Polynomial Systems

In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system.
The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami
structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to
period-2n motions (n = 1, 2, . . . ,∞). On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue
approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the
period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding
differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation
diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram
to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few
layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the
complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the
Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues.
Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly