View source for Linear system of differential equations with periodic coefficients

A system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594901.png" /> linear ordinary differential equations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594903.png" /> is a real variable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594905.png" /> are complex-valued functions, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594907.png" /> is called the period of the coefficients of the system (1). It is convenient to write (1) as one vector equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l0594909.png" /></td> </tr></table>

It is assumed that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949010.png" /> are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949011.png" /> and are measurable and Lebesgue integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949012.png" />, and that the equalities (2) are satisfied almost-everywhere, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949013.png" />. A solution of (3) is a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949014.png" /> with absolutely-continuous components such that (3) is satisfied almost-everywhere. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949016.png" /> are an (arbitrarily) given number and vector. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949017.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949018.png" /> exists and is uniquely determined. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949019.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949020.png" /> with absolutely-continuous entries is called the matrizant (or evolution matrix, or transition matrix, or [[Cauchy matrix|Cauchy matrix]]) of (3) if almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949021.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949022.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949024.png" /> is the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949025.png" /> matrix. The transition matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949026.png" /> satisfies the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949027.png" /></td> </tr></table>

The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949028.png" /> is called the monodromy matrix, and its eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949029.png" /> are called the multipliers of (3). The equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

for the multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949031.png" /> is called the characteristic equation of equation (3) (or of the system (1)). To every eigen vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949032.png" /> of the monodromy matrix with multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949033.png" /> corresponds a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949034.png" /> of (3) satisfying the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949035.png" /></td> </tr></table>

The Floquet–Lyapunov theorem holds: The transition matrix of (3) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949036.png" />-periodic matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949037.png" /> can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949039.png" /> is a constant matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949040.png" /> is an absolutely-continuous matrix function, periodic with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949041.png" />, non-singular for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949042.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949043.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949045.png" /> are matrices with the given properties, then the matrix (5) is the transition matrix of an equation (3) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949046.png" />-periodic matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949047.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949048.png" />, called the indicator matrix, and the matrix function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949049.png" /> in the representation (5) are not uniquely determined. In the case of real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949050.png" /> in (5), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949051.png" /> is a real matrix, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949053.png" /> are complex matrices, generally speaking. For this case there is a refinement of the Floquet–Lyapunov theorem: The transition matrix of (3) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949054.png" />-periodic real matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949055.png" /> can be represented in the form (5), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949056.png" /> is a constant real matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949057.png" /> is a real absolutely-continuous matrix function, non-singular for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949058.png" />, satisfying the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949060.png" /> is a real matrix such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949061.png" /></td> </tr></table>

In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949062.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949065.png" /> are arbitrary matrices with the given properties, then (5) is the transition matrix of an equation (3) with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949066.png" />-periodic real matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949067.png" />.

An immediate consequence of (5) is Floquet's theorem, which asserts that equation (3) has a [[Fundamental system of solutions|fundamental system of solutions]] splitting into subsets, each of which has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949068.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949069.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949071.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949072.png" /> are absolutely-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949073.png" />-periodic (generally speaking, complex-valued) vector functions. (The given subset of solutions corresponds to one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949074.png" />-cell of the Jordan form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949075.png" />.) If all elementary divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949076.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949077.png" /></td> </tr></table>

Formula (5) implies that (3) is reducible (see [[Reducible linear system|Reducible linear system]]) to the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949078.png" /></td> </tr></table>

by means of the change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949079.png" /> (Lyapunov's theorem).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949080.png" /> be the multipliers of equation (3) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949081.png" /> be an arbitrary indicator matrix, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

The eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949084.png" /> are called the characteristic exponents (cf. [[Characteristic exponent|Characteristic exponent]]) of (3). From (6) one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949086.png" />. The characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949087.png" /> can be defined as the complex number for which (3) has a solution that is representable in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949088.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949089.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949090.png" />-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).''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Property of the solutions</td> <td colname="2" style="background-color:white;" colspan="1">Characteristic exponents</td> <td colname="3" style="background-color:white;" colspan="1">Multipliers</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Stability of the trivial solution (boundedness on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949091.png" /> of all solutions)</td> <td colname="2" style="background-color:white;" colspan="1">Real parts non-positive; if zero or purely imaginary characteristic exponents are present, then simple elementary divisors of the indicator matrix correspond to them</td> <td colname="3" style="background-color:white;" colspan="1">Situated inside or on the unit circle; in the latter case simple elementary divisors of the monodromy matrix correspond to them</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Asymptotic stability of the trivial solution (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949092.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949093.png" /> for any solution)</td> <td colname="2" style="background-color:white;" colspan="1">Real parts negative</td> <td colname="3" style="background-color:white;" colspan="1">Situated inside the unit circle</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Boundedness of all solutions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949094.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Purely imaginary with simple elementary divisors of the indicator matrix</td> <td colname="3" style="background-color:white;" colspan="1">Situated on the unit circle; simple elementary divisors of the monodromy matrix correspond to them</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Instability of the trivial solution (existence of solutions unbounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949095.png" />)</td> <td colname="2" style="background-color:white;" colspan="1">There is either a characteristic exponent with positive real part or a purely imaginary one (in particular, zero) with non-simple elementary divisor of the indicator matrix</td> <td colname="3" style="background-color:white;" colspan="1">There is a multiplier either outside the unit circle or one on the unit circle with a non-simple elementary divisor of the monodromy matrix</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949096.png" />-periodic solution</td> <td colname="2" style="background-color:white;" colspan="1">For some characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949097.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949098.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l05949099.png" /> is an integer)</td> <td colname="3" style="background-color:white;" colspan="1">One of the multipliers is equal to one</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Existence of a semi-periodic solution, i.e. a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490100.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490101.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490102.png" /></td> <td colname="2" style="background-color:white;" colspan="1">For some characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490103.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490104.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490105.png" /> is an integer)</td> <td colname="3" style="background-color:white;" colspan="1">There is a multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490106.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490107.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490108.png" /> given). These problems thus reduce to the calculation or estimation of the characteristic exponents (multipliers) of (1).

The equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490111.png" /> are a measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490112.png" />-periodic matrix function and vector function, respectively, that are Lebesgue integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490113.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490115.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490116.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

does not have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490117.png" />-periodic solutions, then (7) has a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490118.png" />-periodic solution. It can be determined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490119.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490121.png" /> is the transition matrix of the homogeneous equation (8), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490123.png" />.

Suppose that (8) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490124.png" /> linearly independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490125.png" />-periodic solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490126.png" />. Then the adjoint equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490127.png" /></td> </tr></table>

also has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490128.png" /> linearly independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490129.png" />-periodic solutions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490130.png" />. The inhomogeneous equation (7) has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490131.png" />-periodic solution if and only if that the orthogonality relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490132.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

hold. If so, an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490133.png" />-periodic solution of (7) has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490134.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490135.png" /> are arbitrary numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490136.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490137.png" />-periodic solution of (7). Under the additional conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490138.png" /></td> </tr></table>

the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490139.png" />-periodic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490140.png" /> is determined uniquely; moreover, there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490141.png" />, independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490142.png" />, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490143.png" /></td> </tr></table>

Suppose one is given an equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490144.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

with a matrix coefficient that holomorphically depends on a complex  "small"  parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490145.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490146.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

Suppose that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490147.png" /> the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490148.png" /></td> </tr></table>

converges, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490149.png" /></td> </tr></table>

which guarantees the (componentwise) convergence of the series (11) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490150.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490151.png" />. Then the transition matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490152.png" /> of (10) for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490153.png" /> is an analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490154.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490155.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490156.png" /> be a constant matrix with eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490158.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490159.png" /> be the multipliers of equation (10), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490160.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490161.png" /> is a multiplier of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490162.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490163.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490164.png" /> are integers. If simple elementary divisors of the monodromy matrix correspond to this multiplier, or, in other words, if to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490166.png" />, correspond simple elementary divisors of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490167.png" /> (for example, if all the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490168.png" /> are distinct), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490169.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490171.png" />-fold characteristic exponent (of equation (10) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490172.png" />) of simple type. It turns out that the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490174.png" /> characteristic exponents of (10) with small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490175.png" /> can be very easily computed to a first approximation. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490177.png" /> be the corresponding normalized eigen vectors of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490178.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490179.png" />;

