Ordinary Differential Equations

198 Pages·2005·0.709 MB·English

by  Hsu S.-B.

ORDINARY
DIFFERENTIAL
EQUATIONS
Sze-Bi Hsu
ii
Contents
1 INTRODUCTION 1
1.1 Introduction: Where ODE comes from . . . . . . . . . . . . . . . . . 1
2 FUNDAMENTAL THEORY 9
2.1 Introduction and Preliminary . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Local Existence and Uniqueness of Solutions of I.V.P. . . . . . . . . 11
2.3 Continuation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Continuous Dependence on Parameters and Initial Conditions . . . . 21
2.5 Differentiability of Initial Conditions and Parameters . . . . . . . . . 24
2.6 Differential Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 LINEAR SYSTEMS 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Fundamental Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Linear Systems with Constant Coefficients . . . . . . . . . . . . . . . 39
3.4 Two Dimensional Linear Autonomous system . . . . . . . . . . . . . 46
3.5 Linear Systems with Periodic Coefficients . . . . . . . . . . . . . . . 50
3.6 Adjoint System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 STABILITY OF NONLINEAR SYSTEMS 65
4.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Saddle Point Property: Stable and unstable manifolds . . . . . . . . 73
4.4 Orbital Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Travelling Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 88
5 METHOD OF LYAPUNOV 99
5.1 An Introduction to Dynamical System . . . . . . . . . . . . . . . . . 99
5.2 Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 TWO DIMENSIONAL SYSTEMS 117
6.1 Poincare´-Bendixson Theorem . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Levinson-Smith Theorem . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7 SECOND ORDER LINEAR EQUATIONS 145
7.1 Sturm’s Comparison Theorem and Sturm-Liouville boundary value
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 Fredholm Alternative for 2nd order linear equations . . . . . . . . . 158
iii
iv CONTENTS
8 THE INDEX THEORY AND BROUWER DEGREE 163
8.1 Index Theory In The Plane . . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Brief introduction to Brouwer degree in Rn . . . . . . . . . . . . . . 170
9 INTRODUCTION TO REGULAR AND SINGULAR PERTUR-
BATION METHODS 177
9.1 Regular Perturbation Methods . . . . . . . . . . . . . . . . . . . . . 177
9.2 Singular Perturbations : Boundary Value Problems . . . . . . . . . . 182
9.3 Singular Perturbation : Initial Value Problem . . . . . . . . . . . . . 187
Chapter 1
INTRODUCTION
1.1 Introduction: Where ODE comes from
Thetheoryofordinarydifferentialequationsisdealingwiththelargetimebehavior
of the solution x(t,x ) of the initial value problem I.V.P of first order system of
0
differential equations:
dx
1 =f (t,x ,x ,···,x )
dt 1 1 2 n
.
.
.
dx
n =f (t,x ,x ,···,x )
dt n 1 2 n
x (0)=x , i=1,2···n
i i0
or in vector notation
dx
= f(t,x), f :D ⊆R×Rn −→Rn, D ⊆R×Rn is open ,
dt
(1.1)
x(0) = x
0
where x=(x ,···x ), f =(f ,···,f )
1 n 1 n
If the right hand side of (1.1) is independent of time t , i.e.
dx
=f(x). (1.2)
dt
Then we say (1.1) is an autonomous system. In this case, we call f is a vector
field on its domain. If the right hand side depends on time t, we say (1.1) is a
nonautonomoussystem. Themostimportantnonautonomoussystemistheperiodic
system i.e., f(t,x) satisfies
f(t+w,x)=f(t,x)
for some w > 0 (w is the period). If f(t,x) = A(t)x where A(t) ∈ Rn×n, then we
say
dx
=A(t)x. (1.3)
dt
1
2 CHAPTER 1. INTRODUCTION
is a linear system of differential equations. It is easy to verify that if ϕ(t),ψ(t) are
solutions of (1.3), then αϕ(t)+βψ(t) is also a solution of linear system (1.3) for
α,β ∈R. The system
dx
=A(t)x+g(t) (1.4)
dt
is called linear system with nonhomogeneous part g(t). If A(t)≡A, then
dx
=Ax (1.5)
dt
is a linear system with constant coefficients. A system (1.1) which is not linear is
called a nonlinear system. Usually it is much harder to analyze nonlinear systems
than the linear ones. The essential difference is that linear systems can be broken
downintoparts. Throughsuperpositionprinciple,Laplacetransform,Fourieranal-
ysis,wefindalinearsystemispreciselyequalstothesumofitsparts. Butnonlinear
