Piecewise linear models for the quasiperiodic transition to chaos

Invariant polygons in systems with grazing-sliding

The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.

Mode locking and Arnold tongues in integrate-and-fire neural oscillators

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 qDelta, where 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.