Linear system of differential equations with periodic coefficients
A system of linear ordinary differential equations of the form
![]() | (1) |
where is a real variable,
and
are complex-valued functions, and
![]() | (2) |
The number is called the period of the coefficients of the system (1). It is convenient to write (1) as one vector equation
![]() | (3) |
where
![]() |
It is assumed that the functions are defined for
and are measurable and Lebesgue integrable on
, and that the equalities (2) are satisfied almost-everywhere, that is,
. A solution of (3) is a vector function
with absolutely-continuous components such that (3) is satisfied almost-everywhere. Suppose that
and
are an (arbitrarily) given number and vector. A solution
satisfying the condition
exists and is uniquely determined. A matrix
of order
with absolutely-continuous entries is called the matrizant (or evolution matrix, or transition matrix, or Cauchy matrix) of (3) if almost-everywhere on
one has
![]() |
and , where
is the unit
matrix. The transition matrix
satisfies the relation
![]() |
The matrix is called the monodromy matrix, and its eigen values
are called the multipliers of (3). The equation
![]() | (4) |
for the multipliers is called the characteristic equation of equation (3) (or of the system (1)). To every eigen vector
of the monodromy matrix with multiplier
corresponds a solution
of (3) satisfying the condition
![]() |
The Floquet–Lyapunov theorem holds: The transition matrix of (3) with -periodic matrix
can be represented in the form
![]() | (5) |
where is a constant matrix and
is an absolutely-continuous matrix function, periodic with period
, non-singular for all
, and such that
. Conversely, if
and
are matrices with the given properties, then the matrix (5) is the transition matrix of an equation (3) with
-periodic matrix
. The matrix
, called the indicator matrix, and the matrix function
in the representation (5) are not uniquely determined. In the case of real coefficients
in (5),
is a real matrix, but
and
are complex matrices, generally speaking. For this case there is a refinement of the Floquet–Lyapunov theorem: The transition matrix of (3) with
-periodic real matrix
can be represented in the form (5), where
is a constant real matrix and
is a real absolutely-continuous matrix function, non-singular for all
, satisfying the relations
![]() |
where is a real matrix such that
![]() |
In particular, . Conversely, if
,
and
are arbitrary matrices with the given properties, then (5) is the transition matrix of an equation (3) with a
-periodic real matrix
.
An immediate consequence of (5) is Floquet's theorem, which asserts that equation (3) has a fundamental system of solutions splitting into subsets, each of which has the form
![]() |
![]() |
![]() |
![]() |
where the are absolutely-continuous
-periodic (generally speaking, complex-valued) vector functions. (The given subset of solutions corresponds to one
-cell of the Jordan form of
.) If all elementary divisors of
are simple (in particular, if all roots of the characteristic equation (4) are simple), then there is a fundamental system of solutions of the form
![]() |
Formula (5) implies that (3) is reducible (see Reducible linear system) to the equation
![]() |
by means of the change of variable (Lyapunov's theorem).
Let be the multipliers of equation (3) and let
be an arbitrary indicator matrix, that is,
![]() | (6) |
The eigen values of
are called the characteristic exponents (cf. Characteristic exponent) of (3). From (6) one obtains
,
. The characteristic exponent
can be defined as the complex number for which (3) has a solution that is representable in the form
![]() |
where is a
-periodic vector-valued function. The main properties of the solutions in which one is usually interested in applications are determined by the characteristic exponents or multipliers of the given equation (see the Table).'
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|
In applications, the coefficients of (1) often depend on parameters; in the parameter space one must distinguish the domains at whose points the solutions of (1) have desired properties (usually these are the first four properties mentioned in the Table, or the fact that with
given). These problems thus reduce to the calculation or estimation of the characteristic exponents (multipliers) of (1).
The equation
![]() | (7) |
where and
are a measurable
-periodic matrix function and vector function, respectively, that are Lebesgue integrable on
(
,
almost-everywhere), is called an "inhomogeneous linear ordinary differential equation with periodic coefficientsinhomogeneous linear ordinary differential equation with periodic coefficients" . If the corresponding homogeneous equation
![]() | (8) |
does not have -periodic solutions, then (7) has a unique
-periodic solution. It can be determined by the formula
![]() |
where and
is the transition matrix of the homogeneous equation (8), where
,
.
