Dynamics and Control of Nonlinear Arrays of Coupled Oscillators

We will discuss the dynamics and control of the translational motion of a coupled array
of nonlinear oscillators. This topic is very generic and addresses many applications in
variety of fields such as laser arrays, friction (tribology) at the atomic scale, soliton
propagation in lattices, arrays of Josephson junctions, and others. In our talk we will
address the topic of friction at the atomic scale. We will argue that friction force/friction
coefficient may be related to the degree of phase synchronization in the propagating array
and consequently, increase in phase synchronization may significantly reduce friction. In
particular, we will discuss the effect of substrate (lattice) irregularities/impurities on array
synchronization and will demonstrate examples where disordered array may be better
phase synchronized thus resulting in significantly lower friction.
We will further discuss experimental results, mechanisms and techniques to control
friction at the atomic scale such as surface oscillations and terminal attractor design. Our
experiments involve an AFM tip sliding on a vibrating surface mounted on a piezoelement.
Experimental results and numerical simulations of the model suggest that the
friction coefficient undergoes a transition to very small values when the amplitude of the
vibration reaches a critical threshold. Finally, we will discuss terminal attractor algorithm
to control frictional dynamics of an array of particles towards pre-assigned values of the
frictional force. This control technique is robust and significantly reduces the transient
time to reach the prescribed behavior.