View source for Variational equations

''system of variational equations, equations in variation''

Linear differential (or difference) equations whose solution is the derivative, with respect to a parameter, of the solution of a differential (or difference) equation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962201.png" /> be a solution of the [[Cauchy problem|Cauchy problem]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962203.png" />, with graph in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962204.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962206.png" /> are continuous. Then for every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962207.png" /> and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962208.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v0962209.png" /> such that for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622010.png" /> having a continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622012.png" /> and satisfying the inequality

<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/v/v096/v096220/v09622013.png" /></td> </tr></table>

the Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622015.png" /> has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622016.png" />, defined in some neighbourhood of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622017.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622018.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622019.png" />. For the difference of these solutions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622020.png" />, there is 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/v/v096/v096220/v09622021.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622022.png" /> is a solution of the linear differential 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/v/v096/v096220/v09622023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622025.png" />, with initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622026.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622027.png" /> is  "little oh"  uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622028.png" />, and the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622029.png" />, by definition, equals

<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/v/v096/v096220/v09622030.png" /></td> </tr></table>

Equation (1) is called the variational equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622031.png" /> along the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622032.png" />.

In the literature a weaker form of this theorem is more often quoted (where instead of Fréchet differentiability a weaker sense of differentiability is used): If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622033.png" /> on the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622034.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622035.png" /> and the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622036.png" /> is continuous and has continuous partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622038.png" /> while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622039.png" /> is continuously differentiable, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622040.png" /> of the Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622042.png" /> is continuously differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622043.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622044.png" />, and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622045.png" /> is a solution of the linear differential equation (the variational equation for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622046.png" /> along the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622047.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/v/v096/v096220/v09622048.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622050.png" />, satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622051.png" />.

The variational equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622052.png" /> is a linear differential (difference) equation whose solution is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622053.png" />-th derivative with respect to a parameter of the solution of a differential (difference) equation. The form of the linear homogeneous equation corresponding to a variational equation of any order is the same (i.e. independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622054.png" />), the difference lies in the inhomogeneity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622055.png" />.

If the right-hand side of the differential equation is not varied (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622056.png" /> in the first formulation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622057.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622058.png" /> in the second), then the variational equation (of the first order) is homogeneous.

The variational equation of an autonomous system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622059.png" /> at a fixed point (i.e. along a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622060.png" />) is a linear system of differential equations with constant coefficients, and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622061.png" /> is not varied, then the system is homogeneous for variations of the first order and  "with quasi-polynomial right-hand side"  for variations of higher orders. Variational equations of autonomous systems along a periodic (almost periodic) solution are linear systems of differential equations with periodic coefficients (respectively, with almost-periodic coefficients, cf. [[Linear system of differential equations with periodic coefficients|Linear system of differential equations with periodic coefficients]]; [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]).

The definition given above applies to equations of any order. For example, the variational equation (if only the initial point in the phase space is varied) for the pendulum equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622062.png" /> in the lower position of equilibrium (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622064.png" />) is the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622065.png" />, called the equation for small oscillations of a pendulum, while in the upper position of equilibrium (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622067.png" />) the equation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622068.png" />. For differential equations on a differentiable manifold the variational equations for the solution are defined similarly to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622069.png" /> treated above; the values of the solution of the variational equations ly in the tangent bundle of the manifold. There are two ways of reduction of the case of an arbitrary differentiable manifold to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622070.png" />, the first consisting of imbedding the manifold in a Euclidean space of sufficiently high dimension and extending the differential equation (vector field) to a neighbourhood, while the second way consists of writing down the differential equation, given on the differentiable manifold, in a neighbourhood of the trajectory in terms of the coordinates of a chart, where the chart is chosen to depend smoothly on the point (e.g. for Riemannian manifolds by using the exponential geodesic mapping). This allows one to write the given equation as a differential equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622071.png" />, having (as in the first reduction) a right-hand side of the same smoothness class as the right-hand side (vector field) of the equation on the manifold. For a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622072.png" /> on a Riemannian manifold the variational equation along the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622073.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622074.png" /> is not varied, may be written 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/v/v096/v096220/v09622075.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622076.png" /> is the [[Covariant derivative|covariant derivative]]. The variational equation of a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622077.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622078.png" /> is a differentiable manifold) along the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622079.png" /> (if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622080.png" /> is not varied) is 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/v/v096/v096220/v09622081.png" /></td> </tr></table>

the value of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622082.png" /> of this equation at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622083.png" /> lies in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622084.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622085.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622086.png" />, and the solution itself is the sequence

