Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.Comment: 7 pages, 5 figure

Hyperbolic Chaos of Turing Patterns

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.Comment: 4 pages, 4 figure

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis.Comment: 24 pages, 13 figure

Fast numerical test of hyperbolic chaos

The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for high-dimensional system this amount is normally much less then the full phase space dimension, this method provide the fast and memory saving way for numerical hyperbolicity test of such systems.Comment: 4 pages and 4 figure

Numerical Investigation of Water Droplets Shape Influence on Mathematical Modeling Results of Its Evaporation in Motion through a High-Temperature Gas

The numerical investigation of influence of a single water droplet shape on the mathematical modeling results of its evaporation in motion through high-temperature gases (combustion products of a typical condensed substance) has been executed. Values of evaporation time, motion velocity, and distance passed by various droplet shapes (cylinder, sphere, hemisphere, cone, and ellipsoid) in a high-temperature gases medium were analyzed. Conditions have been defined when a droplet surface configuration affects the integrated characteristics of its evaporation, besides temperature and combustion products concentration in a droplet trace, insignificantly. Experimental investigations for the verification of theoretical results have been carried out with using of optical diagnostic methods for two-phase gas-vapor-liquid flows

Water Droplet With Carbon Particles Moving Through High-Temperature Gases

An experimental investigation was carried out on the influence of solid inclusions (nonmetallic particles with sizes from a few tens to hundreds of micrometers) on water droplet evaporation during motion through high-temperature gases (more than 1000 K). Optical methods for diagnostics of two-phase (gas and vapor-liquid) flows (particle image velocimetry (PIV) and interferometric particle imaging (IPI)) were used. It was established that introducing foreign solid particles into the water droplets intensifies evaporation rate in high-temperature gas severalfold. Dependence of liquid evaporation on sizes and concentration of solid inclusion were obtained