# 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