading to there are unique eigenvectors (ei ) and (fi ) such that Aei = λi ei , and Bfi = A∗T fi = λ∗i fi The eigenvectors of B = A∗T are usually referred to as the adjoint eigenvectors of A Therefore, < fi , Aei > = < fi , λi ei > = λi < fi , ei > and < Bfj , ei > = < A∗T fj , ei > = < λ∗ fj , ei > = λj < fj , ei > Moreover, according to the definition then < y, Ax > = < By, x > , so λi < fj , ei > = λj < fj , ei > 28 Bachelor thesis PHUNG THI HONG LIEN If i = j, λi = λj this means that < fj , ei > = 0, i = j This is the really important result we need, since after normalizing the eigenvetors it implies that (ei ) and (fi ) can be chosen such that < fj , ei > = δi j Next, suppose that we write any vector x in the basis of eigenvectors n of A, so x = xi ei Then taking the inner product of x with fj gives i=1 n < fj , x > = n xi < fj , ei > = xj and so x = i=1 < fi , x >ei i=1 Proof of theorem: Let A be the Jacobian matrix of f at x = Let λi , ei and fi , i = 1, , n be the eigenvalues, eigenvectors and adjoint eigenvectors We can write n n x= xi ei = i=1 Therefore, dx = dt n i=1 < fi , x >ei i=1 dxi ei = dt n i=1 d < fi , x >ei dt Moreover, dx = Ax + o(|x|) dt and n Ax = n xi (Aei ) = i=1 λi < fi , x >ei , i=1 so d < fi , x > = λi < fi , x > +o(|x|) dt 29 (2.6) Bachelor thesis PHUNG THI HONG LIEN This puts us in a position to define a Lyapunov function Let (vi ), i = 1, , n be a set of strictly positive real numbers and let n vi < fi , x >∗ < fi , x > V (x) = (2.7) i=1 Then V is clearly differentiable (as it is a quadratic function) and positve definte Futhermore, differentiating (2.7) with respect to time we find that V˙ (x) equals n vi [( i=1 d d < fi , x >∗ ) < fi , x > + < fi , x >∗ ( < fi , x >)] dt dt and using (2.6), we can be rewritten as n vi (λ∗ i + λi ) < fi , x > ∗ < fi , x > +o(|x|2 ) V˙ (x) = i=1 Since (λ∗ i + λi ) < 0, i = 1, , n there is a small neighbourhood G, of x = in which the sum dominates o(|x|2 ) terms and hence V˙ is strictly less than zero on G Hence, by “Lyapunov’s second stability theorem”, x = is asymptotically stable Example 2.2.3 We will consider the equations x˙ = y + a( x3 − x) y˙ = −x with a > The  ∂ x˙  ∂x  ∂ y˙ ∂x Jacobian matrix evaluated at x = is  ∂ x˙ ∂y  = x a − a = −a = A ∂ y˙  −1 −1 ∂y 30 Bachelor thesis PHUNG THI HONG LIEN We have A − λI = −a − λ −1 −λ We consider characteristic equation λ2 + aλ + = The eigenvalues of A are λ1 = −a + √ a2 − , λ2 = −a − √ a2 − We see that, two solutions of A have strictly negative real part for all a > 0, and which are distinct provided a2 = Therefore, x = is asymptotically stable for all a > 0, a = Example 2.2.4 Show that every solution of x˙ = −t2 x, y˙ = −ty is asymptotically stable with t > We have the Jacobian  ∂ x˙  ∂x  ∂ y˙ ∂x matrix:  ∂ x˙ ∂y  = ∂ y˙  ∂y B − λI = −t2 0 −t −t2 − λ 0 −t − λ = B We consider characteristic equation (−t2 − λ)(−t − λ) = The eigenvalues of B are λ1 = −t2 , λ2 = −t We see that, two solutions of B have strictly negative real part for all t > Therefore, every solution are asymptotically stable for all t > 31 Bachelor thesis 2.3 PHUNG THI HONG LIEN Orbital stability From the example (1.5.2) show that the definitions of stability introduced so far are not necessarily the most useful If one is interested in an attracting set such as a periodic orbit: two nearby points may move apart due to phase lagging, where the ‘angular velocity’ varies with the distance from the periodic orbit Hence, the periodic orbit, thought of as a set, may attract nearby For this reason it is necessary to define stability in terms of the set of points