Control and synchronization of continuous space-extended systems is realized by means of a finite number of local tiny perturbations. The perturbations are selected by an adaptive technique, and they are able to restore each of the independent unstable patterns present within a space time chaotic regime, as well as to synchronize two space time chaotic states. The effectiveness of the method and the robustness against external noise is demonstrated for the amplitude and phase turbulent regimes of the one-dimensional complex Ginzburg-Landau equation. The problem of the minimum number of local perturbations necessary to achieve control is discussed as compared with the number of independent spatial correlation lengths.

We study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a magnetic field with a constant longitudinal and a time-dependent transverse component using the Landau-Lifshitz-Gilbert equation. We characterize the dynamical behavior of the system through calculation of the Lyapunov exponents, Poincar\'e sections, bifurcation diagrams, and Fourier power spectra. In particular we explore the positivity of the largest Lyapunov exponent as a function of the magnitude and frequency of the applied magnetic field and its direction with respect to the main anisotropy axis of the magnetic particle. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated structure of intricately intermingled chaotic and regular phases.

A derivation is given of the amplitude equations governing pattern formation in surface tension gradient-driven Bénard–Marangoni convection. The amplitude equations are obtained from the continuity, the Navier–Stokes and the Fourier equations in the Boussinesq approximation neglecting surface deformation and buoyancy. The system is a shallow liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. The amplitude equations of the convective modes are equations of the Ginzburg–Landau type with resonant advective non-variational terms. Generally, and in agreement with experiment, above threshold solutions of the equations correspond to an hexagonal convective structure in which the fluid rises in the centre of the cells. We also analytically study the dynamics of pattern formation leading not only to hexagons but also to squares or rolls depending on the various dimensionless parameters like Prandtl number, and the Marangoni and Biot numbers at the boundaries. We show that a transition from an hexagonal structure to a square pattern is possible. We also determine conditions for alternating, oscillatory transition between hexagons and rolls. Moreover, we also show that as the system of these amplitude equations is non-variational the asymptotic behaviour (t→∞) may not correspond to a steady convective pattern. Finally, we have determined the Eckhaus band for hexagonal patterns and we show that the non-variational terms in the amplitude equations enlarge this band of allowable modes. The analytical results have been checked by numerical integration of the amplitude equations in a square container. Like in experiments, numerics shows the emergence of different hexagons, squares and rolls according to values given to the parameters of the system.