Stability of ordinary differential equations

DEPARTMENT OF MATHEMATICS———o0o——— PHUNG THI HONG LIEN STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS BACHELOR THESIS Hanoi – 2019... 10 2 STABILITY OF ORDINARY DIFFERENTIAL EQUA-TIONS 16

DEPARTMENT OF MATHEMATICS

———o0o———

PHUNG THI HONG LIEN

STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS

BACHELOR THESIS

Hanoi – 2019

Thesis Acknowledgement

I would like to express my gratitudes to the teachers of the ment of Mathematics, Hanoi Pedagogical University 2, the teachers inthe analytic group as well as the teachers involved The lecturers have im-parted valuable knowledge and facilitated for me to complete the courseand the thesis

Depart-In particular, I would like to express my deep respect and gratitude

to Dr Tran Van Bang, who has direct guidance, help me complete thisthesis

Due to time, capacity and conditions are limited, so the thesis cannot avoid errors Then, I look forward to receiving valuable commentsfrom teachers and friends

Ha Noi, May 7, 2019

Student

Phung Thi Hong Lien

Thesis Assurance

I assure that the data and the results of this thesis are true and notidentical to other topics I also assure that all the help for this thesis hasbeen acknowledge and that the results presented in the thesis has beenidentified clearly

Ha Noi, May 7, 2019

Student

Phung Thi Hong Lien

Notation iv

1.1 Introduction 2

1.2 Existence and uniqueness theorems 3

1.3 Flows 3

1.4 Limit sets and trajectories 6

1.5 Example 7

1.6 Definitions of stability 10

2 STABILITY OF ORDINARY DIFFERENTIAL EQUA-TIONS 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

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)

un-of any point in a neighbourhood is small enough or it is in a small bourhood (but perhaps, larger) Various standards have been developed

neigh-to demonstrate the stability or instability of an orbit A more generalmethod is the Lyapunov function In fact, any one of the numbers ofdifferent stabilization standards is used Therefore, we will study thestability of ordinary differential equations

in general with the n-order form

∞

P

k=0

(tA)kk! = I + tA +

(tA)22! + +

(tA)kk! +

1.2 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

differen-tiable Then there exists maximal t1 > 0, t2 > 0 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 Cr (r times continuously differentiable) and r ≥ 1,

in some neighbourhood of (x0, t0) Then there exists ε > 0 and δ > 0 such

that if |x0−x0| < ε there is a unique solution x(t) defined on [t0−δ, t0+δ]

with x(t0) = x0 Solutions are depended continuously on x0 and on t

Definition 1.3.3 Suppose that ˙x = f (x) The solution of this

differ-ential 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

dtϕ(x, t) = f (ϕ(x, t))

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 = x0etA Then the flow ϕ(x0, t) = x0etA

We infer the flow ϕ(x, t) = xetA

We will go to check properties of the flow, we have:

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) = 0

Example 1.3.7 Consider the equation







˙x = −xx(0) = x0

x is stationary point ⇔ x = ϕ(x, t)

⇔ x = xe−t, ∀t

⇔ x(e−t − 1) = 0, ∀t

⇔ x = 0

Hence, the flow has an unique stationary point, that is x = 0.

Definition 1.3.8 A point x is periodic of (minimal) period T if and only

if ϕ(x, t + T ) = ϕ(x, t) ,∀t, and ϕ(x, t + s) 6= ϕ(x, t) for all 0 ≤ s < T The curve Γ = {y|y = ϕ(x, t), 0 ≤ t < T } is called a periodic orbit ofthe differential equation and is a closed curve in phase space

Example 1.3.9 Consider the system of differential equations

From the system (1.2)

Definition 1.3.10 A set M is invariant iff for all x ∈ M, ϕ(x, t) ∈ M,

∀t A set is forward (resp backward) invariant if for all x ∈ M, ϕ(x, t) ∈

M for all t > 0 (resp.t < 0).