<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/v/v096/v096220/v09622087.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622088.png" /> is the derivative of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622089.png" />-th power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622090.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622091.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622092.png" /> be a closed differentiable manifold. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622093.png" /> of all diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622094.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622095.png" />, mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622096.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622097.png" />, is equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622098.png" />-topology. The following assertions hold (cf. [[#References|[4]]]): 1) For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v09622099.png" /> the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]

<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/v/v096/v096220/v096220100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220101.png" /> is the Grassmann manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220102.png" />-dimensional vector subspaces of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220103.png" />, is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220104.png" /> of the second Baire class (cf. [[Baire classes|Baire classes]]); 2) in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220105.png" /> there is an everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220106.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220107.png" /> with the properties: a) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220108.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220109.png" /> is upper semi-continuous at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220110.png" />; and b) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220112.png" />, the subspace

<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/v/v096/v096220/v096220113.png" /></td> </tr></table>

is exponentially separated from its algebraic complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220114.png" /> in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220115.png" />, i.e. there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220116.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220118.png" /> and any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220119.png" /> the inequality

<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/v/v096/v096220/v096220120.png" /></td> </tr></table>

holds.

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220121.png" /> of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220122.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220123.png" /> on a closed differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220124.png" /> is equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220125.png" />-topology. A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220126.png" /> induces a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220127.png" /> (the action (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220128.png" />) of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220129.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220130.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220131.png" /> the Lyapunov exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220132.png" /> is by definition equal to the right-hand side of (2).

The following assertions hold:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220133.png" />) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220134.png" /> the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220135.png" /> ly in the second Baire class [[#References|[4]]];

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220136.png" />) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220137.png" />, for every probability distribution that is invariant relative to the dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220138.png" /> induced by the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220139.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220140.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220141.png" />-algebra of which contains all Borel subsets), almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220142.png" /> is such that the variational equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220143.png" /> along the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220144.png" /> is a regular linear system of differential equations (cf. [[#References|[5]]], [[#References|[6]]]).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220145.png" />) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220146.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220147.png" /> denote the set of all vector fields of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220148.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220149.png" />, equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220150.png" />-topology; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220151.png" /> be a probability distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220152.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220153.png" />-algebra of which contains all Borel sets, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220154.png" /> denote the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220155.png" /> consisting of all vector fields for which the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220156.png" /> is invariant relative to the dynamical systems induced by them; then (cf. [[#References|[7]]]):

A) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220158.png" />, the function

<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/v/v096/v096220/v096220159.png" /></td> </tr></table>

(the phase average sum of the highest Lyapunov exponents of the variational equation) is upper semi-continuous;

B) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220160.png" /> there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220161.png" /> an everywhere-dense set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220162.png" /> in which the function

<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/v/v096/v096220/v096220163.png" /></td> </tr></table>

is continuous (for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220164.png" />), i.e. in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096220/v096220165.png" /> continuity is typical for the phase averages of the Lyapunov exponents of the variational equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,   "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Millionshchikov,   "Baire function classes and Lyapunov exponents XII"  ''Differential Eq.'' , '''19''' :  2  (1983)  pp. 155–159  ''Differentsial'nye Uravneniya'' , '''19''' :  2  (1083)  pp. 215–220</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Oseledets,   "A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1969)  pp. 197–232  ''Tr. Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.M. Millionshchikov,   "Metric theory of linear systems of differential equations"  ''Math. USSR Sb.'' , '''6''' :  2  (1968)  pp. 149–158  ''Mat. Sb.'' , '''77'''  (1968)  pp. 163–173</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.M. Millionshchikov,   "Results and unsolved problems in the theory of Lyapunov indices"  ''Differential Eq.'' , '''14''' :  4  (1978)  pp. 543  ''Differentsial'nye Uravneniya'' , '''14''' :  4  (1978)  pp. 759–760</TD></TR></table>



====Comments====
Equations in variation make sense for many more general equations, in particular for partial differential equations.

In [[#References|[a1]]] the phrase equation of first variation is used. See also [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hille,   "Lectures on ordinary differential equations" , Addison-Wesley  (1964)</TD></TR></table>