Solvable model for spatiotemporal chaos

We show that the dynamical behavior of a coupled map lattice where the individual maps are Bernoulli shift maps can be solved analytically for integer couplings. We calculate the invariant density of the system and show that it displays a nontrivial spatial behavior. We also introduce and calculate a generalized spatiotemporal correlation function

Dynamics of Coupled Maps with a Conservation Law

A particularly simple model belonging to a wide class of coupled maps which obey a local conservation law is studied. The phase structure of the system and the types of the phase transitions are determined. It is argued that the structure of the phase diagram is robust with respect to mild violations of the conservation law. Critical exponents possibly determining a new universality class are calculated for a set of independent order parameters. Numerical evidence is produced suggesting that the singularity in the density of Lyapunov exponents at $\lambda=0$ is a reflection of the singularity in the density of Fourier modes (a ``Van Hove'' singularity) and disappears if the conservation law is broken. Applicability of the Lyapunov dimension to the description of spatiotemporal chaos in a system with a conservation law is discussed.Comment: To be published in CHAOS #7 (31 page, 16 figures

Pinning control of spatiotemporal chaos

Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented

Controlling Physical Systems with Symmetries

Symmetry properties of the evolution equation and the state to be controlled are shown to determine the basic features of the linear control of unstable orbits. In particular, the selection of control parameters and their minimal number are determined by the irreducible representations of the symmetry group of the linearization about the orbit to be controlled. We use the general results to demonstrate the effect of symmetry on the control of two sample physical systems: a coupled map lattice and a particle in a symmetric potential.Comment: 6 page

The temperature dependence of the isothermal bulk modulus at 1 bar pressure

It is well established that the product of the volume coefficient of thermal expansion and the bulk modulus is nearly constant at temperatures higher than the Debye temperature. Using this approximation allows predicting the values of the bulk modulus. The derived analytical solution for the temperature dependence of the isothermal bulk modulus has been applied to ten substances. The good correlations to the experiments indicate that the expression may be useful for substances for which bulk modulus data are lacking

Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical description of a complicated physical phenomenon can be learned from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality, and symmetry, a weak formulation of differential equations, and sparse regression. To illustrate this, we extract a system of governing equations describing flows of incompressible Newtonian fluids -- the Navier-Stokes equation, the continuity equation, and the boundary conditions -- from numerical data describing a highly turbulent channel flow in three dimensions. These relations have the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects as well as insight into the quality of the data, serving as a useful diagnostic tool. The approach described here is remarkably robust, yielding accurate results for very high noise levels, and should thus be well-suited to experimental data