Suppose we have the differential equation ˙x = f (x) with the flow ϕ(x, t), x ∈

Rn, t ∈ R, but if this is not the case then the definitions carry over aftersuitable limits are made on the domains of x, t

Definition 1.4.1 The trajectory through x is the set γ(x) = S

t∈R

ϕ(x, t)and the positive semi-trajectory, γ+(x), and the negative semi-trajectory,

Lemma 1.4.2 i) M is invariant iff γ(x) ⊂ M, ∀x ∈ M ;

ii) M is invariant iff Rn\M is invariant;

iii) Let Mi be a countable collection of invariant subsets of Rn Then

∪iMi and ∩iMi are also invariant

Definition 1.4.3 The w−limit set of x, Λ(x), and the α−limit set of x,A(x), are the sets Λ(x) = {y ∈ Rn|∃tn with tn −→ ∞ and ϕ(x, tn) −→ y

as n −→ ∞}, and A(x) = {y ∈ Rn|∃sn with sn −→ −∞ and ϕ(x, sn) −→

y as n −→ ∞} The w−limit set, Λ(x), which is the set of points which xtend to (i.e the limit points of γ+(x)) and the α−limit set, A(x), which

is the set of points that trajector, through x tends to in backward time.Example 1.4.4 Consider B(0, kxk)

Theorem 1.4.6 The set Λ(x) is invariant, and if γ+(x) is bounded thenΛ(x) is non-empty and compact.

Set x(t) = r(t)cosθ(t), y(t) = r(t)sinθ(t) and the usual tion for polar coordinates then x2 + y2 = r2 Differentiating two sidesthrough with respect to time,

transforma-2x ˙x + 2y ˙y = 2r ˙r

y = ˙rsinθ + r ˙sinθi.e ˙y = ˙rsinθ + r ˙θcosθ

Multiplying the second equation by −sinθ, the fourth equation by cosθ

We have

− ˙xsinθ = ˙rsinθcosθ + r ˙θsin2θ,

˙ycosθ = ˙rsinθcosθ + r ˙θcos2θ

The sum of two above equations, we get

y

r +

xr

˙y

r = ˙θi.e.,

with solution r(t) = r0, θ(t) = t + ϕ, where r(0) = r0 and θ(0) = ϕ.This shows directly that solution are circles and the motion has constantangular velocity, 1 The orgin, r = 0 is a stationary point.

Suppose that the equation represents the motion of a particle which

we want position at the origin If it is not in the correct position, it willget any better over time or it remains the same distance from the origin

If we want to say that a stationary point is stable, all nearby initialconditions will tend to it The origin is not stable

However, we can put the particle reasonably close to the origin and asmall error will not affect the outcome of the experiment too much Thesystem is stable, since the particle starts near the origin

On the other hand, the differential equations are an approximation tothe true equations of motion We will be interested in finding whetherthe solutions of slightly different equations give the same basis solutionstructure Thus we could investigate the effect of adding small nonlinearterms to the equation

To see clearly, we will consider the example 1.5.2

Example 1.5.2 According to example 1.5.1

We consider the small perturbation, defined by

˙r = εr2, ˙θ = 1,with ε ≥ 0 small enough and ˙r > 0, so small errors are increased in time.The ˙r equation can be integrated explicitly We have

⇔ −1r(t) = ε(t) + c ⇔ r(t) =

−1ε(t) + c.

Where t = 0 then

r(0) = −1

c ⇔ c = −1

r(0).Hence,

r(t) = 1

1r(0) − εt

We see that r(0) 6= 0 solutions tend to infinity in finite time We caninfer to this idea of stability the system is unstable, not because solutiondiverge, but because the solution of the original equations are different

in character from those of the perburbed equation

From two examples, we have considered three types of stable

Consider the differential equation

˙x = f (x, t), x ∈ Rn.Definition 1.6.1 A point x is Lyapunov stable (start near stay near)iff for all ε > 0, ∃δ > 0 so that if |x − y| < δ then |ϕ(x, t) − ϕ(y, t)| <

ε, ∀t ≥ 0

Definition 1.6.2 A point x is quasi-asymptotically stable (tends toeventually) iff there exsist δ > 0 such that if |x − y| < δ then |ϕ(x, t) −ϕ(y, t)| −→ 0, as t −→ ∞

