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A theory concerning the structure of the space of solutions, and the properties of solutions, of a [[Linear system of differential equations with periodic coefficients|linear system of differential equations with periodic coefficients]]
 
A theory concerning the structure of the space of solutions, and the properties of solutions, of a [[Linear system of differential equations with periodic coefficients|linear system of differential equations with periodic coefficients]]
  
<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/f/f040/f040640/f0406401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
x  ^  \prime  = \
 +
A ( t) x,\ \
 +
t \in \mathbf R ,\ \
 +
x \in \mathbf R  ^ {n} ;
 +
$$
  
the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406402.png" /> is periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406403.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406404.png" /> and is summable on every compact interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406405.png" />.
+
the matrix $  A ( t) $
 +
is periodic in $  t $
 +
with period $  \omega > 0 $
 +
and is summable on every compact interval in $  \mathbf R $.
  
1) Every [[Fundamental matrix|fundamental matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406406.png" /> of the system (1) has a representation
+
1) Every [[Fundamental matrix|fundamental matrix]] $  X $
 +
of the system (1) has a representation
  
<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/f/f040/f040640/f0406407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
X ( t)  = F ( t) \
 +
\mathop{\rm exp} ( tK),
 +
$$
  
called the Floquet representation (see [[#References|[1]]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406408.png" /> is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f0406409.png" />-periodic matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064010.png" /> is some constant matrix. There is a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064011.png" /> of the space of solutions of (1) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064012.png" /> has Jordan form in this basis; this basis can be represented in the form
+
called the Floquet representation (see [[#References|[1]]]), where $  F ( t) $
 +
is some $  \omega $-
 +
periodic matrix and $  K $
 +
is some constant matrix. There is a basis $  x _ {1} \dots x _ {n} $
 +
of the space of solutions of (1) such that $  K $
 +
has Jordan form in this basis; this basis 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/f/f040/f040640/f04064013.png" /></td> </tr></table>
+
$$
 +
x _ {i}  = \
 +
( \psi _ {1i}  \mathop{\rm exp} ( \alpha _ {i} t) \dots
 +
\psi _ {ni}  \mathop{\rm exp} ( \alpha _ {i} t)),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064014.png" /> are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064016.png" />-periodic coefficients, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064017.png" /> are the characteristic exponents (cf. [[Characteristic exponent|Characteristic exponent]]) of the system (1). Every component of a solution of (1) is a linear combination of functions of the form (of the Floquet solutions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064018.png" />. In the case when all the characteristic exponents are distinct (or if there are multiple ones among them, but they correspond to simple elementary divisors), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064019.png" /> are simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064020.png" />-periodic functions. The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064022.png" /> in the representation (2) are, generally speaking, complex valued. If one restricts oneself just to the real case, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064023.png" /> does not have to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064024.png" />-periodic, but must be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064025.png" />-periodic.
+
where $  \psi _ {ki} $
 +
are polynomials in $  t $
 +
with $  \omega $-
 +
periodic coefficients, and the $  \alpha _ {i} $
 +
are the characteristic exponents (cf. [[Characteristic exponent|Characteristic exponent]]) of the system (1). Every component of a solution of (1) is a linear combination of functions of the form (of the Floquet solutions) $  \psi _ {ki}  \mathop{\rm exp} ( \alpha _ {i} t) $.  
 +
In the case when all the characteristic exponents are distinct (or if there are multiple ones among them, but they correspond to simple elementary divisors), the $  \psi _ {ki} $
 +
are simply $  \omega $-
 +
periodic functions. The matrices $  F ( t) $
 +
and $  K $
 +
in the representation (2) are, generally speaking, complex valued. If one restricts oneself just to the real case, then $  F ( t) $
 +
does not have to be $  \omega $-
 +
periodic, but must be $  2 \omega $-
 +
periodic.
  
