arXiv subjects

Mitsuru Shibayama

Publications and source records attributed to Mitsuru Shibayama.

10 recordsLinked to original sources

arXiv2026

A Minimax Approach to Relative Periodic Orbits in Symmetric Three-Degree-of-Freedom Hamiltonian Systems

We study three-degree-of-freedom Hamiltonian systems that are invariant under rotations about the $z$-axis and under reflection across the $xy$-plane. Fixing the angular momentum, such systems reduce to Hamiltonian systems with two degrees of freedom. We focus on the range of energy values for which the corresponding Hill regions are compact. First, under suitable assumptions on the topology of these compact Hill regions, we prove the existence of periodic solutions on each prescribed energy surface of the reduced system by means of a variational minimax method. These periodic solutions are obtained as saddle points of the Maupertuis functional. The resulting solutions are either nontrivial spatial periodic solutions or trivial planar brake solutions in the reduced system. Next, by computing the Morse index, we provide a sufficient condition ensuring that the periodic solutions obtained are nontrivial. Finally, we apply our results to the isosceles three-body problem and to the spatial anisotropic Kepler problem. In both cases, we verify the sufficient condition for nontriviality and thereby establish the existence of nontrivial periodic solutions.

math.DS↗

arXiv2026

Existence of Really Perverse Central Configurations in the Spatial $N$-Body Problem

We construct explicit examples of really perverse central configurations in the spatial Newtonian $N$-body problem. A central configuration is called really perverse if it satisfies the central configuration equations for two distinct mass distributions having the same total mass. While such configurations were previously known only in the planar case for large $N$, we prove the existence of spatial really perverse central configurations for $N=27,\dots,55$.

math.DS↗

arXiv2026

Invariant Curves and the Variational Structure in Tubular Origami Dynamical Systems

We present a theoretical and numerical dynamical-systems analysis of tubular origami tessellations by identifying the inverse module number, $N^{-1}$, as a perturbation parameter within the framework of Kolmogorov--Arnold--Moser (KAM) theory. In the large-module limit ($N \to \infty$), we show that the conservative dynamics formally converges to an integrable map with a variational structure, whose generating function corresponds to the total discrete mean curvature. From the viewpoint of KAM theory, nonresonant invariant curves of the integrable limit are expected to persist for sufficiently large $N$. Consistent with this expectation, numerical computations with increasing $N$ show that large regions of the phase space are filled with structures that appear to be invariant curves. By adjusting mountain-valley fold assignments and fold lengths, the system can be transformed into a nontwist map that exhibits multiple zero frequencies. The frequency profile in the integrable limit and the persistence of invariant curves allow us to control the number and arrangement of stable folding regions appearing as coexisting elliptic islands. These islands provide a phase-space interpretation of distinct folding modes, separated by invariant curves that act as geometric barriers to continuous deformation and obstruct transitions without self-intersection. Finally, we analyze the expanding and contracting dynamics of the origami structure within the framework of conformally symplectic systems. By introducing a virtual auxiliary fold as a drift control mechanism, we numerically confirm the existence of stable quasi-periodic attractors.

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arXiv2025

Hamiltonian systems and monotone twist mappings for braids

In 1986, Moser showed that for a given area-preserving map, there exists a Hamiltonian system that realizes it on the Poincaré section. Using his technique, we show that for any braid, there exists a Hamiltonian system whose orbits realize the given braid. In particular, when the braid is pseudo-Anosov, so is the Poincaré map of the corresponding Hamiltonian.

math.DS↗

arXiv2025

Variational Construction of Homoclinic and Heteroclinic Orbits in the Planar Sitnikov Problem

The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and has been studied by symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). We study the limiting case of the Sitnikov problem as the eccentricity of the massive particles tends to 1. By variational method, we show the existence of infinitely many homoclinic and heteroclinic solutions in the planar Sitnikov problem. In a previous work, for certain periodic symbolic sequences, the second author showed the existence of periodic solutions realizing them. In this paper, we show the existence of homoclinic and heteroclinic solutions between some of these periodic orbits which realize certain non-periodic symbolic sequences.

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arXiv2025

A study of braids arising from simple choreographies of the planar Newtonian N-body problem

We study periodic solutions of the planar Newtonian $N$-body problem with equal masses. Each periodic solution traces out a braid with $N$ strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch factor greater than 1, which reflects the complexity of the corresponding periodic solution. For each $N \ge 3$, Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian $N$-body problem. We prove that braids arising from Yu's periodic solutions are of pseudo-Anosov types, except in the special case where all particles move along a circle. We also identify the simple choreographies whose braid types have the largest and smallest stretch factors, respectively.

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arXiv2022

Braids, metallic ratios and periodic solutions of the $2n$-body problem

Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar $2n$-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios. New numerical periodic solutions of the planar $2n$-body problem are also provided.

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arXiv2018

Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler's third law: a numerical test

We test numerically the recently proposed linear relationship between the scale-invariant period $T_{\rm s.i.} = T |E|^{3/2}$, and the topology of an orbit, on several hundred planar Newtonian periodic three-body orbits. Here $T$ is the period of an orbit, $E$ is its energy, so that $T_{\rm s.i.}$ is the scale-invariant (s.i.) period, or, equivalently, the period at unit energy $|E| = 1$. All of these orbits have vanishing angular momentum and pass through a linear, equidistant configuration at least once. Such orbits are classified in ten algebraically well-defined sequences. Orbits in each sequence follow an approximate linear dependence of $T_{\rm s.i.}$, albeit with slightly different slopes and intercepts. The orbit with the shortest period in its sequence is called the "progenitor": six distinct orbits are the progenitors of these ten sequences. We have studied linear stability of these orbits, with the result that 21 orbits are linearly stable, which includes all of the progenitors. This is consistent with the Birkhoff-Lewis theorem, which implies existence of infinitely many periodic orbits for each stable progenitor, and in this way explains the existence and ensures infinite extension of each sequence.

physics.class-ph↗

arXiv2013

Variational proof of the existence of the super-eight orbit in the four-body problem

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881--901) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver have numerically found a similar periodic solution called "super-eight" in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the technique established by Tanaka (1993 Ann. Inst. H. Poincar'e Anal. Non Lin'eaire 10, 215--238, 1994 Proc. Amer. Math. Soc. 122, 275--284).

math.DS↗

arXiv2013

Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be chosen from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.

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