View source for Central exponents

''of a linear system of ordinary differential equations''

Quantities defined by the formulas

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(the upper central exponent) and

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

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211704.png" /> is the [[Cauchy operator|Cauchy operator]] of the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211706.png" /> is a mapping

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that is summable on every interval. The central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c0211709.png" /> may be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117010.png" />; the inequalities

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

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then its central exponents are finite numbers. The central exponents are connected with the Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117014.png" /> and with the [[Singular exponents|singular exponents]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117015.png" /> by the inequalities

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For a system (1) with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117017.png" /> the central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117019.png" /> are equal, respectively, to the maximum and minimum of the real parts of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117020.png" />. For a system (1) with periodic coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117022.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117024.png" /> being the smallest period) the central exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117026.png" /> are equal, respectively, to the maximum and minimum of the logarithms of the moduli of the [[Multipliers|multipliers]] divided by the period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117027.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117028.png" /> is an almost-periodic mapping (see [[Linear system of differential equations with almost-periodic coefficients|Linear system of differential equations with almost-periodic coefficients]]), then the central exponents of (1) coincide with the singular exponents:

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(Bylov's theorem).

For every fixed system (1) the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117030.png" /> is sufficient for the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117031.png" /> such that for every system

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

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the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117034.png" /> is asymptotically stable (Vinograd's theorem). The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117036.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117037.png" />) on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117038.png" /> of the system (1) with bounded continuous coefficients (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117039.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117040.png" />), endowed with the metric

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is upper (respectively lower) semi-continuous, but neither of these functions is continuous everywhere. For every system (1), in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117042.png" /> one can find another system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

arbitrarily close to it (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117044.png" />) such that

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117048.png" />, are the largest (highest) and the smallest (least) characteristic Lyapunov exponents of the system (2).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117049.png" /> is a uniformly-continuous mapping and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117050.png" />, then for almost-every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117051.png" /> (in the sense of every normalized invariant measure of shift dynamical systems (cf. [[Shift dynamical system|Shift dynamical system]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117052.png" />, concentrated on the closure of the trajectory of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117053.png" />; the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117055.png" /> are regarded as points of the space of the shift dynamical system) the upper (lower) central exponent of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117056.png" /> is equal to the largest (smallest) characteristic Lyapunov exponent of this system:

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Suppose that a dynamical system on a smooth closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117058.png" /> is given by a smooth vector field. Then for almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117059.png" /> (in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variations along the trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021170/c02117060.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  V.M. Millionshchikov,   "Typical properties of conditional exponential stability VI"  ''Differential equations'' , '''20''' :  6  pp. 707–715  ''Differentsial'nye Uravneniya'' , '''20''' :  6  (1984)</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>