32 research outputs found
Normal form transforms separate slow and fast modes in stochastic dynamical systems
Modelling stochastic systems has many important applications. Normal form
coordinate transforms are a powerful way to untangle interesting long term
macroscale dynamics from detailed microscale dynamics. We explore such
coordinate transforms of stochastic differential systems when the dynamics has
both slow modes and quickly decaying modes. The thrust is to derive normal
forms useful for macroscopic modelling of complex stochastic microscopic
systems. Thus we not only must reduce the dimensionality of the dynamics, but
also endeavour to separate all slow processes from all fast time processes,
both deterministic and stochastic. Quadratic stochastic effects in the fast
modes contribute to the drift of the important slow modes. The results will
help us accurately model, interpret and simulate multiscale stochastic systems
Poder germinativo de sementes de capim-annoni-2 (Eragrostis plana Ness) recuperadas em fezes de bovinos
A normal form of thin fluid film equations solves the transient paradox
Imagine two constant thickness, thin films of fluid colliding together: the transient flow forms a hump where they collide; thereafter they slowly relax. But, apparently reliable lubrication models expressed only in the thickness of the fluid forecast that precisely nothing happens. How can we resolve this paradox? Dynamical systems theory constructs a normal form of the Navier–Stokes equations for the flow of a thin layer of fluid upon a solid substrate. These normal form equations illuminate the fluid dynamics by decoupling the interesting long-term ‘lubrication’ flow from the rapid viscous decay of transient shear modes. The normal form clearly shows the slow manifold of the lubrication model and demonstrates that
the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness, but instead is modified by any initial
lateral shear flow. With these initial conditions, the lubrication model makes better forecasts. This dynamical systems approach could similarly illuminate other models of complicated dynamics