Valery V. Kozlov
Dept. of Theoretical Mechanics
Moscow State University
Stanislav D.Furta
Dept. of Theoretical Mechanics
Moscow Aviation Institute
THE FIRST LYAPUNOV METHOD FOR STRONGLY NON-LINEAR
SYSTEMS OF DIFFERENTIAL EQUATIONS
1
o
. We will consider a certain class of dynamical systems which can be
described by means of a smooth vector field
v
(
x
)
and which has an
equilibrium at the origin
x
0
( ),
,
( )
x
v x
x
v
n
R
0
0
(1)
Using the first Lyapunov method, one can explicitly construct families of
solutions of (1) entering the equilibrium position
x
0
as
t
or
t
in a form of series. The behavior of the above trajectory of
dynamical systems contains a lot of important information about the
structure of the phase portrait of the system in a small neighborhood of
x
0
. In particular, the existence of trajectories entering the equilibrium
position as
t
implies instability of the latter equilibrium.
Let
v
x
( )
0
be the Jacobian of the vector field
v
(
x
)
evaluated at the
equilibrium. Let us assume that the characteristic equation
det
I
0
(2)
possesses
p
roots
1
,...,
p
with negative (positive) real parts. Then system
(1) has a
p
-parametric family of solutions going to the origin as
t
(
t
)
. Those solutions can be expanded into the following series [1]
x t
x
t
j
j
t
j
j
p
p
j
j
p
p
( )
( )exp((
...
) )
...
,...,
1
1
1
1
0
(3)
Here
j
j
p
1
1
...
and coefficients
x
j
j
p
1
...
are polynomials in
t
and depend on
p
arbitrary parameters which have to be small enough to ensure
convergence of series (3). The first partial sum of (3) (
j
j
p
1
1
...
) is
obviously a linear combination of particular solutions of the ‘truncated’
linear system
x
x
(4)
1

It is however worth noticing that in a general case formula (3) represents
complex solutions of system (1). To construct real solutions one needs
more complicated and refined formulae.
The first Lyapunov method in its classical ‘quasi-linear’ setting actually
consists of three main steps:
a)
to simplify the original system by neglecting some terms and obtaining
a cut system;
b) to construct a particular solution or a family of particular solutions of
the cut system;
c) to build the above solutions of the cut system up to the particular
solutions of the entire system in a form of series.
Before explaining the notion of strongly non-linear systems and advanced
studying of the subject, let us consider a couple of examples on the
classical first Lyapunov method. If the characteristic equation (2) has roots
which do not lie on the imaginary axis, system (1) has particular solutions
with specific asymptotic properties. Hence, it is very useful to know

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- Summer '20
- Linear Algebra, Vector Space, Complex number, Vector field