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490180.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490181.png" /></td> </tr></table>

let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490182.png" /></td> </tr></table>

be the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490183.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490184.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490185.png" /> are the numbers from (12). Then for the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490186.png" /> characteristic exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490188.png" />, of (10), which become <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490189.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490190.png" />, one has series expansions in fractional powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490191.png" />, starting with terms of the first order:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490192.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490193.png" /> are the roots (written as many times as their multiplicity) of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490194.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490195.png" /> are natural numbers equal to the multiplicities of the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490196.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490198.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490199.png" />). If the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490200.png" /> is simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490201.png" /> and the corresponding function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490202.png" /> is analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490203.png" />. From (13) it follows that cases are possible in which the  "unperturbed"  (that is, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490204.png" />) system is stable (all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490205.png" /> are purely imaginary and simple elementary divisors correspond to them), but the  "perturbed"  system (small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490206.png" />) is unstable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490207.png" /> for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490208.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490209.png" /> be the distinct multipliers of equation (3) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490210.png" /> be their multiplicities, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490211.png" />. Suppose that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490212.png" /> on the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490213.png" />-plane are surrounded by non-intersecting discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490214.png" /> and that a cut, not intersecting these discs, is drawn from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490215.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490216.png" />. Suppose that with each multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490217.png" /> is associated an arbitrary integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490218.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490219.png" /> is the transition matrix of (10). The branches of the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490220.png" /> are determined by means of the cut. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490221.png" /> ( "matrix logarithmmatrix logarithm" ) can be defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490222.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490223.png" /> is the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490224.png" />. The set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490225.png" /> determines a branch of the matrix logarithm. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490226.png" /> for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490227.png" />. Generally speaking, formula (14) for all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490228.png" /> does not cover all the values of the matrix logarithm, that is, all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490229.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490230.png" />. However, the solution given by (14) has the important property of holomorphy: The entries of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490231.png" /> in (14) are holomorphic functions of the entries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490232.png" />. For equation (10), formula (5) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490233.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490235.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490236.png" /> is determined in accordance with (14), then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490237.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>

are series that converge for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490238.png" />. The main information about the behaviour of the solutions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490239.png" /> which is usually of interest in applications is contained in the indicator matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490240.png" />. Below a method for the asymptotic integration of (10) is given, that is, a method for successively determining the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490241.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490242.png" /> in (16).

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490243.png" /> in (11). Although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490244.png" />, generally speaking there is no branch of the matrix logarithm such that the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490245.png" /> is analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490246.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490247.png" />. This branch of the logarithm will exist in the so-called non-resonance case, when among the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490248.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490249.png" /> there are no numbers for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490250.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490251.png" /> is an integer). In the resonance case (when such eigen values exist) equation (10) reduces by a suitable change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490252.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490253.png" />, to an analogous equation for which the non-resonance case holds. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490254.png" /> can be determined from the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490255.png" />.

In (16), in the non-resonance case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490256.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490257.png" />, and the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490258.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490259.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490260.png" /> are found from the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490261.png" /></td> </tr></table>

after equating coefficients at the same powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490262.png" /> in this equation. To determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490263.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490264.png" /> one obtains a matrix equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490265.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490266.png" />. The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490267.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490268.png" /> are found, and moreover uniquely (the non-resonance case), from (17) and the periodicity condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059490/l059490269.png" />.

For special cases of the system (1) see [[Hamiltonian system, linear|Hamiltonian system, linear]] and [[Hill equation|Hill equation]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.Z. Shtokalo,   "Linear differential equations with variable coefficients: criteria of stability and unstability of their solutions" , Hindushtan Publ. Comp.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Erugin,   "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Yakubovich,   V.M. Starzhinskii,   "Linear differential equations with periodic coefficients" , Wiley  (1975)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.W. Brockett,   "Finite dimensional linear systems" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR></table>