Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem

Simultaneous Binary Collisions for Collinear Four-Body Problem

In this paper, we use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions for the collinear four- body problem in both the decoupled case and the coupled case. All the solutions are found and more importantly, we describe the relationship between the decoupled solutions and the coupled solutions.

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.

On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem

The singularity at a simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than $ C^{8/3} $ differentiable with respect to initial conditions. Through a blow-up of the singularity, this loss of differentiability is investigated and a new proof of the $ C^{8/3} $ regularity is provided. In the process, it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold…

3D Filippov系统在Cusp-Fold奇点附近的分岔分析

Bifurcation Analysis of 3D Filippov Systems around Cusp-Fold Singularities

本文研究了3D Filippov系统在cusp-fold奇点附近的局部动力学，分析了不可见cusp-fold奇点的穿越动力学与分岔特性，并完成了升压变换器系统的双参数分岔集。

Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L1,..,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper …