Diﬀerential Equations
Massoud Malek
Nonlinear Systems of Ordinary Diﬀerential Equations
♣
Dynamical System
.
A dynamical system has a state determined by a collection of real
numbers, or more generally by a set of points in an appropriate state space. Small
changes in the state of the system correspond to small changes in the numbers.
The evolution rule of the dynamical system is a ﬁxed rule that describes what future
states follow from the current state. The rule is deterministic: for a given time interval
only one future state follows from the current state.
The mathematical models used to describe the swinging of a clock pendulum, the ﬂow
of water in a pipe, or the number of ﬁsh each spring in a lake are examples of dynamical
systems.
♣
Autonomous System
.
An autonomous diﬀerential equation is a system of ordinary dif-
ferential equations which does not depend on the independent variable. It is of the form
d
dt
X
(
t
) =
F
(
X
(
t
))
,
where
X
takes values in n-dimensional Euclidean space and t is usually time.
It is distinguished from systems of diﬀerential equations of the form
d
dt
X
(
t
) =
G
(
X
(
t
)
, t
)
,
in which the law governing the rate of motion of a particle depends not only on the
particle’s location, but also on time; such systems are not autonomous.
Autonomous systems are closely related to dynamical systems. Any autonomous sys-
tem can be transformed into a dynamical system and, using very weak assumptions, a
dynamical system can be transformed into an autonomous systems.
♣
Jacobian Matrix
.
Consider the function
F
:
IR
n
-→
IR
m
,
where
F
(
x
1
, x
2
, . . . , x
n
) =
f
1
(
x
1
, x
2
, . . . , x
n
)
f
2
(
x
1
, x
2
, . . . , x
n
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
f
m
(
x
1
, x
2
, . . . , x
n
)
.
The partial derivatives of
f
1
(
x
1
, ..., x
n
)
, . . . , f
m
(
x
1
, ..., x
n
)
(if they exist) can be organized in
an
m
×
n
matrix.
The Jacobian matrix of
F
(
x
1
, x
2
, . . . , x
n
)
denoted by
J
F
is as follows:
J
F
(
x
1
, . . . , x
n
) =
∂f
1
∂x
1
···
∂f
1
∂x
n
.
.
.
.
.
.
.
.
.
∂f
m
∂x
1
···
∂f
m
∂x
n
.
Its importance lies in the fact that it represents the best linear approximation to a
diﬀerentiable function near a given point.
California State University, East Bay