The relation between recurrent behavior and the absence of a stable periodic orbit is discussed for unimodal maps. Almost all periodic maps are non-recurrent and almost all non-periodic are recurrent. Still one feels that the critical orbit cannot accumulate to quickly on the critical point, if it wants to avoid an asymptotic periodic behavior. We obtain exponential bounds of this type in terms...

Maslov indices in periodic-orbit theory are investigated using phase space path integral. Based on the observation that the Maslov index is the multi-valued function of the monodromy matrix, we introduce a generalized monodromy matrix in the universal covering space of the symplectic group and show that this index is uniquely determined in this space. The stability of the orbit is shown to dete...

The problem of robust stabilization of unstable periodic solutions of nonlinear dynamic systems subject to parametric uncertainties is treated. The challenge is to achieve robust stability of orbital motion when both the period and the location of the periodic orbit are uncertain. The technical approach used here is to extend the method of Generalized Sampled-data Hold Function control of linea...

A planetary body moving on an eccentric orbit around the primary is subject to a periodic perturbing potential, affecting its internal mass distribution. In a previous paper (Rappaport et al., 1997, Icarus 126, 313), we have calculated the periodic modulation of the gravity coefficients of degree 2, for a body on a synchronous orbit. Here, the previous analysis is extended by considering also n...

In our previous paper, it was confirmed that the unstable periodic orbit embedded in the chaotic attractor in magneto-elastic beam system can be stabilized by delayed feedback control experimentally. It seems an advantage that the control method does not require any exact model of the system. However, the application of the control raises the problem that we cannot predict the stabilized unstab...

A non-standard bifurcation, similar to a transcritical one, in a model of a bioreactor has been detected. This happens in a periodically-forced system with restrictions on the state space. The bioreactor is periodically fed with substrate. In the mathematical model, a periodic orbit approaches (without hitting) the restriction surface as a bifurcation parameter is varied. The way the orbit appr...