Chapter 2
Linear and Perturbed Linear Systems
One of the important topics in dierential equations is concerned with the local theory; that is, the behavior of solutions near a given solution t ( ). If we let
x = t + y , then the discussion reduces to dete
1.6. Autonomous systems.
A dierential equation is said to be autonomous if the vector eld does not
depend upon the independent variable t. In this section, we give a few general but
very important properties of autonomous systems.
If U lRd is an open set,
1.7. Stability and attractors.
Consider the autonomous dierential equation
(7.1)
x = f ( x) ,
where f C r (lRd , lRd ), r 1.
For notation, for any x lRd , c lR, we let B (x, c) = cfw_ lRd : | x| < c .
Suppose that x0 is an equilibrium point of (7.1). We s
1.5. Linear equation with constant coecients.
In this section, we present the basic theory of a linear system of dierential
equations with constant coecients. The interest in such systems arises through the
linear variational equation near a constant solu
1.4. Dierential inequalities.
Let Dr denote the right hand derivative of a function. If (t, u) is a scalar
function of the scalars t, u in some open connected set , we say that a function
v (t), a t < b, is a solution of the dierential inequality
(4.1)
Dr
General Properties of Dierential Equations
1.1. Initial value problem.
Let D be an open set of lR lRd and f : D lRd , (t, x) f (t, x), be continuous.
A dierential equation is a relation
(1.1)
x(t) = f (t, x(t) or
x = f (t, x) ,
d
where x(t) = dt x(t). We
1.2. Continuation of solutions.
It is convenient to let the pair (x, I ) denote a solution of the initial value problem
(1.2) on the interval I. We say that a solution (y, J ) is an extension of (x, I ) if I J
and x(t) = y (t) on I. We say that (x, I ) is
1.3. Uniqueness, continuous dependence and dierentiability.
In this section, we prove uniqueness of the solution of the initial value problem
(1.2) under the hypotheses that f is locally Lipschitzian and give also some dierentiability results when the fun
1.8. Liapunov functions.
On several occassions, we have gained information about the behavior of the
solutions of a dierential equation by considering the rate of change of a scalar function
along the solutions of the equation. In particular, in Exercises
1.9. The principle of Wazewski.
To motivate the discussion of the results of this section, let us rst introduce the
following concepts for the dierential equation
(9.1)
x = f ( x)
in lRd .
Deinition 9.1. Suppose that U is a given set in lRd . A set A is s
2.4. Linear Periodic Systems.
If A C 0 (lR, lRdd ) (or A C 0 (lR, Cdd ) is a d d matrix function and there is
l
a constant p > 0 such that A(t + p) = A(t) for all t, then we say that the dierential
equation (1.1) is a periodic system or a p-periodic syste
2.5. Nonhomogeneous Linear Systems.
If A C 0 (lR, lRdd ) (or A C 0 (lR, Cdd ) is a d d matrix function and g
l
d
d
C (lR, lR ) (or g C (lR, C ), we consider the linear equation
l
(5.1)
x = A ( t) x + g ( t)
and refer to it as a linear nonhomogeneous equa
2.3. Stability.
In this section, we discuss stability properties of solutions of (1.1) in terms of a
fundamental matrix solution of (1.1). The concepts of stability are the same for linear
systems as for nonlinear systems. As a consequence, we rst present
2.2. Liouvilles Theorem.
We recall a few elementary facts from linear algebra. If B = (bij ) is a d d
matrix, let 1 , 2 , . . . , d be the not necessarily distinct eigenvalues of B . Then we
d
know that det B = j =1 j and Tr B = d=1 bjj = d=1 j . Also, if
1.10. Discrete systems.
We rst present some abstract results for discrete dynamical systems and then
make some remarks on applications to nonautonomous dierential equations which
are periodic in time, periodic orbits of autonomous equations, ows on a cyli
1.11. The Poincar Bendixson Theorem.
e
For an autonomous dierential equation in the plane, the -limit set and -limit
set of bounded orbits have a very simple structure as was observed by Poincar and
e
Bendixson at the turn of the century. In this section,
Exercise 1
Consider the dierential equation
x = f (x, t)
(1)
with initial condition x(t0 ) = x0 . Assume that f C 1 (Rn+1 , Rn ). Given
h > 0 we call xh (t) (Euler approximation) the function dened by
xh (nh + t) = xh (nh) + f (xh (nh), nh)t for n 0 and 0