Orbital stability asks that a disturbance remain close to an invariant orbit or symmetry family, rather than to one fixed representative. In a complex Landau amplitude equation with , , , the circle attracts radial disturbances while every phase on it remains stationary. The family is attracting as a set, but its individual points do not have individual asymptotic stability because their phase is neutral.
A periodic orbit has orbital asymptotic stability if nearby trajectories remain close and their distance to the orbit tends to zero. A persistent phase shift along the orbit is allowed.
Forsuppose satisfies . Then is a circular periodic orbit of period . Linearizing its radial equation givesso its transverse Floquet multiplier is . It attracts radially if and repels if ; a vanishing derivative requires nonlinear terms.
For example, and give , radial exponent , and an attracting limit cycle. The angular direction is neutral because shifting the initial angle changes the phase of the same orbit. Attraction is therefore orbital stability with convergence to the orbit, rather than asymptotic convergence to one moving point. The argument uses only differentiability of and does not assume that it is a polynomial.
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Orbital stability refers to the stability of the orbits of celestial bodies under the influence of gravitational forces. In astrodynamics and celestial mechanics, it is an important concept that describes whether an orbiting body will remain in a stable orbit or if it is likely to change its trajectory significantly over time, possibly leading to escape from a gravitational influence, collision with another body, or spiraling into a star or planet.