We notice that the given system has an invariant plane and can be reduced to the family of the planar systems, which we construct positive invariant sets and provide the equilibrium points for.
System (2) has the invariant plane [alpha]y+z = [alpha][y.sub.1] + [z.sub.1] for any [y.sub.1] [member of] R, [z.sub.1] [greater than or equal to] 0, [alpha][y.sub.1] + [z.sub.1] < [alpha](A+1)T.
Rodney Gomes of the Brazilian Observatorio Nacional said his team's calculations give an "elegant explanation for the current tilt between the invariant plane
of the inner giant planets and the solar equator, and add new constraints to the orbital elements of Planet Nine".
More precisely, if c > 0 then when t [right arrow][bar.[alpha]]the orbit of system (1) having maximal interval of definition ([bar.[alpha]], [bar.[omega]]), under the assumption of Corollary 2, tends to the invariant plane cz + d = 0, and studying the dynamics on this invariant plane we can determine the [alpha]-limit sets.
(b) Eventually [e.sup.g] can be an exponential factor, coming from the multiplicity of the infinite invariant plane.
Due to a variety of perturbing effects, the so-called six orbital elements describing the orbit increase or decrease with time, and the satellite orbit could not be considered as an ellipse strictly in an invariant plane. As a result, the state vectors of sensor and target will be different from those in a strict elliptical orbit .
As a result, the six orbital elements are affected by perturbation force, and the satellite trajectory is not a simple ellipse in an invariant plane. The [J.sub.2], [J.sub.3] and [J.sub.4] perturbation terms and their effect on the Doppler parameters are analyzed in this paper.
The Face [T.sub.3] is on the invariant plane Z = 0.
The Face [T.sub.2] is on the invariant plane C = 0.