Chapter 3
Contractions
In this chapter we discuss contractions which are of fundamental importance for the eld of
analysis, and essential tools for proving properties of ODEs. Before we discuss them, we rst
need to introduce some background on the setting
M2AA1 Dierential Equations
Exercise sheet 6 answers
1. We write f (x, ) with x = (x, y ) R2 . It is also easier to treat f (x, ) R2 as a column vector. Then the
21
1
Jacobian for x = 0 at = 0 is given by D1 f (0, 0) =
, so that we have kerD1 f (0, 0) =
21
from 2008 exam
4.
Consider the following model of the chemical reaction between two substances whose
concentrations are denoted by x and y , respectively:
dx
4xy
= ax
,
dt
1 + x2
y
dy
.
= x 1
dt
1 + x2
Here a is a positive parameter. Note also that as x a
M2AA1 Dierential Equations
7
Exercise sheet 9 answers
1. [HSD, chap 11, q2] In the context of pred-prey we focus on x, y 0. We note that the x- and y -axis are
ow invariant, but the ow on the y -axis has the problem that limx0 y . The nullclines are
x = 0
M2AA1 progress test 9 February 2009, 16:00-17:00
Please attempt all parts of the question (since they are often unrelated).
.
Consider the ODE
dx
= Ax,
(1)
dt
with A gl(2, R) (2 2 matrix with real coecients) and x R2 , whose ow t : R2 R2
is given by
t (x)
M2AA1 progress test 16 March 2009, 16:00-17:00
Please attempt all parts of the questions (since they are often unrelated).
1.
Consider the ODE
dx
= f (x, ) = sin() + cos(x) + x2 ,
dt
with x R, and R a parameter.
(a) (i)
(ii)
Show that this curve may be ap
M2AA1 Dierential Equations
Exercise sheet 1
1. Calculate the ow t of the ODE x = 0.5x with x R and show the image of any subinterval of the real
line under the forward ow (t > 0) is another smaller subinterval.
2. Derive the ow t for the linear ODE x = Lx
M2AA1 Dierential Equations
Exercise sheet 1 answers
1. Flow: t : R R, t (x) = e0.5t x. |t ([a, b])| = |e0.5t [a, b]| < |[a, b]| if t > 0.
2. General answer t (x) = exp(tL)x. If you work this out (either by using Taylor expansion of exp, or by
solving ODE
M2AA1 Dierential Equations
Exercise sheet 2 answers
dx
dt
1. (a) Let (x, y ) Rm R and rewrite the ODE as the system
(b) We have x(t) = t,t0 (x(t0 ) and
dx(t)
dt
dy
dt
= f (x, y ) and
= 1.
= f (x, t). By denition, we have
dx
x(t0 + ) x(t0 )
(t0 ) = lim
,
0
M2AA1 Dierential Equations
Exercise sheet 3
1. Show that from the conditions on the distance function d (in the denition of a metric space) it follows that
the distance is positive denite: d(x, y ) 0 for all x, y X .
2. Decide, with proof, which of the fo
1.
[entirely bookwork]
(a) (i)
[3] f (ax1 + bx2 ) = af (x1 ) + bf (x2 ) for all x1 , x2 R2 and all a, b R.
(ii)
d
[4] The ow t : R2 R2 of the ODE satises dt t (x) = f (t (x) for all t,
with t1 t2 = t1 +t2 (and in particular 0 = Id), so that if x(0) := x t
Chapter 5
The ow near an equilibrium
It is important to study the ow of an ODE. In the absence of general methods that give direct
global insight (this absence lies at the basis of what now is known as chaotic dynamics), it has
proven to be a rather succe
1
Inner products on Cm
On the complex vector space Cm , an inner product is a positive denite conjugate-symmetric sesquilinear
(or Hermitian ) form , : Cm Cm C:
positive denite z, z > 0 if z = 0.
conjugate-symmetric y, z = z, y .
sesquilinear (linear i
Chapter 1
Introduction
1.1
Ordinary Dierential Equations
This course deals with a very important class of dierential equations, so-called ordinary differential equations (ODEs). These have the form
dx
= f (x, , t).