systems which phenomena is almost our everyday life, does not have superposition
principle. Inthefollowingwepresentsomeimportantexamplesofdifferentialequa-
tions from Physics, Chemistry and Biology.
Example 1.1.1 mx¨+cx˙ +kx=0
This describes the motion of a spring with damping and restoring force. Apply
Newton’s law, F =ma, we have
ma=mx¨=F =−cx˙ −kx= Friction + restoring force.
Let y =x˙. Then
(cid:189)
x˙ =y
y˙ =−cy− kx
m m
Example 1.1.2 mx¨+cx˙ +kx=F coswt. Then we have
(cid:181) (cid:182) (cid:181) (cid:182)
x˙ y
=
y˙ cy− kx+F coswt
m m
(cid:112)
If c=0 and w = k/m, then we have ”resonance”.
Example 1.1.3 Electrical Networks
Let Q(t) be the charge on the capacitor at time t. Use the following
Kirchoff’s 2nd law: In a closed circuit, the impressed voltage equals the sum of the
voltage drops in the rest of the circuit.
1.1. INTRODUCTION: WHERE ODE COMES FROM 3
Fig.1.1
1. The voltage drop across a resistance of R ohms equals RI (Ohm’s law)
2. The voltage drop across an inductance of L henrys equals LdI
dt
3. The voltage drop across a capacitance C farads equals Q/C
Hence
dI Q
E(t)=L +RI +
dt C
Since I(t)= dQ, it follows that
dt
d2Q dQ 1
L +R + Q=E(t)
dt2 dt C
Example 1.1.4 Van der Pol Oscillator [K] p.481, [HK] p.172
u(cid:48)(cid:48) +(cid:178)u(cid:48)(u2−1)+u=0, 0<(cid:178)(cid:191)1
Let E(t)= u(cid:48)2 + u2 be the energy. Then
2 2
E(cid:48)(t) = u(cid:48)u(cid:48)(cid:48) +uu(cid:48) =u(cid:48)(−(cid:178)u(cid:48)(u2−1)−u)+uu(cid:48)
(cid:189)
= −(cid:178)(u(cid:48))2(u2−1)= <0, |u|>1
>0, |u|<1
Hence the oscillator is “self-excited”.
Example 1.1.5 Van der Pol Oscillator with periodic forcing
u(cid:48)(cid:48) +(cid:178)u(cid:48)(u2−1)+u=Acoswt
The is the equation Cartwright, Littlewood, Levinson studied in 1940-1950 and
Smale constructed Smale’s horseshoe in 1960. It is one of the model equations in
chaotic dynamics.
4 CHAPTER 1. INTRODUCTION
Example 1.1.6 Second order conservative system
x¨+g(x)=0
or equivalently
x˙ =x
1 2
x˙ =−g(x )
2 1
(cid:82)
The energy E(x ,x )= 1x2+V(x ), V(x )= x1g(s)ds satisfies
1 2 2 2 1 1 0
d
E =0
dt
Example 1.1.7 Duffing’s equation
x¨+(x3−x)=0
The potential V(x)=−(1/2)x2+(1/4)x4 is a double-well potential.
Fig.1.2
Example 1.1.8 Duffing’s equation with damping and periodic forcing.
x¨+βx˙ +(x3−x)=Acoswt
This is also a typical model equation in chaotic dynamcis.
Example 1.1.9 Simple pendulum equation
d2θ g
+ sinθ =0
dt2 (cid:96)
1.1. INTRODUCTION: WHERE ODE COMES FROM 5
Fig.1.3
F = ma
d2θ
−mgsinθ = m(cid:96)·
dt2
Example 1.1.10 Lorentz equation [S] p.301, [V]
1. x˙ =σ(y−x)
2. y˙ =rx−y−xz σ,r,b>0
3. z˙ =xy−bz
When σ = 10, b = 8, r = 28, (x(0),y(0),z(0)) ≈ (0,0,0), we have butterfly
3
phenomenon.
Example 1.1.11 Michaelis-Menten Enzyme Kinetics ([K] p.511),([LS] p.302)
Consider the conversion of a chemical substrate S to a product P by enzyme catal-
ysis. The reaction scheme
E+S (cid:42)(cid:41)k1 ES →k2 E+P
k−1
was proposed by Michaelis and Menten in 1913. By the law of Mass action. We
have the follows equations
d[E]=−k [E][S]+k [ES]+k [ES]
dt 1 −1 2
d[S]=−k [E][S]+k [ES]
dt 1 −1
d[ES]=k [E][S]−k [ES]−k [ES]
dt 1 −1 2
d[P]=k [ES]
dt 2
6 CHAPTER 1. INTRODUCTION
subject to initial concentration
[E](0)=E , [S](0)=S , [ES](0)=[P](0)=0.
0 0
Since
d d
[ES]+ [E]=0
dt dt
Hence
[ES]+[E]≡E
0
Let
(cid:193) (cid:193)
1. u=[ES] C , v =[S] S , τ =k E t
0 0 1 0
(cid:193)
(cid:177) (cid:177)
2. κ=(k +k ) k S , (cid:178)=C S , λ= k−1 , C =E 1+κ
−1 2 1 0 0 0 k−1+k+2 0 0
Then we have the equation of singular perturation
du (v+κ)u
(cid:178) =v− , 0<(cid:178)(cid:191)1
dτ 1+κ
(cid:193)
dv
=−v+(v+κλ)u (1+κ)
dt
u(0)=0, v(0)=1
We shall study this example in chapter 9.
Example 1.1.12 Belousov-Zhabotinskii Reaction [M]
dx
(cid:178) = qy−xy+x(1−x)
dt
dy
δ = −qy−xy+2fz
dt
dz
= x−z
dt
where (cid:178),δ,q are small, f ≈0.5. This is an important oscillator in chemistry.
Example 1.1.13 Logistic equation. Let x(t) be the population of a species.
(cid:179) (cid:180)
dx x
= rx−bx2 =rx 1−
dt K
K = carrying capacity
Example 1.1.14 The Lotka-Volterra model for Predator-Prey interaction. Let
x(t),y(t) be the population of prey and predator at time t respectively. ([M],p.124
and p.62). Then we have

 dx =ax−bxy
dt
a,b,c,d>0

dy =cxy−dy
dt
or 
 dx =rx(1− x)−bxy
dt k

dy =cxy−dy
dt