International Journal of Advanced Engineering Technology
E-ISSN 0976-3945
Research Article
STABILITY PROBLEMS FOR DIFFERENTIAL EQUATIONS
1
Tailor Ravi M., 2Bhathawala P.H.
1
Address for Correspondence
Assistant professor of Laxmi institute of technology, sarigam, valsad.
2
Principal of S.S Agrawal Institute of Management and Technology, Navsari.
ABSTRACT
In this paper, consider problems of stability of differential eqations. We study the existence of eigenvalue for differential
system. Derive system of homogeneous and nonhomogeneous differential equations in terms of eigenvalue problems
. An
matrix
can be in the Jordan canonical form then find complete solution of differential
equation. On this solution, check the stability of differential system.
KEYWORDS: Differential Equations, Eigenvalues, Stability, Jordan Canonical Form, Equilibrium Points.
INTRODUCTION
A homogeneous linear system of differential
equation with constant coefficient of the form
=
Where
are the eigenvalues of A
are the corresponding
and
eigenvectors.
In the general case, the general solution of (1)
with
is given by
=
,
.
.
.
The eigenvalues
and the corresponding
eigenvectors are appear in the computation of
=
.
For example, if A has the Jordan canonical form,
or in matrix form
Where,
where
and
with
Then
arises in a wide variety variety of physical and
engineering system. The solution of this system
is intimately related to the eigenvalue problems
for the matrix A.
To see this, assume that the system (1) has
If
, where
is the order of
, then
solution
, where v is not dependent
on t. then from (1), we must have
That is
Showing that � is an eigenvalues of A and
is
corresponding eigenvectors. Thus the eigenpairs
of A can be used to compute a solution
of (1). If A has n linearly independent
eigenvectors (when all the eigenvalues of A are
distinct), then the general solution of the system
can be written as
IJAET/Vol.II/ Issue III/July-September, 2011/148-149
Thus, the system of differential equation (1) is
completely solved by knowing the eigenvalues
and eigenvectors of the system matrix A.
Furthermore, many interesting and desirable
properties of physical and engineering system
can be studied just by knowing the location or
the nature of the eigenvalues of the system
matrix A. stability is one such property. The
stability is defined with respect to an equilibrium
solution.
International Journal of Advanced Engineering Technology
Stability Criteria for Equilibrium Solution.
An equilibrium solution of the system
,
Is the vector
is an equilibrium solution
if and only if A is non singular.
is said to be stable
An equilibrium solution
if, for every
stable if it is stable and there exist a
as
such
whenever
.
System (1) is asymptotically stable if the
equilibrium solution
is asymptotically
stable.
An asymptotically stable system is necessarily
stable, but the converse is not true.
A system is called marginally stable if it is stable
but not asymptotically stable.
Mathematical Criteria for Asymptotic Stability
Theorem
1:
Stability
Theorem
for
Homogeneous System for Differential Equations.
A necessary and sufficient condition for the
equilibrium
solution
of
the
homogeneous system (1) to be asymptotically
stable is that the eigenvalues of the matrix A all
have negative real parts.
An equilibrium solution is unstable if at least one
eigenvalue has positive real parts.
as
Proof: It is enough to prove that
. Because the general solution of the
system
is
given
by
then
=
=
=
Thus,
if and only if
.
Theorem (2) follows from theorem (1).
Theorem 2: Stability Theorem for a
Nonhomogeneous System for Differential
Equations.
An equilibrium solution of (2) to be
asymptotically stable is that the eigenvalues of
the matrix A all have negative real parts.
An equilibrium solution is unstable if at least one
eigenvalue has positive real parts.
CONCLUSIONS
We presented an approach for the solution of
homogeneous and nonhomogeneous differential
equation by used Jordan canonical form.
The systems of differential equation are solved
by knowing the eigenvalues and eigenvectors.
Stability of differential equation is defined over
equilibrium solutions of the system.
Stability of system of differential equation is
depending upon the nature of the eigenvalues.
Stability of differential system is most important
in study of any types of population modal or
growth model.
ACKNOWLEDGEMENT
The author is grateful to the referee for valuable
comments which led to an improved version of
the paper.
REFERENCES
1.
2.
, the proof follows from (3) to
3.
(6).
,
Note that if
then
, and
, when
, if and only if
.
Stability of a Nonhomogeneous System
A nonhomogeneous linear system of differential
equation with constant coefficient of the form
Where b is a constant vector. The stability of
such a system is also governed by the
eigenvalues of A. In fact, the stability of any
solution of (8) is equivalent to the stability of
of
the
equilibrium
solution
homogeneous system (7). This can be seen as
follows.
IJAET/Vol.II/ Issue III/July-September, 2011/148-149
be an equilibrium solution of (8).
Define
, there exists a real numbers
such that
whenever
.
System (1) is stable if the equilibrium solution
is stable.
is asymptotically
An equilibrium solution
that
Let
E-ISSN 0976-3945
4.
Boyce, William. Elementary Differential
Equations. New York: John Wiley & Sons.
Cullen, Michael & Zill, Dennis. Differential
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Matrix analysis by Roger A. Horn, Charles R.
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Numerical linear algebra By Lloyd Nicholas
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