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Register a new userOn the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem
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Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which are emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hills equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd $(2p,p)$-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even $(2p,p)$-periodic solutions are elliptic and linearized stable. These are the first analytical results on the stability of nonconstant periodic orbits of the elliptic Sitnikov problem.
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It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2in [0,9]$ and the eccentricity $ein [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(bb,e)$ range $[0,9]times [0,1)$ via the $om$-index theory of symplectic paths for $om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $om$-index decreasing property of the solutions in $bb$ for fixed $ein [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $om$-degeneracy curves for $om ot=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.
In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable for us to consider the existence of generalized solutions and non-collision solutions with Mixed-symmetries or with Double-Eight constraints, arising from Figure-Eight, for N-body problem. All of the orbits we found numerically in Section ref{se7} have not been obtained by other authors as far as we know. To prove the existence of these new periodic solutions, the variational approach and critical point theory are applied to the classical N-body equations. And along the line used in this paper, one can construct other symmetric constraints on N-body problems and prove the existence of periodic solutions for them.
Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism.
We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces.
We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.
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