We study circle exchange transformations on three subintervals, that reverse orientation on at least one of the subintervals (a flip). We prove that if such a transformation has exactly one flip, then it has a periodic orbit and thus no dense orbit. For the cases of two and three flips, we construct uniquely ergodic minimal examples by finding a periodic point of an adapted Rauzy–Veech renormal...

In this paper, we first present numerical methods that allow us to compute accurately periodic orbits in high dimensional mappings and demonstrate the effectiveness of our methods by computing orbits of various stability types. We then use a terminology for the different stability types, which is perfectly suited for systems with many degrees of freedom, since it clearly reflects the configurat...

The average lifetime (τ(H)) it takes for a randomly started trajectory to land in a small region (H) on a chaotic attractor is studied. τ(H) is an important issue for controlling chaos. We point out that if the region H is visited by a short periodic orbit, the lifetime τ(H) strongly deviates from the inverse of the naturally invariant measure contained within that region (μN (H) −1). We introd...

In this thesis we show how the concept of Pyragas control can be adapted to equivariant dynamical systems near Hopf bifurcation. We introduce a control term which uses the spatio-temporal symmetry of the unstable periodic orbit. We give explicit necessary and sufficient conditions for the stabilization of the periodic orbit near equivariant Hopf bifurcation. For the remaining eigenvalues, we ca...

(a) (b) (c) (d) Abstract We identify all the periodic orbits of the integrable swinging Atwood's machine by calculating the rotation number of each orbit on its invariant tori in phase space, and also providing explicit formulas for the initial conditions needed to generate each orbit.

Time-delay autosynchronization (TDAS) can be used to stabilize unstable periodic orbits in dynamical systems. The technique involves continuous feedback of signals delayed by the orbit's period so that the feedback signal vanishes on the target orbit and hence the later is a solution of the original dynamical system. Furthermore, this control method only requires the knowledge of the period of ...

We identify all the periodic orbits of the integrable swinging Atwood's machine by calculating the rotation number of each orbit on its invariant tori in phase space, and also providing explicit formulas for the initial conditions needed to generate each orbit.

We study the planar problem of two satellites attracted by a center of force. Assuming that the center of mass of the two-satellite system is on a circular orbit around the center of force and using Levi-Civita regularization we prove the existence of an almost periodic orbit with an infinite number of collision between the satellites.

We apply periodic orbit theory to a two-dimensional nonintegrable billiard system whose boundary is varied smoothly from a circular to an equilateral triangular shape. Although the classical dynamics becomes chaotic with increasing triangular deformation, it exhibits an astonishingly pronounced shell effect on its way through the shape transition. A semiclassical analysis reveals that this shel...