Order and the ubiquitous occurrence of chaos

Dynamical systems on infinitely sheeted Riemann surfaces

“…The value of E determines uniquely the four period vectors, which are defined generically through the contour integrals (18). For the special case of the pentagonal curve the four periods vectors are simply given by the formula (21). As explained in Section 3 the allowed periods V 2 and V 4 define a quasi-elliptic period lattice on the plane while the forbidden periods V 1 and V 3 define the windows.…”

“…Combining analytical and numerical results for a simple ODE, Bountis and Fokas [21] identified a chaotic behaviour with the property that the singularities or branch points of their solutions are dense. In their work, however, it is not too clear whether the criterion is that such points should be dense on the Riemann surface itself or that the projection of the branch points is dense on the complex plane.…”

Relationship Between Singularities in Classical Mechanics for Complex Initial Conditions and for Complex Time

“…Despite their semiclassical importance, however, much of the current interest in complex-time singularities is due to their relevance for the issue of the integrability of classical systems [22][23][24][25][26][27]. This relevance emerges from the Painlevé-Kowaleskaya conjecture that systems having the Painlevé property are integrable.…”

“…These occur when the time t of evolution is made complex [20][21][22][23][24][25][26][27]. It is then found that the coordinate and momentum of a typical system become singular at a particular complex value of the time t * , even when initial conditions are chosen to be real.…”

“…and their distribution [22][23][24][25][26][27]. In such studies, the initial conditions for the trajectories have usually been taken (at least implicitly) to be fixed and real.…”