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
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 pro
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 .
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 throu
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 <
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
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
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
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
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
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
eq
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 +
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
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 th
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,
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
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