Island myriads in periodic potentials.

A phenomenon of emergence of stability islands in phase space is reported for two periodic potentials with tiling symmetries, one square and the other hexagonal, inspired by bidimensional Hamiltonian models of optical lattices. The structures found, here termed as island myriads, resemble web-tori with notable fractality and arise at energy levels reaching that of unstable equilibria. In general, the myriad is an arrangement of concentric island chains with properties relying on the translational and rotational symmetries of the potential functions. In the square system, orbits within the myriad come in isochronous pairs and can have different periodic closure, either returning to their initial position or jumping to identical sites in neighbor cells of the lattice, therefore impacting transport properties. As seen when compared to a more generic case, i.e., the rectangular lattice, the breaking of square symmetry disrupts the myriad even for small deviations from its equilateral configuration. For the hexagonal case, the myriad was found but in attenuated form, mostly due to extra instabilities in the potential surface that prevent the stabilization of orbits forming the chains.

Direct observation of a "devil's staircase" in wave-particle interaction.

We report the experimental observation of a "devil's staircase" in a time-dependent system considered as a paradigm for the transition to large-scale chaos in the universality class of Hamiltonian systems. A test electron beam is used to observe its non-self-consistent interaction with externally excited wave(s) in a traveling wave tube (TWT). A trochoidal energy analyzer records the beam energy distribution at the output of the interaction line. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the slow wave structure (a 4 m long helix) of the TWT. The resonant velocity domain associated to a single wave is observed, as well as the transition to large-scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a "devil's staircase" behavior for increasing excitation amplitude, due to the nonlinear forcing by the second wave on the pendulum-like motion of a charged particle in one electrostatic wave.

Electron climbing a "devil's staircase" in wave-particle interaction.

Numerous nonlinear driven systems display spectacular responses to forcing, including chaos and complex phase-locking plateaus characterized by "devil's staircase," Arnold tongues, and Farey trees. In the universality class of Hamiltonian systems, a paradigm is the motion of a charged particle in two waves, which inspired a renormalization group method for its description. Here we report the observation of the underlying "devil's staircase" by recording the beam velocity distribution function at the outlet of a traveling wave tube versus the amplitude of two externally induced waves.