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 $ x ( \cdot ) : ( \alpha , \beta ) \rightarrow \mathbf R ^ {n} $ be a solution of the Cauchy problem $ \dot{x} = f ( x , t ) $, $ x ( t _ {0} ) = x _ {0} $, with graph in a domain $ G $ in which $ f $ and $ f _ {x} ^ { \prime } $ are continuous. Then for every interval $ [ p , s ] \subset ( \alpha , \beta ) $ and for every $ \epsilon > 0 $ one can find a $ \delta > 0 $ such that for any continuous function $ g : G \rightarrow \mathbf R ^ {n} $ having a continuous derivative $ g _ {x} ^ \prime $ in $ G $ and satisfying the inequality

$$ \| g - f \| _ {C ( G) } = \ \sup _ {( x , t ) \in G } \ | g ( x , t ) - f ( x , t ) | < \delta , $$

the Cauchy problem $ \dot{y} = g ( y , t ) $, $ y ( t _ {0} ) = y _ {0} $ has a solution $ y ( \cdot ) $, defined in some neighbourhood of the interval $ [ p , s ] $ for every $ y _ {0} \in \mathbf R ^ {n} $ satisfying $ | y _ {0} - x _ {0} | < \delta $. For the difference of these solutions, $ y ( \cdot ) - x ( \cdot ) $, there is the formula

$$ y ( t) - x ( t) = \ z ( t) + o ( | y _ {0} - x _ {0} | + \| g - f \| _ {C ^ {1} ( G) } ) , $$

where $ z ( \cdot ) $ is a solution of the linear differential equation

$$ \tag{1 } \dot{z} = A ( t) z + h ( t) $$

in which $ A ( t) = f _ {x} ^ { \prime } ( x ( t) , t ) $, $ h ( t) = g ( x ( t) , t ) - f ( x ( t) , t ) $, with initial value $ z ( t _ {0} ) = y ( t _ {0} ) - x ( t _ {0} ) $; here $ o ( \cdot ) $ is "little oh" uniformly in $ t \in [ p , s ] $, and the norm $ \| g - f \| _ {C ^ {1} ( G) } $, by definition, equals

$$ \sup _ {( x , t ) \in G } \ \{ | g ( x , t ) - f ( x , t ) | + \| g _ {x} ^ \prime ( x , t ) - f _ {x} ^ { \prime } ( x , t ) \| \} . $$

Equation (1) is called the variational equation for $ \dot{x} = f ( x , t ) $ along the solution $ x ( \cdot ) $.

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 $ f ( x , t , \mu ) : G \times ( a , b ) \rightarrow \mathbf R ^ {n} $ on the product $ G \times ( a , b ) $ of a domain $ G \subset \mathbf R ^ {n} \times \mathbf R $ and the interval $ ( a , b ) \subset \mathbf R $ is continuous and has continuous partial derivatives $ f _ {x} ^ { \prime } $, $ f _ \mu ^ { \prime } $ while the function $ x _ {0} ( \cdot ) : ( a , b ) \rightarrow \mathbf R ^ {n} $ is continuously differentiable, then the solution $ x ( \cdot , \mu ) $ of the Cauchy problem $ \dot{x} = f ( x , t , \mu ) $, $ x ( t _ {0} ) = x _ {0} ( \mu ) $ is continuously differentiable with respect to $ \mu $ in the interval $ ( a , b ) $, and its derivative $ x _ \mu ^ \prime ( \cdot , \mu ) $ is a solution of the linear differential equation (the variational equation for the equation $ \dot{x} = f ( x , t , \mu ) $ along the solution $ x ( \cdot , \mu ) $)

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

where $ A ( t) = f _ {x} ^ { \prime } ( x ( t , \mu ) , t , \mu ) $, $ h ( t) = f _ \mu ^ { \prime } ( x ( t , \mu ) , t , \mu ) $, satisfying the initial condition $ z ( t _ {0} ) = x _ {0 \mu } ^ \prime ( \mu ) $.

