Central exponents

of a linear system of ordinary differential equations

Quantities defined by the formulas

(the upper central exponent) and

(the lower central exponent); sometimes the lower central exponent is defined as

Here is the Cauchy operator of the system

(1)

where is a mapping

that is summable on every interval. The central exponents and may be ; the inequalities

hold, which imply that if the system (1) satisfies the condition

then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) and with the singular exponents by the inequalities

For a system (1) with constant coefficients the central exponents and are equal, respectively, to the maximum and minimum of the real parts of the eigen values of . For a system (1) with periodic coefficients ( for all and some , being the smallest period) the central exponents and are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the multipliers divided by the period .

If is an almost-periodic mapping (see Linear system of differential equations with almost-periodic coefficients), then the central exponents of (1) coincide with the singular exponents:

(Bylov's theorem).

For every fixed system (1) the condition is sufficient for the existence of a such that for every system

satisfying the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem and the condition

the solution is asymptotically stable (Vinograd's theorem). The condition in Vinograd's theorem is not only sufficient but also necessary. (The necessity remains valid when asymptotic stability is replaced by Lyapunov stability.)

The function (respectively, ) on the space of the system (1) with bounded continuous coefficients (so that is continuous and ), endowed with the metric

is upper (respectively lower) semi-continuous, but neither of these functions is continuous everywhere. For every system (1), in one can find another system

(2)

arbitrarily close to it (in ) such that

where and , , are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).

If is a uniformly-continuous mapping and if , then for almost-every mapping (in the sense of every normalized invariant measure of shift dynamical systems (cf. Shift dynamical system), , concentrated on the closure of the trajectory of the point ; the mappings and are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system is equal to the largest (smallest) characteristic Lyapunov exponent of this system:

Suppose that a dynamical system on a smooth closed manifold is given by a smooth vector field. Then for almost-every point (in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of coincides with its largest (smallest) characteristic Lyapunov exponent. Generic properties of the central exponent (from the point of view of the Baire categories) have been studied, see .

References

[1]	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)
[2]	N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1974) pp. 46–96 Itogi Nauk. Mat. Anal , 12 pp. 71–146
[3a]	V.M. Millionshchikov, "Typical properties of conditional exponential stability II" Differential equations , 19 : 9 pp. 1126–1132 Differentsial'nye Uravneniya , 19 : 9 (1983) pp. 1503–1510
[3b]	V.M. Millionshchikov, "Typical properties of conditional exponential stability VI" Differential equations , 20 : 6 pp. 707–715 Differentsial'nye Uravneniya , 20 : 6 (1984)
[3c]	V.M. Millionshchikov, "Typical properties of conditional exponential stability VII" Differential equations , 20 : 8 pp. 1005–1013 Differentsial'nye Uravneniya , 20 : 8 (1984) pp. 1366–1376

References

[a1]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)