Suppose that (8) has linearly independent
-periodic solutions
. Then the adjoint equation
![]() |
also has linearly independent
-periodic solutions,
. The inhomogeneous equation (7) has a
-periodic solution if and only if that the orthogonality relations
![]() | (9) |
hold. If so, an arbitrary -periodic solution of (7) has the form
![]() |
where are arbitrary numbers and
is a
-periodic solution of (7). Under the additional conditions
![]() |
the -periodic solution
is determined uniquely; moreover, there is a constant
, independent of
, such that
![]() |
Suppose one is given an equation
![]() | (10) |
with a matrix coefficient that holomorphically depends on a complex "small" parameter :
![]() | (11) |
Suppose that for the series
![]() |
converges, where
![]() |
which guarantees the (componentwise) convergence of the series (11) for in the space
. Then the transition matrix
of (10) for fixed
is an analytic function of
for
. Let
be a constant matrix with eigen values
,
. Let
be the multipliers of equation (10),
. If
is a multiplier of multiplicity
, then
![]() | (12) |
where are integers. If simple elementary divisors of the monodromy matrix correspond to this multiplier, or, in other words, if to each
,
, correspond simple elementary divisors of the matrix
(for example, if all the numbers
are distinct), then
is called an
-fold characteristic exponent (of equation (10) with
) of simple type. It turns out that the corresponding
characteristic exponents of (10) with small
can be very easily computed to a first approximation. Namely, let
and
be the corresponding normalized eigen vectors of the matrices
and
;
![]() |
![]() |
let
![]() |
be the Fourier series of , and let
![]() |
where are the numbers from (12). Then for the corresponding
characteristic exponents
,
, of (10), which become
for
, one has series expansions in fractional powers of
, starting with terms of the first order:
![]() | (13) |
Here the are the roots (written as many times as their multiplicity) of the equation
![]() |
and are natural numbers equal to the multiplicities of the corresponding
(
,
for
). If the root
is simple, then
and the corresponding function
is analytic for
. From (13) it follows that cases are possible in which the "unperturbed" (that is, with
) system is stable (all the
are purely imaginary and simple elementary divisors correspond to them), but the "perturbed" system (small
) is unstable (
for at least one
). This phenomenon of stability loss for an arbitrary small periodic change of parameters (with time) is called parametric resonance. Similar but more complicated formulas hold for characteristic exponents of non-simple type.
Let be the distinct multipliers of equation (3) and let
be their multiplicities, where
. Suppose that the points
on the complex
-plane are surrounded by non-intersecting discs
and that a cut, not intersecting these discs, is drawn from the point
to the point
. Suppose that with each multiplier
is associated an arbitrary integer
and that
is the transition matrix of (10). The branches of the logarithm
are determined by means of the cut. The matrix
( "matrix logarithmmatrix logarithm" ) can be defined by the formula
![]() | (14) |
where is the circle
. The set of numbers
determines a branch of the matrix logarithm. Also,
for small
. Generally speaking, formula (14) for all possible
does not cover all the values of the matrix logarithm, that is, all solutions
of the equation
. However, the solution given by (14) has the important property of holomorphy: The entries of the matrix
in (14) are holomorphic functions of the entries of
. For equation (10), formula (5) takes the form
![]() | (15) |
where ,
. If
is determined in accordance with (14), then
![]() | (16) |
are series that converge for small . The main information about the behaviour of the solutions as
which is usually of interest in applications is contained in the indicator matrix
. Below a method for the asymptotic integration of (10) is given, that is, a method for successively determining the coefficients
and
in (16).
Suppose that in (11). Although
, generally speaking there is no branch of the matrix logarithm such that the matrix
is analytic for
and
. This branch of the logarithm will exist in the so-called non-resonance case, when among the eigen values
of
there are no numbers for which
![]() |
( is an integer). In the resonance case (when such eigen values exist) equation (10) reduces by a suitable change of variable
, where
, to an analogous equation for which the non-resonance case holds. The matrix
can be determined from the matrix
.
In (16), in the non-resonance case ,
, and the matrices
,
,
are found from the equation
![]() |
after equating coefficients at the same powers of in this equation. To determine
and
one obtains a matrix equation of the form
![]() | (17) |
where . The matrices
and
are found, and moreover uniquely (the non-resonance case), from (17) and the periodicity condition
.
For special cases of the system (1) see Hamiltonian system, linear and Hill equation.
References
[1] | I.Z. Shtokalo, "Linear differential equations with variable coefficients: criteria of stability and unstability of their solutions" , Hindushtan Publ. Comp. (1961) (Translated from Russian) |
[2] | N.P. Erugin, "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press (1966) (Translated from Russian) |
[3] | V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , Wiley (1975) (Translated from Russian) |
Comments
References
[a1] | R.W. Brockett, "Finite dimensional linear systems" , Wiley (1970) |
[a2] | J.K. Hale, "Ordinary differential equations" , Wiley (1969) |
Linear system of differential equations with periodic coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_system_of_differential_equations_with_periodic_coefficients&oldid=16408