The variational equation of order $ k $ is a linear differential (difference) equation whose solution is the $ k $-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 $ k $), the difference lies in the inhomogeneity $ h ( t) $.

If the right-hand side of the differential equation is not varied ( $ g = f $ in the first formulation, $ f ( x , t , \mu ) $ does not depend on $ \mu $ in the second), then the variational equation (of the first order) is homogeneous.

The variational equation of an autonomous system $ \dot{x} = f ( x) $ at a fixed point (i.e. along a solution $ x ( \cdot ) = x _ {0} $) is a linear system of differential equations with constant coefficients, and, if $ f ( \cdot ) $ 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 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 $ \ddot{x} + \omega ^ {2} \sin x = 0 $ in the lower position of equilibrium ( $ x = 0 $, $ \dot{x} = 0 $) is the equation $ \ddot{x} + \omega ^ {2} x = 0 $, called the equation for small oscillations of a pendulum, while in the upper position of equilibrium ( $ x = \pi $, $ \dot{x} = 0 $) the equation is $ \ddot{x} - \omega ^ {2} x = 0 $. For differential equations on a differentiable manifold the variational equations for the solution are defined similarly to the case of $ \mathbf R ^ {n} $ 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 $ \mathbf R ^ {n} $, 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 $ \mathbf R ^ {n} $, 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 $ \dot{x} = F ( x) $ on a Riemannian manifold the variational equation along the trajectory $ x ( t) $, if $ F $ is not varied, may be written in the form

$$ \nabla _ {F ( x ( t) ) } \mathfrak x = \ \nabla _ {\mathfrak x} F ( x ( t) ) , $$

where $ \nabla _ {a} $ is the covariant derivative. The variational equation of a differentiable mapping $ f : V ^ {n} \rightarrow V ^ {n} $ (where $ V ^ {n} $ is a differentiable manifold) along the trajectory $ \{ f ^ { t } x \} _ {t \in \mathbf Z } $ (if the mapping $ f $ is not varied) is the equation

$$ \mathfrak x ( t + 1 ) = d f _ {f ^ { t } x } \mathfrak x ( t) ; $$

the value of the solution $ \mathfrak x ( \cdot ) $ of this equation at the point $ t $ lies in the tangent space $ T _ {f ^ { t } x } V ^ {n} $ of $ V ^ {n} $ at the point $ f ^ { t } x $, and the solution itself is the sequence

$$ \{ d ( f ^ { t } ) _ {x} \mathfrak x \} _ {t \in \mathbf Z } ,\ \ \mathfrak x \in T _ {x} V ^ {n} , $$

where $ d ( f ^ { m } ) _ {x} $ is the derivative of the $ m $-th power of $ f $ at $ x $.

Let $ V ^ {n} $ be a closed differentiable manifold. The set $ S $ of all diffeomorphisms $ f $ of class $ C ^ {1} $, mapping $ V ^ {n} $ onto $ V ^ {n} $, is equipped with the $ C ^ {1} $-topology. The following assertions hold (cf. [4]): 1) For every $ k \in \{ 1, \dots, n \} $ the Lyapunov characteristic exponent

$$ \tag{2 } \lambda _ {n-} k+ 1 ( f , x ) = \ \inf _ {\mathbf R ^ {k} \in G _ {k} ( T _ {x} V ^ {n} ) } \ \sup _ {\mathfrak x \in \mathbf R ^ {k} } \ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ \frac{1}{t} \mathop{\rm ln} | d f ^ { t } \mathfrak x | , $$

where $ G _ {k} ( T _ {x} V ^ {n} ) $ is the Grassmann manifold of $ k $-dimensional vector subspaces of the tangent space $ T _ {x} V ^ {n} $, is a function $ \lambda _ {n- k+ 1} ( \cdot ) : S \times V ^ {n} \rightarrow \mathbf R $ of the second Baire class (cf. Baire classes); 2) in the space $ S \times V ^ {n} $ there is an everywhere-dense set $ D $ of type $ G _ \delta $ with the properties: a) for every $ k \in \{ 1, \dots, n \} $ the function $ \lambda _ {k} ( \cdot ) : S \times V ^ {n} \rightarrow \mathbf R $ is upper semi-continuous at every point of $ D $; and b) for every $ ( f , x ) \in D $, $ \lambda \in \mathbf R $, the subspace

