Rounding border effects at the escape point of open integrable billiards are analyzed via the escape times statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but same escape times. As the rounding effects increase, backgammon stripes start to overlap and the escape times statistics obeys the power law decay and anomalous diffusion is expected. Tiny rounded borders (around 0.1% from the whole billiard size) are shown to be sufficient to generate the sticky motion, while borders larger than 10% are enough to produce escape times with chaotic decay.

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.

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