2) The system (1) can be reduced to a differential equation with a constant matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064026.png" />, by means of the Lyapunov transformation
+
2) The system (1) can be reduced to a differential equation with a constant matrix, $  y  ^  \prime  = Ky $,  
 +
by means of the Lyapunov transformation
  
<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/f/f040/f040640/f04064027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
= F ( t) y,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064029.png" /> are the matrices from the Floquet representation (2) (see [[#References|[2]]]). The combination of representation (2) together with the substitution (3) is often called the Floquet–Lyapunov theorem.
+
where $  F ( t) $
 +
and $  K $
 +
are the matrices from the Floquet representation (2) (see [[#References|[2]]]). The combination of representation (2) together with the substitution (3) is often called the Floquet–Lyapunov theorem.
  
3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064030.png" /> be the spectrum of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064031.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064034.png" />, in view of (2) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064035.png" /> splits into the direct sum of two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064037.png" />
+
3) Let $  \{ \alpha _ {1} \dots \alpha _ {l} \} $
 +
be the spectrum of the matrix $  K $.  
 +
For every $  \alpha \in \mathbf R $
 +
such that $  \alpha \neq  \mathop{\rm Re}  \alpha _ {j} $,  
 +
$  j = 1 \dots l $,  
 +
in view of (2) the space $  \mathbf R  ^ {n} $
 +
splits into the direct sum of two subspaces $  S _  \alpha  $
 +
and $  U _  \alpha  $
  
<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/f/f040/f040640/f04064038.png" /></td> </tr></table>
+
$$
 +
( \mathbf R  ^ {n}  = \
 +
S _  \alpha  + U _  \alpha  ,\ \
 +
S _  \alpha  \cap U _  \alpha  = \emptyset )
 +
$$
  
 
such that
 
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/f/f040/f040640/f04064039.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow + \infty } \
 +
\mathop{\rm exp} (- \alpha t)
 +
V ( t) x ( 0)  = 0
 +
\  \iff \ \
 +
x ( 0)  \in  S _  \alpha  ,
 +
$$
  
<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/f/f040/f040640/f04064040.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow - \infty }  \mathop{\rm exp} (- \alpha t) V ( t) x
 +
( 0)  = 0 \  \iff \  x ( 0)  \in  U _  \alpha  ;
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064041.png" /> is the fundamental matrix of (1) normalized at zero. This implies exponential [[Dichotomy|dichotomy]] of (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064042.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040640/f04064043.png" />.
+
here $  V ( t) $
 +
is the fundamental matrix of (1) normalized at zero. This implies exponential [[Dichotomy|dichotomy]] of (1) if $  \mathop{\rm Re}  \alpha _ {j} \neq 0 $
 +
for any $  j = 1 \dots l $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Floquet,  ''Ann. Sci. Ecole Norm. Sup.'' , '''12''' :  2  (1883)  pp. 47–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. Lyapunov,  "Problème général de la stabilité du mouvement" , ''Collected works'' , '''2''' , Princeton Univ. Press , Moscow-Leningrad  (1956)  pp. 7–263  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Yakubovich,  V.M. Starzhinskii,  "Linear differential equations with periodic coefficients" , Wiley  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.L. Massera,  J.J. Shäffer,  "Linear differential equations and function spaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[6]</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Floquet,  ''Ann. Sci. Ecole Norm. Sup.'' , '''12''' :  2  (1883)  pp. 47–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. Lyapunov,  "Problème général de la stabilité du mouvement" , ''Collected works'' , '''2''' , Princeton Univ. Press , Moscow-Leningrad  (1956)  pp. 7–263  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Yakubovich,  V.M. Starzhinskii,  "Linear differential equations with periodic coefficients" , Wiley  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.L. Massera,  J.J. Shäffer,  "Linear differential equations and function spaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[6]</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,  "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,  "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


A theory concerning the structure of the space of solutions, and the properties of solutions, of a linear system of differential equations with periodic coefficients

$$ \tag{1 } x ^ \prime = \ A ( t) x,\ \ t \in \mathbf R ,\ \ x \in \mathbf R ^ {n} ; $$

the matrix $ A ( t) $ is periodic in $ t $ with period $ \omega > 0 $ and is summable on every compact interval in $ \mathbf R $.

1) Every fundamental matrix $ X $ of the system (1) has a representation

$$ \tag{2 } X ( t) = F ( t) \ \mathop{\rm exp} ( tK), $$

called the Floquet representation (see [1]), where $ F ( t) $ is some $ \omega $- periodic matrix and $ K $ is some constant matrix. There is a basis $ x _ {1} \dots x _ {n} $ of the space of solutions of (1) such that $ K $ has Jordan form in this basis; this basis can be represented in the form

$$ x _ {i} = \ ( \psi _ {1i} \mathop{\rm exp} ( \alpha _ {i} t) \dots \psi _ {ni} \mathop{\rm exp} ( \alpha _ {i} t)), $$

where $ \psi _ {ki} $ are polynomials in $ t $ with $ \omega $- periodic coefficients, and the $ \alpha _ {i} $ are the characteristic exponents (cf. Characteristic exponent) of the system (1). Every component of a solution of (1) is a linear combination of functions of the form (of the Floquet solutions) $ \psi _ {ki} \mathop{\rm exp} ( \alpha _ {i} t) $. In the case when all the characteristic exponents are distinct (or if there are multiple ones among them, but they correspond to simple elementary divisors), the $ \psi _ {ki} $ are simply $ \omega $- periodic functions. The matrices $ F ( t) $ and $ K $ in the representation (2) are, generally speaking, complex valued. If one restricts oneself just to the real case, then $ F ( t) $ does not have to be $ \omega $- periodic, but must be $ 2 \omega $- periodic.

2) The system (1) can be reduced to a differential equation with a constant matrix, $ y ^ \prime = Ky $, by means of the Lyapunov transformation

$$ \tag{3 } x = F ( t) y, $$

where $ F ( t) $ and $ K $ are the matrices from the Floquet representation (2) (see [2]). The combination of representation (2) together with the substitution (3) is often called the Floquet–Lyapunov theorem.

3) Let $ \{ \alpha _ {1} \dots \alpha _ {l} \} $ be the spectrum of the matrix $ K $. For every $ \alpha \in \mathbf R $ such that $ \alpha \neq \mathop{\rm Re} \alpha _ {j} $, $ j = 1 \dots l $, in view of (2) the space $ \mathbf R ^ {n} $ splits into the direct sum of two subspaces $ S _ \alpha $ and $ U _ \alpha $

$$ ( \mathbf R ^ {n} = \ S _ \alpha + U _ \alpha ,\ \ S _ \alpha \cap U _ \alpha = \emptyset ) $$

such that

$$ \lim\limits _ {t \rightarrow + \infty } \ \mathop{\rm exp} (- \alpha t) V ( t) x ( 0) = 0 \ \iff \ \ x ( 0) \in S _ \alpha , $$

$$ \lim\limits _ {t \rightarrow - \infty } \mathop{\rm exp} (- \alpha t) V ( t) x ( 0) = 0 \ \iff \ x ( 0) \in U _ \alpha ; $$

here $ V ( t) $ is the fundamental matrix of (1) normalized at zero. This implies exponential dichotomy of (1) if $ \mathop{\rm Re} \alpha _ {j} \neq 0 $ for any $ j = 1 \dots l $.

References

[1] G. Floquet, Ann. Sci. Ecole Norm. Sup. , 12 : 2 (1883) pp. 47–88
[2] A.M. Lyapunov, "Problème général de la stabilité du mouvement" , Collected works , 2 , Princeton Univ. Press , Moscow-Leningrad (1956) pp. 7–263 (In Russian)
[3] B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[4] V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , Wiley (1975) (Translated from Russian)
[5] J.L. Massera, J.J. Shäffer, "Linear differential equations and function spaces" , Acad. Press (1966)
[6] N.P. Erugin, "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press (1966) (Translated from Russian)

Comments

References

[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1969)
[a2] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
How to Cite This Entry:
Floquet theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Floquet_theory&oldid=46944
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article