Period-1 Motions to Twin Spiral Homoclinic Orbits in the Rössler System

In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.