Теория устойчивости и теория управления движение

Vasilieva, E. V.

BSU Digital Library

36 «Шестые Богдановские чтения по обыкновенным дифференциальным уравнениям»
x˙2 = −x1 + y2, y˙2 = − 3x2‖x‖3 − x2 − y1 + u2,
x˙3 = y3, y˙3 = − 3x3‖x‖3 − x3 + u3,
where x = 〈x1;x2;x3〉 is coordinate vector of the celestial body, y = 〈y1;y2;y3〉 is an impulse
vector, and u = 〈u1;u2;u3〉 is the acceleration vector [1]. The center of mass of the Earth
coincides with the origin of the coordinate system, while the Ox1 axis is directed along the
axis connecting the centers of mass of the Earth and the Sun. ‖ · ‖ is the Euclidean norm of a
vector. The libration points L1 and L2 in the rotating coordinate system are stationary and have
coordinates x∗ = 〈1;0;0〉, y∗ = 〈0;1;0〉 and x∗∗ = 〈−1;0;0〉, y∗∗ = 〈0;−1;0〉, respectively.
It is known that the libration points L1 and L2 are unstable. Hence, the question of retention
of the celestial bodies motion in the neighborhood of these points is relevant. In some cases,
instability may be a positive factor that helps reduce energy costs while maneuvering [2]. To
intercept maneuvers in near-Earth space in [3] the developed scheme maneuvering is described.
This scheme involves the selection and calculation of the orbit of expectations, and the construction
of the active part of the trajectory.
For example, to solve the comet and asteroid hazard problem explores the idea of impact
on a potentially dangerous object to alter its trajectory and thus prevent the threat of collision.
To achieve this goal in [4] proposed to use gravity assist in near-Earth space. It uses properties
of unstable trajectory collision. In this study, the use of another type of instability is proposed,
namely, the instability property of the collinear libration points [3].
The study presents the results of research and numerical construction of the set of trajectories
to intercept potentially dangerous objects in the near-Earth space.
Aknowledgement. The author acknowledge Saint-Petersburg State University for a research
grant 9.37.345.2015
Refrences
1. Shmyrov A., Shmyrov V. Controllable orbital motion in a neighborhood of collinear libration point
// Applied Mathematical Sciences. 2014. Vol. 8 (9–12). P. 487–492.
2. Shymanchuk D.V., Shmyrov A. S. Construction of trajectory of the return in the neighborhood of the
collinear libration point of the Sun-Earth system // Bulletin of St.-Petersburg University. Ser. 10. Release
2. 2013. P. 76–85. (in Russian)
3. Shmyrov A., Shymanchuk D. Maneuvering in near-Earth space with the use of the collinear libration
points // 2015 International Conference on Mechanics — Seventh Polyakhov’s Reading, art. no. 7106777.
2015.
4. Eismont N.A. et al. On the possibility of the guidance of small asteroids to dangerous celestial bodies
using the gravity-assist maneuver // Solar System Research. 2013. Vol. 47, no. 4. P. 325–333.
ON THE STABILITY OF PERIODIC POINTS
OF THE THREE-DIMENSIONAL DIFFEOMORPHISMS
E.V. Vasilieva
Saint Petersburg State University, St. Petersburg, Russia
ekvas1962@mail.ru
Smooth Cr -diffeomorphisms of the three-dimensional space with a hyperbolic fixed point at
the origin are considered, where r > 1 (case r =∞ is included). The existence of non-transversal
homoclinic point is assumed; i.e. intersection of the stable and unstable manifolds contains a point,
referred to as a homoclinic point, and this point is a point of tangency of these manifolds.
Теория устойчивости и теория управления движением 37
It follows from results of Sh. Newhouse, B. F. Ivanov, L. P. Shil’nikov, S. V. Gonchenko and
other authors (papers [1, 2, 3]) that, when the stable and unstable manifolds are tangent in a
certain way, a neighborhood of the homoclinic point may contain stable periodic points, but at
least one of the characteristic exponents for such points tends to zero with increasing the period.
Let f be a self-diffeomorphism of three-dimensional space of class Cr (1 6 r 6 ∞) with
fixed hyperbolic fixed point at the origin. Obviously, there are two cases:
1) real matrix Df(0) have real eigenvalues only;
2) real matrix Df(0) have complex eigenvalues.
In papers [4, 5] the first case is considered. It is shown that under certain conditions imposed
mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the
homoclinic point contains an infinite set of stable periodic points whose characteristic exponents
are negative and bounded away from zero. An example of such a diffeomorphism was considered
in the monograph of V. A. Pliss [6].
The main purpose of the talk is to show that this result can be extended to the second case.
Aknowledgement. The work is partially supported by RFBR.
References
1. Newhouse Sh. Diffeomorphisms with finitely many sinks // Topology. 1973. Vol. 12. P. 9–18.
2. Ivanov B. F. Stability of Trajectories that do not Leave a Neighborhood of a Homoclinic Curve
// Differ. Uravn. 1979. Vol. 15, no. 8. P. 1411–1419.
3. Gonchenko S.V., Turaev D.V., Shil’nikov L. P. Dynamical Phenomena in Multidimensinal Systems
with a Structurally Unstable Poincare Curve // Dokl. RAN. 1993. Vol. 330, no. 2. P. 144–147.
4. Vasilieva E.V. Multidimensional Diffeomorphisms with Stable Periodic Points // Dokl. Acad. Nauk.
2011. Vol. 441, no. 3. P. 299–301.
5. Vasilieva E.V. Diffeomorphisms of Multidimensional Space with Infinite Set of Stable Perodic Points
// Vestnik St. Petersburg University. Mathematics. 2012. Vol. 45, no. 3. P. 115–124.
6. Pliss V.A. Integral’nye mnozhestva perodicheskikh system differentsial’nykh uravnenii. (Integral Sets
of Periodic Systems of Differential Equations). Moscow: Nauka, 1977. 304 pp.


On the stability of periodic points of the three-dimensional diffeomorphisms

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On the stability of periodic points of the three-dimensional diffeomorphisms

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