Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable equilibrium point at the origin based on the adaptive ...

This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semi-conjugacy between an attractor for the Swift-Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of com...

We consider a class of differential equations, ẍ + γẋ + g(x) = f(ωt), with ω ∈ R, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x) = x, p ∈ N, we show that, when the dissipation coefficient is large enough, there is only one such...

We investigate the bifurcation mechanism for the loss of transverse stability of the chaotic attractor in an invariant subspace in an asymmetric dynamical system. It is found that a direct transition to global riddling occurs through a transcritical contact bifurcation between a periodic saddle embedded in the chaotic attractor on the invariant subspace and a repeller on its basin boundary. Thi...

In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3d Navier-Stokes equations have a global attractor provided that all weak solutions are continuous from (0,∞) into L2. In this paper we adapt his framework to treat stochastic equations: w...

We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity...

We consider the motion of an incompressible fluid confined to a shallow basin with varying bottom topography. A two-dimensional shallow water model has been derived from a three-dimensional anisotropic eddy viscosity model and has been shown to be globally well posed in [15]. The dynamical system associated with the shallow water model is studied. We show that this system possesses a global att...

The current threat of global warming and the public demand for confident projections of climate change pose the ultimate challenge to science: predicting the future behaviour of a system of such overwhelming complexity as the Earth's climate. This Theme Issue addresses two practical problems that make even prediction of the statistical properties of the climate, when treated as the attractor of...

CONTENTS Introduction 151 § 1. Maximal attractors of semigroups generated by evolution equations 156 § 2. Examples of parabolic equations and systems having a maximal attractor 158 § 3. The Hausdorff dimension of invariant sets 164 § 4. Estimate of the change in volume under the action of shift operators generated by linear evolution equations 167 § 5. An upper bound for the Hausdorff dimension...