on the periodic orbit Suppose that x0 ∈ Rn is on a periodic orbit of period T for the system x˙ = f (x), so ϕ(x0 , T ) = x0 Let Γ = {x0 ∈ Rn |x = ϕ(x0 , t) for some ≤ t < T } Hence, we can define a neighbourhood N (Γ, ε) of Γ This means N (T, ε) = {x ∈ Rn |∃y ∈ Γ s.t |x − y| < ε} (see Fig 2.3) Figure 2.3: Orbital stability Definition 2.3.1 A periodic orbit, Γ, is Lyapunov orbitally stable iff for all ε > there exists δ > such that if x ∈ N (Γ, δ) then ϕ(x, t) ∈ N (Γ, ε) for all t ≥ 32 Bachelor thesis 2.4 PHUNG THI HONG LIEN Bounding functions The first lemma of this chapter (lemma 2.1.2, the Bounding Lemma) gives a simple argument which shows that the existence of a Liaponuov function on some open domain G implies that at least some trajectories remain in G for all t ≥ As we know that, in many examples there is no stable stationary point and yet all trajectories tend to some bounded region of phase space What happens in this region can be very complicated (e.g chaotic) or relatively simple (e.g periodic), but it is clearly important to identify such regions if possible It is possible to modify the arguments used in the previous sections to prove this type of result: the function that takes the place of the Lyapunov function is called a bounding function Given a function g : Rn −→ R, the gradient of g is normal to surface of constant g(x) Theorem 2.4.1 Suppose x˙ = f (x), x ∈ Rn , there is a continuously differentiable function g : Rn → R such that the set D = {x ∈ Rn |g(x) < 0} is a simply connected bounded domain with smooth boundary ∂D If (∇g).f < on ∂D then for all x ∈ D, ϕ(x, t) ∈ D for all t ≥ Theorem 2.4.2 Let D ⊂ Rn a simply connected, compact domain and V : Rn → R a continuously differentiable function Suppose that for each k > 0, Vk = {x ∈ Rn |V (x) < k} is a simply connected bounded domain with Vk ⊂ Vk if k < k If there exists k > such that D ⊂ Vk and δ > such that V˙ (x) ≤ −δ < for all x ∈ Rn \ D then for all x there exists t(x) ≥ such that ϕ(x, t) ∈ Vk for all t > t(x) Proof If x ∈ Vk then ϕ(x, t) ∈ Vk for all t ≥ by theorem (2.16)( put g(x) = V (x) − k and note that f.∇g < = V˙ ≤ −δ < 0) Suppose that y ∈ Rn \ Vk Since V is continuous, there exists m < k 33 Bachelor thesis PHUNG THI HONG LIEN such that D ⊂ Vm ⊂ Vk and so V˙ (x) ≤ −δ for all x ∈ Rn \ Vm Hence t V˙ dt ≤ −δt, ∀t V (ϕ(y, t)) − V (y) = such that ϕ(y, t) ∈ Rn \Vm Since V (y) > m this implies that there exists t(y) ≤ (V (y) − m) ϕ such that ϕ(y, t(y)) ∈ ∂Vm ( and is also in Vk ) Therefore ϕ(y, t) ∈ Vk for all t ≥ t(y) Example 2.4.3 The Lorenz equations, x˙ = σ(−x + y) y˙ = rx − y − xz z˙ = −bz + xy We see that, the constants σ, r, b are positive We aim to show that all solutions of this equation tend to some bounded ellipsoid in R3 Consider the functions V (x, y, z) given by 2V (x, y, z) = rx2 + σy + σ(z − 2r)2 Surfaces V (x, y, z) = k are bounded ellipsoids for all finite k > and Vk ⊂ Vk if < k < k Differentiating we have V˙ (x, y, z) = rσx(−x + y) + σy(rx − y − xz) + σ(−bz + xy)(z − 2r) = −σ(rx2 + y + bz − 2brz) = −σ(rx2 + y + b(z − r)2 − br2 ) Next, let D be the bounded ellipsoid D = {(x, y, z)|rx2 + y + b(z − r)2 < br2 } D ⊂ Vk with k is sufficiently large and V˙ (x, y, z) < for (x, y, z) ∈ 34 Bachelor thesis PHUNG THI HONG LIEN Rn \ D Choosing k large enough we can arrange for V˙ (x, y, z) < −1 for (x, y, z) ∈ R3 \ Vk and D ⊂ Vk Therefore, all trajectories eventually enter and never leave the bounded ellipsoid Vk by theorem 2.4.2 With a little more work it is possible to estimate the size of k with more care, and choose a smaller region which all