Singular exponents

of a linear system of ordinary differential equations

The quantities defined by:

(the upper singular exponent) and

(the lower singular exponent), where is the Cauchy operator (i.e. the fundamental solution or principal solution) of the system

(1)

where is a mapping that is summable on every interval.

The singular exponents can be equal to ; if for a certain ,

(1prm)

then the singular exponents are numbers.

For a system (1) with constant coefficients , the singular exponents and are equal to, respectively, the maximum and minimum of the real parts of the eigenvalues of the operator . For a system (1) with periodic coefficients ( for all for a certain ), the singular exponents and are equal to, respectively, the maximum and minimum of the logarithms of the absolute values of the multipliers, divided by the period . The singular exponents are sometimes also called general exponents (see [4]).

The following definitions are equivalent to those mentioned above if (1prm) holds for a certain : The singular exponent is equal to the greatest lower bound of the set of those numbers for each of which there is a number such that for any solution of the system (1) the inequality

is fulfilled; the singular exponent is equal to the least upper bound of the set of those numbers for each of which a number exists such that for every solution of the system (1) the inequality

is fulfilled.

For the singular exponents and for the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent), for each the inequalities

hold. For linear systems with constant or periodic coefficients,

but there exist systems for which the corresponding inequalities are strict (see Uniform stability).

The singular exponent (respectively, ), as a function on the space of all systems of the form (1) with bounded continuous coefficients (the mapping is continuous and ) provided with the metric

is semi-continuous from above (respectively, from below) but is not continuous everywhere.

If the mapping is uniformly continuous and

then the shift dynamical system has invariant normalized measures and concentrated on the closure of the trajectory of the point such that, for almost all (in the sense of the measure ), the upper singular exponent of the system

(2)

is equal to its largest (leading) Lyapunov characteristic exponent,

and for almost all (in the sense of the measure ), the lower singular exponent of the system (2) is equal to its smallest Lyapunov characteristic exponent,

For almost-periodic mappings (see Linear system of differential equations with almost-periodic coefficients) the measures and are identical and coincide with the unique normalized invariant measure concentrated on the restriction of the shift dynamical system to the closure of the trajectory of the point , which in this case exists.

Let a dynamical system on a smooth, closed -dimensional manifold be defined by a smooth vector field. Then there exist normalized invariant measures and for this system such that for almost every point (in the sense of the measure ) the upper singular exponent and the leading Lyapunov characteristic exponent of the system of variational equations (equations in variations, linearized equations) along the trajectory of the point coincide, and for almost every point (in the sense of the measure ) the lower singular exponent and the smallest Lyapunov characteristic exponent of the system of variational equations along the trajectory of the point coincide. The definitions of singular exponents, Lyapunov characteristic exponents, etc., retain their meaning for systems of variational equations of smooth dynamical systems defined on arbitrary smooth manifolds. The system of variational equations of such a dynamical system along the trajectory of a point can be written in the form (1) by using, for example, that basis in the tangent space to at every point of the trajectory of which is obtained by a parallel transfer along this trajectory (in the sense of the Riemannian connection induced by the smooth Riemannian metric) of some basis of the tangent space of at .

References

[1]	P. Bohl, "Ueber Differentialgleichungen" J. Reine Angew. Math. , 144 (1913) pp. 284–318
[2]	K. Persidskii, "First approximation kinetic stability" Mat. Sb. , 40 : 3 (1933) pp. 284–293 (In Russian)
[3]	B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)
[4]	Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[5]	N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1974) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146