$$ l _ \lambda ( f , x ) = \ \left \{ { \mathfrak x \in T _ {x} V ^ {n} } : { \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ \frac{1}{t} \mathop{\rm ln} | d f ^ { t } \mathfrak x | \leq \lambda } \right \} $$

is exponentially separated from its algebraic complement $ l _ \lambda ^ {c} $ in the tangent space $ T _ {x} V ^ {n} $, i.e. there exist $ \alpha , \beta > 0 $ such that for all $ \mathfrak x \in l _ \lambda ^ {c} $, $ \mathfrak y \in l _ \lambda ( f , x ) $ and any integers $ t \geq s \geq 0 $ the inequality

$$ | d f ^ { t } \mathfrak x | \cdot | d f ^ { s } \mathfrak y | \geq \ \alpha | d f ^ { s } \mathfrak x | \cdot | f ^ { t } \mathfrak y | \ \mathop{\rm exp} ( \beta ( t - s ) ) $$

holds.

The set $ S $ of vector fields $ F $ of class $ C ^ {1} $ on a closed differentiable manifold $ V ^ {n} $ is equipped with the $ C ^ {1} $-topology. A vector field $ F \in S $ induces a dynamical system $ f ^ { t } $ (the action (of class $ C ^ {1} $) of the group $ \mathbf R $) on $ V ^ {n} $. For every $ k \in \{ 1, \dots, n \} $ the Lyapunov exponent $ \lambda _ {n - k + 1 } ( F , x ) $ is by definition equal to the right-hand side of (2).

The following assertions hold:

$ \alpha $) for each $ k \in \{ 1 \dots n \} $ the functions $ \lambda _ {k} ( \cdot ) : S \times V ^ {n} \rightarrow \mathbf R $ ly in the second Baire class [4];

$ \beta $) for every $ F \in S $, for every probability distribution that is invariant relative to the dynamical system $ f ^ { t } $ induced by the vector field $ F $ on $ V ^ {n} $ (the $ \sigma $-algebra of which contains all Borel subsets), almost-every point $ x $ is such that the variational equation $ \dot{x} = F ( x) $ along the trajectory $ \{ f ^ { t } x \} $ is a regular linear system of differential equations (cf. [5], [6]).

$ \gamma $) for every $ m \in \mathbf N $, let $ S ^ {( m)} $ denote the set of all vector fields of class $ C ^ {m} $ on $ V ^ {n} $, equipped with the $ C ^ {m} $-topology; let $ P $ be a probability distribution on $ V ^ {n} $, the $ \sigma $-algebra of which contains all Borel sets, and let $ S _ {P} ^ {( m)} $ denote the subspace of $ S ^ {( m)} $ consisting of all vector fields for which the distribution $ P $ is invariant relative to the dynamical systems induced by them; then (cf. [7]):

A) for every $ m \in \mathbf N $, $ k \in \{ 1, \dots, n \} $, the function

$$ \sum _ { i= 1} ^ { k } \int\limits _ {V ^ {n} } \lambda _ {i} ( \cdot , x ) d P ( x) : \ S _ {P} ^ {( m)} \rightarrow \mathbf R $$

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

B) for every $ m \in \mathbf N $ there is in $ S _ {P} ^ {( m)} $ an everywhere-dense set of type $ G _ \delta $ in which the function

$$ \int\limits _ {V ^ {n} } \lambda _ {k} ( \cdot , x ) d P ( x) : \ S _ {P} ^ {( m)} \rightarrow \mathbf R $$

is continuous (for every $ k \in \{ 1, \dots, n \} $), i.e. in $ S _ {P} ^ {( m)} $ continuity is typical for the phase averages of the Lyapunov exponents of the variational equations.

References

[1]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2]	E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[3]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[4]	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
[5]	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
[6]	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
[7]	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

Comments

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

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

References

[a1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a2]	E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1964)