Siyuan Xing and Albert C. J. Luo, Period-1 Motions to Twin Spiral Homoclinic Orbits in the Rössler System, J. Comput. Nonlinear Dynam. Aug 2023, 18(8): 081008. https://doi.org/10.1115/1.4062201

​The Rössler system was proposed as one of the simplest three-dimensional nonlinear systems in order to exhibit chaotic behaviors in 1976. Since then, extensive qualitative and quantitative research on such a system has been conducted to study the mathematical structures of periodic orbits in the vicinity of homoclinic orbits. Many advances in this field have been achieved through the linearized, approximate theory and numerical studies, but it is still far away to completely solve the problem. In this paper, the periodic motions of the Rössler system are predicted through an implicit mapping method, and the corresponding stability of periodic motions is determined by eigenvalue analysis. The semi-analytical bifurcation tree from periodic motions to two spiral homoclinic orbits is obtained. The unstable periodic motions with fast-slow movements are accurately predicted. Homoclinic orbits are identified through infinitely large eigenvalues. The approximate homoclinic orbits with different periods are presented for a better understanding of the spiral homoclinic orbits spiraling out from the saddle-focus equilibriums.  

Ships navigating heavy seas are susceptible to various types of wave excitation, which may lead to dynamic instabilities and, in the worst case, capsizing. Compared to the pitch and yaw motions, the roll motion of a ship has the least amount of damping. We study a harmonically excited, single-degree-of-freedom time-delay system with cubic and quintic nonlinearities. This system describes the direct resonance of a ship with an actively controlled anti-roll tank (ART) that is subjected to beam waves, or waves that are approximately perpendicular to the ship’s heading. A proportional–derivative (PD) controller with a constant time delay is assumed to operate the pump in the active ART system. A key result is obtained by deriving the stability boundary of the system, in the parametric space of the control gain and the delay, from the characteristic equation of the linearized system. As shown here, a smaller controller delay is not always better since, in some cases, reducing the delay will reduce the maximum controller gain for which the system is stable. We conduct further analyses using the spectral Tau method, the method of multiple scales, the method of harmonic balance, continuation techniques, and direct numerical simulation.
​