Cauchy operator

of a system of ordinary differential equations

$$ \tag{1 } \dot{x} = \ f (t, x),\ \ x \in \mathbf R ^ {n} $$

The operator $ X ( \theta , \tau ) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $, depending on two parameters $ \theta $ and $ \tau $, which, given the value of any solution $ x (t) $ of the system at the point $ t = \tau $, gives the value of that solution at the point $ t = \theta $:

$$ X ( \theta , \tau ) x ( \tau ) = x ( \theta ). $$

If (1) is a linear system, i.e.

$$ \tag{2 } \dot{x} = A (t) x, $$

where $ A ( \cdot ) $ is a mapping $ ( \alpha , \beta ) \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ (or $ ( \alpha , \beta ) \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $), summable over every interval, then for any $ \theta , \tau \in ( \alpha , \beta ) $ the Cauchy operator is a non-singular linear mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ (or $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $), and for any $ \theta , \tau , \eta \in ( \alpha , \beta ) $, it satisfies

$$ \tag{3 } \left . \begin{array}{c} X ( \theta , \theta ) = I,\ \ X ( \theta , \tau ) = \ X ^ {-1} ( \tau , \theta ), \\ X ( \theta , \eta ) X ( \eta , \tau ) = \ X ( \theta , \tau ) \\ \end{array} \right \} $$

and the inequality

$$ \| X ( \theta , \tau ) \| \leq \mathop{\rm exp} \ \left |\, \int\limits _ \tau ^ \theta \| A (t) \| dt \, \right | . $$

(The equations (3) are also valid for a non-linear system satisfying the conditions of the existence and uniqueness theorem for solutions of the Cauchy problem, with the necessary stipulations concerning the domains of definition of the operators figuring therein.) The general solution of the system

$$ \dot{x} = \ A (t) x + h (t), $$

where $ h ( \cdot ) $ is a mapping

$$ ( \alpha , \beta ) \rightarrow \mathbf R ^ {n} \ \ ( \textrm{ or } \ ( \alpha , \beta ) \rightarrow \mathbf C ^ {n} ) $$

summable on every interval, is described in terms of the Cauchy operator $ X ( \theta , \tau ) $ of the system (2) by the formula of variation of constants:

$$ x (t) = \ X (t, \tau ) x ( \tau ) + \int\limits _ \tau ^ { t } X (t, \theta ) h ( \theta ) d \theta . $$

The Cauchy operator of the system (2) satisfies the Liouville–Ostrogradski formula:

$$ \mathop{\rm det} X ( \theta , \tau ) = \ \mathop{\rm exp} \int\limits _ \tau ^ \theta \mathop{\rm tr} A ( \xi ) d \xi , $$

where $ \mathop{\rm tr} A ( \xi ) $ is the trace of the operator $ A ( \xi ) $.

The derivative of the Cauchy operator $ X ( \theta , \tau ) $ of the system at a point $ x \in \mathbf R ^ {n} $ is equal to the Cauchy operator of the system of equations in variations along the solution $ x (t) $ of the system the value of which at $ t = \tau $ is $ x $ (on the assumption that, for all $ t $ in the interval with end points $ \theta $ and $ \tau $, the graph of $ x (t) $ lies in a domain $ G \subset \mathbf R ^ {n + 1 } $ such that $ f $ is a continuous mapping $ G \rightarrow \mathbf R ^ {n} $ with continuous derivative $ f _ {x} ^ { \prime } $ in $ G $; this is one formulation of a theorem asserting the differentiability of the solution with respect to the initial value).

For a linear system (2) with constant coefficients ( $ A (t) \equiv A $), the Cauchy operator is defined by

$$ \tag{4 } X ( \theta , \tau ) = \ \mathop{\rm exp} ( ( \theta - \tau ) A) $$

(given a linear operator $ B $, $ \mathop{\rm exp} B $ is defined as $ \sum _ {k = 0 } ^ \infty B ^ {k} /k! $; adopting a different approach, one can define $ \mathop{\rm exp} A $ via formula (4), putting $ \theta = \tau + 1 $). It is evident from (4) that the Cauchy operator depends only on the difference $ \theta - \tau $ of the parameters:

$$ X ( \theta + t, \tau + t) = \ X ( \theta , \tau ). $$

This equation is a consequence of the autonomy of the system — a property valid for every autonomous system

$$ \tag{5 } \dot{x} = f (x),\ \ x \in \mathbf R ^ {n} . $$

Denoting the Cauchy operator $ X ( \theta , \tau ) $ of the system (5) by $ f ^ {\theta - \tau } $, one obtains the following formulas from (3):

$$ f ^ {0} = I; \ \ (f ^ {t} ) ^ {-1} = \ f ^ {-t} ; \ \ f ^ {t} f ^ \tau = \ f ^ {t + \tau } $$

(see also Dynamical system; Action of a group on a manifold).

For a linear system (2) with periodic coefficients, i.e.

$$ A (t + T) = \ A (t) $$

for some $ T > 0 $ and all $ t \in \mathbf R $, one has the identity

$$ X ( \theta + T, \tau + T) = \ X ( \theta , \tau ) $$

for all $ \theta , \tau \in \mathbf R $; in this case the operator $ X ( \tau + T, \tau ) $, where $ \tau \in \mathbf R $ is arbitrary, is called the monodromy operator. The matrix defining the operator $ X ( \tau + T, \tau ) $ (or, say, $ X (T, 0) $) relative to some basis is called the monodromy matrix. All monodromy operators of a fixed linear system with periodic coefficients are similar to one another:

$$ X ( \theta + T, \theta ) = \ X ( \theta , \tau ) X ( \tau + T, \tau ) X ^ {-1} ( \theta , \tau ), $$

therefore, the spectrum of the monodromy operator $ X ( \tau + T, \tau ) $ is independent of $ \tau $. The eigen values of the monodromy operator are called the multipliers of the system; one can express conditions for stability and conditional stability of the system in terms of the multipliers (see Lyapunov characteristic exponent; Lyapunov stability; Stability theory). If the system (2) has periodic complex coefficients,

$$ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ),\ \ A (t + T) = A (t) $$

for some $ T > 0 $ and all $ t \in \mathbf R $, one has the theorem of Lyapunov

$$ X ( \theta , \tau ) = \ S _ \tau ( \theta ) \mathop{\rm exp} (( \theta - \tau ) B _ \tau ), $$

where $ B _ \tau = (1/T) \mathop{\rm ln} X ( \tau + T, \tau ) $, and $ S _ \tau ( \theta ) $ is a non-singular linear operator $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ for any $ \theta , \tau \in \mathbf R $, which is a periodic function of $ \theta $:

$$ S _ \tau ( \theta + T) = \ S _ \tau ( \theta ). $$

Different names are sometimes used for the Cauchy operator (e.g. "matrizant" for a linear system, or "operator of translation along trajectories" ).

Comments

The operator $ X ( \theta , \tau ) $ does not usually have the name Cauchy attached to it in the Western literature, and in fact is usually not given any particular name at all. In Section 2.1 of [a2], Cauchy's role in the analysis of (1) is sketched. The Liouville–Ostrogradski formula is better known as Liouville's formula. [a1] contains a proof of this formula.

References

[a1]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[a2]	E. Hille, "Ordinary differential equations in the complex domain" , Wiley (Interscience) (1976)
[a3]	M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974)