Central exponents

of a linear system of ordinary differential equations

Quantities defined by the formulas

$$ \Omega ( A) = \ \lim\limits _ {T \rightarrow + \infty } \ \overline{\lim\limits}\; _ {k \rightarrow + \infty } \ { \frac{1}{kT} } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X (( i + 1) T, iT) \| $$

(the upper central exponent) and

$$ \omega ( A) = \ \lim\limits _ {T \rightarrow + \infty } \ \overline{\lim\limits}\; _ {k \rightarrow + \infty } \ \frac{- 1 }{kT } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X ( iT, ( i + 1) T) \| $$

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

$$ \lim\limits _ {T \rightarrow + \infty } \ \lim\limits _ {\overline{ {k \rightarrow + \infty }}\; } \ \frac{- 1 }{kT } \sum _ {i = 0 } ^ { {k } - 1 } \mathop{\rm ln} \| X ( iT, ( i + 1) T) \| . $$

Here $ X ( \theta , \tau ) $ is the Cauchy operator of the system

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

where $ A ( \cdot ) $ is a mapping

$$ \mathbf R ^ {+} \rightarrow \ \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $$

that is summable on every interval. The central exponents $ \Omega ( A) $ and $ \omega ( A) $ may be $ \pm \infty $; the inequalities

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| \ d \tau \geq \Omega ( A) \geq \ \omega ( A) \geq $$

$$ \geq \ - \lim\limits _ {\overline{ {t \rightarrow + \infty }}\; } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| d \tau $$

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

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \| A ( \tau) \| \ d \tau < + \infty , $$

then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) $ \lambda _ {1} ( A) \dots \lambda _ {n} ( A) $ and with the singular exponents $ \Omega ^ {0} ( A), \omega ^ {0} ( A) $ by the inequalities

$$ \Omega ^ {0} ( A) \geq \ \Omega ( A) \geq \ \lambda _ {1} ( A) \geq \dots \geq \ \lambda _ {n} ( A) \geq \ \omega ( A) \geq \ \omega ^ {0} ( A). $$

For a system (1) with constant coefficients $ ( A ( t) \equiv A) $ the central exponents $ \Omega ( A) $ and $ \omega ( A) $ are equal, respectively, to the maximum and minimum of the real parts of the eigen values of $ A $. For a system (1) with periodic coefficients ( $ A ( t + \theta ) = A ( t) $ for all $ t \in \mathbf R $ and some $ \theta > 0 $, $ \theta $ being the smallest period) the central exponents $ \Omega ( A) $ and $ \omega ( A) $ are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the multipliers divided by the period $ \theta $.

If $ A ( \cdot ) $ 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:

$$ \Omega ( A) = \ \Omega ^ {0} ( A),\ \ \omega ( A) = \ \omega ^ {0} ( A) $$

(Bylov's theorem).

For every fixed system (1) the condition $ \Omega ( A) < 0 $ is sufficient for the existence of a $ \delta > 0 $ such that for every system

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

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

$$ | g ( x, t) | < \delta | x | , $$

the solution $ x = 0 $ is asymptotically stable (Vinograd's theorem). The condition $ \Omega ( A) < 0 $ 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 $ \Omega ( A) $( respectively, $ \omega ( A) $) on the space $ M _ {n} $ of the system (1) with bounded continuous coefficients (so that $ A ( \cdot ) $ is continuous and $ \sup _ {t \in \mathbf R ^ {+} } \| A ( t) \| < + \infty $), endowed with the metric

$$ d ( A, B) = \ \sup _ {t \in \mathbf R ^ {+} } \ \| A ( t) - B ( t) \| , $$

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

$$ \tag{2 } \dot{x} = B _ {i} ( t) x,\ \ i = 1, 2, $$

arbitrarily close to it (in $ M _ {n} $) such that

$$ \lambda _ {1} ( B _ {1} ) = \ \Omega ( A),\ \ \lambda _ {n} ( B _ {2} ) = \ \omega ( A), $$

where $ \lambda _ {1} ( B _ {i} ) $ and $ \lambda _ {n} ( B _ {i} ) $, $ i = 1, 2 $, are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).

If $ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ is a uniformly-continuous mapping and if $ \sup _ {t \in \mathbf R } \| A ( t) \| < + \infty $, then for almost-every mapping $ \widetilde{A} $( in the sense of every normalized invariant measure of shift dynamical systems (cf. Shift dynamical system), $ S = \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $, concentrated on the closure of the trajectory of the point $ A $; the mappings $ \widetilde{A} $ and $ A $ are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system $ \dot{x} = \widetilde{A} ( t) x $ is equal to the largest (smallest) characteristic Lyapunov exponent of this system:

$$ \Omega ( \widetilde{A} ) = \ \lambda _ {1} ( \widetilde{A} ),\ \ \omega ( \widetilde{A} ) = \ \lambda _ {n} ( \widetilde{A} ). $$

Suppose that a dynamical system on a smooth closed manifold $ V ^ {n} $ is given by a smooth vector field. Then for almost-every point $ x \in V ^ {n} $( in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of $ x $ 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)