dt
(1.1.1)
In this equation, x represent
M2AA1 Dierential Equations
Exercise sheet 6
1. Use Lyapunov-Schmidt reduction to nd an expression (or approximation) of the set of equilibria (as a
function of external parameter R) of the planar vector eld f (x, y, ) = ( +2x + y x2 , 2x +(1+ )y xy )
near
M2AA1 Dierential Equations
Exercise sheet 5 answers
1. (a) If |B | then all the matrix elements bij of B satisfy |bij | . (This can be veried from the fact
that if b := max bij then there exists a vector x with |x| = 1 so that |B x| b.) Now the eigenvalue
Chapter 4
Existence and uniqueness of solutions
for nonlinear ODEs
In this chapter we consider the existence and uniqueness of solutions for the initial value problem
for general nonlinear ODEs. Recall that it is this property that underlies the existence
Chapter 1
Introduction
1.1
Ordinary Dierential Equations
This course deals with a very important class of dierential equations, so-called ordinary differential equations (ODEs). These have the form
dx
= f (x, , t).
dt
(1.1.1)
In this equation, x represent
M2AA1 progress test
10 February 2010 09:00-09:50am
.
Consider the linear ODE
dx
= Ax,
(1)
dt
with A gl(3, R) (3 3 matrix with real coecients) and x R3 , whose ow t : R3 R3
is given by
t (x) = et B + et C + tet D,
(2)
with non-zero B, C, D gl(3, R).
(a) (i
M2AA1 progress test
17 March 2010 09:00-09:50am
1.
Provide an example of a smooth (C 1 ) map F : U U , where U is a metric space consisting of
a subset of R equipped with the usual (Euclidean) metric, for each of the described properties
as listed below.
M2AA1 Dierential Equations
Exercise sheet 1
1. Calculate the ow t of the ODE x = 0.5x with x R and show the image of any subinterval of the real
line under the forward ow (t > 0) is another smaller subinterval.
2. Derive the ow t for the linear ODE x = Lx
M2AA1 Dierential Equations
Exercise sheet 1 answers
1. Flow: t : R R, t (x) = e0.5t x. |t ([a, b])| = |e0.5t [a, b]| < |[a, b]| if t > 0.
2. General answer t (x) = exp(tL)x. If you work this out (either by using Taylor expansion of exp, or by
solving ODE
M2AA1 Dierential Equations
Exercise sheet 2
1. Consider a nonautonomous ODE x = f (x, t) with x Rm .
(a) Show that by extending the phase space from Rm to Rm R this ODE can be rewritten as an autonomous
ODE.
(b) Derive a relationship between the vector el
M2AA1 Dierential Equations
Exercise sheet 2 answers
dx
dt
1. (a) Let (x, y ) Rm R and rewrite the ODE as the system
dx(t)
dt
(b) We have x(t) = t,t0 (x(t0 ) and
= f (x, y ) and
dy
dt
= 1.
= f (x, t). By denition, we have
dx
x(t0 + ) x(t0 )
(t0 ) = lim
,
0
M2AA1 Dierential Equations
Exercise sheet 3
1. Show that from the conditions on the distance function d (in the denition of a metric space) it follows that
the distance is positive denite : d(x, y ) 0 for all x, y X .
2. Decide, with proof, which of the f
M2AA1 Dierential Equations
Exercise sheet 3 answers
1. We have, d(x, y) = d(y, x), d(x, y) = 0 if and only if x = y and the triangle inequality d(x, y) + d(y, z)
1
d(x, z). From the latter it follows that d(y, z) = 1 (d(z, y) + d(y, z) 2 d(z, z) = 0 for
M2AA1 Dierential Equations
Exercise sheet 5
1. (a) Prove that if A gl(m, R) is hyperbolic, there exists a > 0 such that A + B is also hyperbolic for all
B gl(m, R) with |B | .
(b) Argue that cfw_A + B | A, B gl(m, R), |B | < is a neighbourhood of A in gl
Chapter 2
Linear autonomous ODEs
2.1
Linearity
Linear ODEs form an important class of ODEs. They are characterized by the fact that the
vector eld f : Rm Rp R Rm is linear (at constant value of the parameters and time),
that is for all x, y Rm and a, b R