Orbit stability

orbital stability

A property of a trajectory $ \xi $( of a solution $ x( t) $) of an autonomous system of ordinary differential equations

$$ \tag{* } \dot{x} = f( x),\ x \in \mathbf R ^ {n} , $$

consisting of the following: For every $ \epsilon > 0 $ there is a $ \delta > 0 $ such that every positive half-trajectory beginning in the $ \delta $- neighbourhood of the trajectory $ \xi $ is contained in the $ \epsilon $- neighbourhood of the trajectory $ \xi $. Here, a trajectory is the set of values of a solution $ x( t) $, $ t \in \mathbf R $, of the system (*), while a positive half-trajectory is the set of values of a solution $ x( t) $ when $ t \geq 0 $. If the solution $ x( t) $ is stable according to Lyapunov (cf. Lyapunov stability), then its trajectory is orbital stable.

The trajectory $ \xi $ is called asymptotically orbital stable if it is orbital stable and if, furthermore, there is a $ \delta _ {0} > 0 $ such that the trajectory of every solution $ x( t) $ of the system (*) starting in the $ \delta _ {0} $- neighbourhood of the trajectory $ \xi $( i.e. $ d( x( 0), \xi ) < \delta _ {0} $) moves, when $ t \rightarrow + \infty $, towards the trajectory $ \xi $, i.e.

$$ \lim\limits _ {t \rightarrow + \infty } d( x( t), \xi ) = 0, $$

where

$$ d( x, \xi ) = \inf _ {y \in \xi } d( x, y) $$

is the distance from the point $ x $ to the set $ \xi $( and $ d( x, y) $ is the distance between the points $ x $ and $ y $).

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 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 $ x _ {0} ( t) $ be a bounded solution of the system (*); moreover, let

$$ \inf _ {t \geq 0 } | \dot{x} _ {0} ( t) | > 0, $$

and let the system of variational equations along $ x _ {0} ( t) $ be regular (see Regular linear system), while all its Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent), except one, are negative; then the trajectories of the solution $ x _ {0} ( t) $ are asymptotically orbital stable.

References

[1]	A.A. Andronov, "Collected works" , Moscow (1956) (In Russian)
[2]	A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)
[3a]	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
[3b]	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

Comments

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

References

[a1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a2]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227