ABSTRACT
The effect of stochastic perturbations on nearly homoclinic pulse trains is considered for three model systems: a Duffing oscillator, the Lorenz-like Shimizu-Morioka model, and a co-dimension-three normal form. Using the Duffing model as an example, it is demonstrated that the main effect of noise does not originate from the neighbourhood of the fixed point, as is commonly assumed, but due to the perturbation of the trajectory outside that region. Singular perturbation theory is used to quantify this noise effect and is applied to construct maps of pulse spacing for the Shimizu-Morioka and normal form models. The dynamics of these stochastic maps is then explored to examine how noise influences the sequence of bifurcations that take place adjacent to homoclinic connections in Lorenz-like and Shilnikov-type flows.
ABSTRACT
Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.
