q-breathers in Fermi-Pasta-Ulam chains: existence, localization, and stability.

The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit, each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes, and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

Cluster synchronization and spatio-temporal dynamics in networks of oscillatory and excitable Luo-Rudy cells.

We study collective phenomena in nonhomogeneous cardiac cell culture models, including one- and two-dimensional lattices of oscillatory cells and mixtures of oscillatory and excitable cells. Individual cell dynamics is described by a modified Luo-Rudy model with depolarizing current. We focus on the transition from incoherent behavior to global synchronization via cluster synchronization regimes as coupling strength is increased. These regimes are characterized qualitatively by space-time plots and quantitatively by profiles of local frequencies and distributions of cluster sizes in dependence upon coupling strength. We describe spatio-temporal patterns arising during this transition, including pacemakers, spiral waves, and complicated irregular activity.

q-Breathers in finite two- and three-dimensional nonlinear acoustic lattices.

In their celebrated experiment, Fermi, Pasta, and Ulam (FPU) [Los Alamos Report No. LA-1940, 1955] observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined q-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs are enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.

q-Breathers and the Fermi-Pasta-Ulam problem.

The Fermi-Pasta-Ulam (FPU) paradox consists of the non-equipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.