trajectories enter 2.5 Non-autonomous equations Suppose x˙ = f (x, t), x ∈ Rn , and f (0, t) = 0, ∀t ≥ Therefore, x = is a stationary point of the flow The definitions of Lyapunov stability, quasiasymptotic stability and asymptotic stability apply equally well in this non-autonomous case Therefore, we should be able to define Lyapunov function as before However, V depends on both x and t then the region V (x, t) < µ may become unbounded time increasing We prevent this possibility by underpinning V with a positive definite function U (x) Definition 2.5.1 Suppose x˙ = f (x, t), x ∈ Rn , and f (0, t) = 0, ∀t ≥ Hence, V (x, t) is a Lyapunov function on some neighbourhood G of x = iff i) V is continuously differentiable in both x and t for (x, t) ∈ G × R; ii) V (0, t) = and V (x, t) ≥ U (x) > for x = 0; ∂V iii) V˙ (x, t) = + f.∇V ≤ for x ∈ G and t ≥ ∂t Theorem 2.5.2 Suppose x˙ = f (x, t) and f (0, t) = 0, ∀t ∈ R If a Lyapunov function V (x, t) exists on some neighbourhood G of x = then x = is Lyapunov stable Proof Given ε > and t0 we need to find δ > such that if x ∈ B(0, δ) when t = t0 then ϕ(x, t) ∈ B(0, ε) ∀t ≥ t0 35 Bachelor thesis PHUNG THI HONG LIEN Choose ε small enough such that B(0, ε) ⊂ G, let µ = min{U (x)|x ∈ ∂B(0, ε)} Since ∂B(0, ε) is compact, there exists y ∈ B(0, ε) with U (y) = µ and µ > as y = V is continuous in x so for µ > we can find δ > such that if x ∈ B(0, δ) then V (x, t0 ) < µ Moreover, V is non-increasing along trajectories, so for x ∈ B(0, δ) V (ϕ(x, t), t) < µ and hence U (ϕ(x, t), t)) < µ, ∀t ≥ t0 But this implies that ϕ(x, t) ∈ B(0, ε), ∀t ≥ t0 The result is proved Theorem 2.5.3 Suppose x˙ = f (x, t) and x = is a stationary point If a Lyapunov function V (x, t) can be defined on a neighbourhood G of x = and −V˙ (x, t) ≥ W (x) > for x = then x = is asymptotically stable Example 2.5.4 We will consider equation x˙ = Solution of this equation x = c We see that, if x = then according to theorem 2.5.2, the origin is Lyapunov stable We can define a Lyapunov function V (x, t) =|x|2 (1 + ) 2+t ∂V V˙ (x, t) = + x.∇V ˙ = −|x|2 < ∂t (2 + t)2 However, the origin is Lyapunov stable but not asymptotically stable 36 Bachelor thesis PHUNG THI HONG LIEN CONCLUSION Bachelor thesis research about Stability of ordinary differential equations The results of the thesis include: Recall the definition and basic properties of some important concepts in flows, limit sets and trajectories and stability We have analyzed some exercises more closely and clarified on Taylor expansion with many variables The thesis has presented some examples to illustrate the applications of the Lyapunov’s second stability theorem, strong linear stability, non-autonomous equations 37 Bibliography [A] References in Vietnamese [1] Nguyễn Thế Hồn - Phạm Phu, Cơ sở phương trình vi phân lí thuyết ổn định, NXB Giáo dục Việt Nam, 2010 [2] Cung Thế Anh, Cơ sở lí thuyết phương trình vi phân, NXB Đai học sư phạm, 2015 [B] References in English [3] Paul Glendinning, Stability and Instability and Chaos, USA, 1994 38 ... stability of ordinary differential equations Chapter PRELIMINARIES 1.1 Introduction In this thesis, we explore the stability of ordinary differential equations And we will consider differential equations. .. 1.6 Definitions of stability 10 STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS 16 2.1 Lyapunov functions 16 2.2 Strong linear stability ... set of x A(x) The α-limit set of x Γ Simple closed curve (or periodic orbit) T Period iv Bachelor thesis PHUNG THI HONG LIEN Preface Differential equations are one of the most important areas of