Definition 1.6.3 A point x is asymptotically stable (tends to directly)iff it is both Lyapunov stable and quasi-asymptotically stable

Example 1.6.4 Consider the system below

˙r = 0, ˙θ = 1 + r,

Figure 1.1: (a) illustrate for deffinition (1.6.1); (b) illustrate for deffinition (1.6.2); (c) illustrate for deffinition(1.6.3).

with solutions r(t) = r0 and θ(t) = (1 + r0)t + θ0

Similarly as example (1.5.1) show that the origin, r = 0 is Lyapunovstable but not quasi-asymptotically stable That solutions are concentriccircles about the origin These circles are also stable in that if one startsnear a given circle with radius r0 the solution will stay near that circles.Moveover, there is a phase lag on each circle that means that no point

on the circle are stable in either a Lyapunov or a quasi-asymptotic sence.There are two nearby initial conditions (r0, 0) and (r0 + δ, 0) We seethat after a time t the difference in angle (or phase) between the twosolutions in 4θ = (1 + r0)t − (1 + r0+ δ)t = t + r0t − t − r0t − δt = −δt

Note: If a stationary point is asympotically stable then there mustexists a neighbourhood of the point such that all points in this neigh-

bourhood tend to the stationary point The largest neighbourhood forwhich this is true is called the domain of (asymptotic) stability of thispoint.

Definition 1.6.5 Let x be an asymptotically stable stationary point ofthe equation ˙x = f (x), so for all  > 0, there exists δ > 0 such that

|y − x| < δ ⇒ |ϕ(y, t) − x| < , ∀t ≥ 0,and there exists δ > 0 such that

Example 1.6.6 Consider the equations

We will use two equations ˙r = 1

r(x ˙x + y ˙y) and ˙θ =

x ˙y − y ˙x

r2 to rewriteequations (1.7), (1.8) in polar coordinates

x2 + y2+yx+y2−y2(x2−y2)− x

2yp

To study the behaviour of system first note that ˙r = 0 when r = 0 or

r = 1 and ˙θ = 0 when θ = 0 And ˙θ > 0 for all θ 6= 0 Hence θ = 0 is aninvariant half-line, trajectories move around to approach this half-linefrom below

We see that the r direction trajectories tend to r = 1 (unless r = 0initially) We can infer the system has two stationary points There arethe orgin (r = 0) and the point (r, θ) = (1, 0) There are two invariantcurves, θ = 0, r = 1 There has almost all trajectories eventually tend

to the (1, 0), we can see in figure (1.2)

Figure 1.2: The phase portrait for example (1.6.6)

Next, we consider a small neighbourhood of (1, 0) We see that points

in θ ≤ 0 will tend to (1, 0), without leaving this neighbourhood, butpoints in θ > 0 will make a circuit around the invariant curve r = 1before tending to (1, 0) from θ < 0

Moreover, a trajectory starting close to (1, 0) on r = 1 with θ > 0will pass through the diametrically opposite point of the circle (1, π)before tending to (1, 0) from θ < 0 To sum up, although (1, 0) is quasi-

asymptotically stable, but it is not Lyapunov stable.

Definition 2.1.1 Suppose that the origin, x = 0, is a stationary pointfor the differential equation ˙x = f (x), x ∈ Rn Let G be an open neigh-bourhood of 0 and V : cl(G) −→ R a continuously differentiable func-tion Then we can define the derivative of V along trajectories by differ-entiating V with respect to time using the chain rule, so

ii) V (x) > 0, ∀x ∈ cl(G)\{0};

iii) ˙V ≤ 0, ∀x ∈ G

Lemma 2.1.2 (Bounding lemma)

