Recent Developments in Dynamical Systems: Three Perspectives

Abstract

The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, nonautonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors.

Trajectory attractor for a non-autonomous magnetohydrodynamic equation of non-Newtonian fluids

“…We can find a comparison of the two first and the three methods in [13] and [32], respectively. One can also find a recent review on the three methods in [4]. In the present work we use the third approach and we show the existence of the trajectory attractor of weak solutions to (7) and give some partial results related to its structure.…”

Role of logistic and Ricker’s maps in appearance of chaos in autonomous quadratic dynamical systems

“…Let g 1G ≥ 0, g 22G , h 1G , h 12G , and h 22G ≥ 0 be the greatest values on the sphere S of the functions g 1 …”

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

“…Miller (1965) and Sell (1967aSell ( , b, 1971 developed the notion of skew product flows and their associated cocycle property. These ideas are further described from a pedagogical point of view in the recent review article by Balibrea et al (2010). With descriptions of "time evolution" appropriate to nonautonomous differential equations at hand, the building blocks of a geometrical theory can be developed.…”