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''orbital stability''

A property of a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700301.png" /> (of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700302.png" />) of an [[Autonomous system|autonomous system]] of ordinary differential equations

<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/o/o070/o070030/o0700303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

consisting of the following: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700304.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700305.png" /> such that every positive half-trajectory beginning in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700306.png" />-neighbourhood of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700307.png" /> is contained in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700308.png" />-neighbourhood of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o0700309.png" />. Here, a trajectory is the set of values of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003011.png" />, of the system (*), while a positive half-trajectory is the set of values of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003012.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003013.png" />. If the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003014.png" /> is stable according to Lyapunov (cf. [[Lyapunov stability|Lyapunov stability]]), then its trajectory is orbital stable.

The trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003015.png" /> is called asymptotically orbital stable if it is orbital stable and if, furthermore, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003016.png" /> such that the trajectory of every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003017.png" /> of the system (*) starting in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003018.png" />-neighbourhood of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003019.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003020.png" />) moves, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003021.png" />, towards the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003022.png" />, i.e.

<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/o/o070/o070030/o07003023.png" /></td> </tr></table>

where

<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/o/o070/o070030/o07003024.png" /></td> </tr></table>

is the distance from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003025.png" /> to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003026.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003027.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003029.png" />).

The use of the concept of asymptotic orbital stability is based on the following facts. A periodic solution of (*) is never asymptotically stable. But if the moduli of all [[Multipliers|multipliers]] of the periodic solution of this system, except one, are less than 1, then the trajectory of this periodic solution is asymptotically orbital stable (the Andronov–Witt theorem). There is also the more general Demidovich theorem (see ): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003030.png" /> be a bounded solution of the system (*); moreover, let

<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/o/o070/o070030/o07003031.png" /></td> </tr></table>

and let the system of variational equations along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003032.png" /> be regular (see [[Regular linear system|Regular linear system]]), while all its Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]), except one, are negative; then the trajectories of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070030/o07003033.png" /> are asymptotically orbital stable.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,   "Collected works" , Moscow  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Andronov,   A.A. Vitt,   A.E. Khaikin,   "Theory of oscillators" , Dover, reprint  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  B.P. Demidovich,   "Orbital stability of bounded solutions of an autonomous system I"  ''Differential Eq.'' , '''4'''  (1968)  pp. 295–301  ''Differensial'nye Uravneniya'' , '''4''' :  4  (1968)  pp. 575–588</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  B.P. Demidovich,   "Orbital stability of bounded solutions of an autonomous system II"  ''Differential Eq.'' , '''4'''  (1968)  pp. 703–709  ''Differensial'nye Uravneniya'' , '''4''' :  8  (1968)  pp. 1359–1373</TD></TR></table>



====Comments====
One also considers orbital stability from the inside (or outside) of a periodic orbit.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>