Suppose that G is some open bounded domain in Rn with boundary

∂G, and that V : cl(G) −→ R is Lyapunov function If there exists

x0 ∈ G such that V (x) > V (x0), for all x ∈ ∂G then

S(x0) = {x ∈ cl(G)|V (x) ≤ V (x0)}

is bounded set in G and ϕ(x0, t) ∈ S(x0), for all t ≥ 0

Proof Since V is continuous in G and V (x) > V (x0) on ∂G We seethat S(x0) is in G, x0 ∈ S(x0) infering to S(x0) is non-empty By theexistence results of the previous chaper (1.2) We see that there are threepossible fates for a trajectory through x0

Either

A) There exists ˙t ≥ 0 such that lim

t→ ˙t

ϕ(x0, t) is infinite; orB) There exists ˙t ≥ 0 such that ϕ(x0, ˙t) ∈ ∂G; or

C) ϕ(x0, t) ∈ G for all t ≥ 0

V is a continuous function on cl(G), ϕ(x0, t) is a continuous function

of t on [0, ˙t) And the non-increasing property of V We infer to that

V (ϕ(x0, ˙t)) = lim

t→ ˙t

V (ϕ(x0, t)) ≤ V (x0)

Therefore, (B) gives a contradiction as V (x) > V (x0) on ∂G, (A) gives

a contradiction as G is bounded, ϕ(x0, t) is continuous for t < ˙t However,

if ϕ(x0, t) tends to infinty, there must exists ¨t so that ϕ(x0, ¨t) ∈ ∂G.Hence, C is the most possibility and V (ϕ(x0, t)) ≤ V (x0), ∀t ≥ 0 and sothe trajectory through x0 stays in S(x0)

Next, to prove that a stationary point is Lyapunov stable by a punov function given Then, we need to show that ε > 0, there is someregion B(0, δ) If a trajectory starts in B(0, δ), it will stay in B(0, ε) Ac-cording to the Bounding Lemma, we must find B(0, δ) so that for all x0

Lya-in B(0, δ), S(x0) ⊂ B(0, ε) This will give us Lyapunov’s First Stabilitytheorem

Theorem 2.1.3 (Lyapunov’s first stability theorem)

Suppose that a Lyapunov function can be defined on a neighbourhood

of the origin, x = 0, which is a stationary point of differential equation

˙x = f (x) Then the origin is Lyapunov stable

Proof Choose ε > 0 small enough such that B(0, ε) ⊂ G We need tofind δ > 0 such that x ∈ B(0, δ) then ϕ(x, t) ∈ B(0, ε) for all t ≥ 0.Let µ = min{V (x)|x ∈ ∂B(0, ε)} V is a Lyapunov function on G,

ε > 0 and B(0, ε) ⊂ G, µ > 0 Moveover, ∂B(0, ε) is compact and V iscontinuous there exists y ∈ ∂B(0, ε) such that V (y) = µ (Fig 2.1)

Figure 2.1: C is the shaded region

Let C = B(0, ε) ∩ {x|V (x) < µ} Since µ > 0 and V (0) = 0, so theorigin is containd in C Choose µ > 0 so that B(0, δ) ⊂ C and applythe Bounding Lemmma with x0 ∈ B(0, δ) Then S(x0) must be inside

C, which lies inside B(0, ε)

Example 2.1.4 Consider the nonlinear oscillator

¨

x + c ˙x + ax + bx3 = 0 (2.1)with a, b, c > 0 We can rewritten equation (2.1) as a differential equation

in two variables by setting y = ˙x, then ˙y = ¨x giving

˙

We see that the only stationary point of this system is (x, y) = (0, 0),

so we want to find a Lyapunov function for the origin We will try afunction of the form

˙

V = −2γcy2 < 0

Hence, ˙V (x, y) is Lyapunov function and (0, 0) is Lyapunov stable.Example 2.1.5 We consider

˙x = −x + x2 − 2xy

˙

y = −2y − 5xy + y2Question: The origin (0, 0) is a stationary, so is it Lyapunov stable? Thefirst, we will try to guess a Lyapunov function V (x, y) = 1

Figure 2.2: Wedge-shaped region

If we have a Lyapunov function, can we prove a stationary point isquasi-asymptotically stable? (and hence asymptotically stable, since itmust be Lyapunov stable if a Lyapunov function exists)