We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier–Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier–Stokes system with damping our estimate...

We give bounds for the global attractor of the delay differential equation ẋ(t) = −μx(t) + f(x(t−τ)), where f is unimodal and has negative Schwarzian derivative. If f and μ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behavio...

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one o...

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG eq...

This paper gives sufficient conditions for persistence of deterministic population processes. It has been conjectured that any population process whose network graph is weakly reversible (has strongly connected components) is persistent. We prove this conjecture for a class of systems. An important application of this work pertains to chemical reaction systems that are “complex-balancing.” For ...

In this paper, we consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. For obtaining these surfaces, we are guided by the integrals of motion that exist for particular values of the parameters of the ...