Periodic Orbits with Least Period Three on the Circle

Periodic Orbits with Least Period Three on the Circle

Three-periodic orbits on hyperbolic plane

Consider what is called the classical (Birkhoff) billiard problem: a particle moves in the planar convex domain D along the straight lines interacting with the boundary ∂D according to the Fermat’s law“ angle of reflection is equal to the angle of incidence”. In mathematical physics it is important to understand how many periodic orbits can be present in a planar billiard. Periodic orbits are t...

Chaining Periodic Three-Body Orbits

This paper studies how to construct orbit transfers, chains, and complex periodic orbits in the planar circular restricted three-body problem using the invariant manifolds of unstable three-body orbits. It is shown that the Poincaré map is a useful tool to identify and construct transfer trajectories from one orbit to another, i.e., homoclinic and heteroclinic orbit connections, with applicatio...

Spatially Periodic Orbits in Coupled Sine Circle Maps

We study spatially periodic orbits for a coupled map lattice of sine circle maps with nearest neighbour coupling and periodic boundary conditions. The stability analysis for an arbitrary spatial period k is carried out in terms of the independent variables of the problem and the stability matrix is reduced to a neat block diagonal form. For a lattice of size kN , we show that the largest eigenv...

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2π/3 or 4π/3, or J has a periodic point of least period n for each n ∈ N∗.