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We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for non-orthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Non-orthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the non-orthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell., Comment: 22 pages, 15 figures; added additional inscribed solid that was missed

Synchronisation and stability under periodic oscillatory driving are well-understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counter-intuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronisation where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilisation phenomenon is numerically observed. Our findings help support the case that in general, deterministic non-autonomous perturbation is a very good candidate for stabilising complex dynamics., to be published in Phys. Rev. E

An analysis of mode-locked solutions that may arise in periodically forced integrate-and-fire (IF) neural oscillators is introduced based upon a firing map formulation of the dynamics. A $q:p$ mode-locked solution is identified with a spike train in which p firing events occur in a period $q\ensuremath{\Delta},$ where $\ensuremath{\Delta}$ is the forcing period. A linear stability analysis of the map of firing times around such solutions allows the determination of the Arnold tongue structure for regions in parameter space where stable solutions exist. The analysis is verified against direct numerical simulations for both a sinusoidally forced IF system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current. This approach is extended to the case of two synaptically coupled IF oscillators, showing that mode-locked states can exist for some self-consistently determined common period of repetitive firing. Numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating. The influence of the synaptic current upon the Arnold tongue structure is explored in the regime of weak coupling.

We present experimental data on the limiting behavior of an interreality system comprising a virtual horizontally driven pendulum coupled to its real-world counterpart, where the interaction time scale is much shorter than the time scale of the dynamical system. We present experimental evidence that, if the physical parameters of the simplified virtual system match those of the real system within a certain tolerance, there is a transition from an uncorrelated dual reality state to a mixed reality state of the system in which the motion of the two pendula is highly correlated. The region in parameter space for stable solutions has an Arnold tongue structure for both the experimental data and a numerical simulation. As virtual systems better approximate real ones, even weak coupling in other interreality systems may produce sudden changes to mixed reality states.

4 pages.-- PACS numbers: 05.45.Xt, 42.55.Px, 42.65.Sf, 05.45.Vx.-- Final full-text version of the paper available at: http://dx.doi.org/10.1103/PhysRevE.73.047201., We theoretically investigate the nonlinear dynamics and synchronization properties of two self-oscillating semiconductor lasers subject to optoelectronic mutual-coupling and feedback loops. The model is based on modified single-mode rate equations for the intracavity photon and carrier densities. The optoelectronic interactions take into account the finite propagation time of the optical and electrical signals. We provide a detailed stability and bifurcation analysis of the fixed points upon variation of the coupling and feedback strengths as well as the associated delay times. This allows us to predict a new scenario for the "death by delay" phenomenon where the delay time in the interaction between the physical units is not essential. We consider both oscillatory and chaotic behavior of the solitary lasers by changing the feedback conditions. We find different types of synchronization including identical, anti-phase and localized synchronization even for perfectly symmetric systems. When the lasers operate as two chaotic oscillators, we find that the isochronal (zero-lagged) state is stable within a large region in the parameter space. The width of the frequency lockbands (Arnold tongues) of two slightly mismatched lasers shows an interesting dependence with the coupling delay time., R.V., J.M., and C.R.M. acknowledge funding from the MCyT (Spain) and FEDER (Project No. FIS2004-00953). R.V. and C.R.M. were also funded by the Secretaría de Estado de Educación y Universidades, MEC, Spain. The work by J.M. is supported by the Spanish CSIC through the program I3P-2003. The work of S.T. and J.M.L. was supported by the U.S. Army Research Office under Contract No. DAAG55-98-1-0269.