Characterization of the domain chaos convection state by the largest Lyapunov exponent

Statistics of phase space localization measures and quantum chaos in the kicked top model

Quantum chaos plays a significant role in understanding several important questions of recent theoretical and experimental studies. Here, by focusing on the localization properties of eigenstates in phase space (by means of Husimi functions), we explore the characterizations of quantum chaos using the statistics of the localization measures, that is the inverse participation ratio and the Wehrl entropy. We consider the paradigmatic kicked top model, which shows a transition to chaos with increasing the kicking strength. We demonstrate that the distributions of the localization measures exhibit a drastic change as the system undergoes the crossover from integrability to chaos. We also show how to identify the signatures of quantum chaos from the central moments of the distributions of localization measures. Moreover, we find that the localization measures in the fully chaotic regime apparently universally exhibit the beta distribution, in agreement with previous studies in the billiard systems and the Dicke model. Our results contribute to a further understanding of quantum chaos and shed light on the usefulness of the statistics of phase space localization measures in diagnosing the presence of quantum chaos, as well as the localization properties of eigenstates in quantum chaotic systems.

Complex network analysis of the gravity effect on premixed flames propagating in a Hele-Shaw cell

We study the effect of gravity on spatiotemporal flame front dynamics in a Hele-Shaw cell from the viewpoint of complex networks. The randomness in flame front dynamics significantly increases with the gravitational level when the normalized Rayleigh number R_{a} is negative. This is clearly identified by two network entropies: the flame front network entropy and the transition network entropy. The irregular formation of large-scale wrinkles driven by the Rayleigh-Taylor instability plays an important role in the formation of high-dimensional deterministic chaos at R_{a}<0, resulting in the increase in the randomness of flame front dynamics.

Boolean logic by convective obstacle flows

We present a potential new mode of natural computing in which simple, heat-driven fluid flows perform Boolean logic operations. The system comprises a two-dimensional single-phase fluid that is heated from below and cooled from above, with two obstacles placed on the horizontal mid-plane. The obstacles remove all vertical momentum that flows into them. The horizontal momentum extraction of the obstacles is controlled in a binary fashion, and constitutes the 2-bit input. The output of the system is a thresholded measure of the energy extracted by the obstacles. Due to the existence of multiple attractors in the phase space of this system, the input–output relationships are equivalent to those of the OR, XOR or NAND gates, depending on the threshold and obstacle separation. The ability to reproduce these logical operations suggests that convective flows might have the potential to perform more general computations, despite the fact that they do not involve electronics, chemistry or multiple fluid phases.

Correlations between the leading Lyapunov vector and pattern defects for chaotic Rayleigh-Bénard convection

We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh-Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic's ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.

Convective flow in the presence of a small obstacle: Symmetry breaking, attractors, hysteresis, and information

This work explores the stability and hysteresis effects that occur when a small sink of momentum is introduced into a heat-driven, two-dimensional convective flow. As per standard fluid mechanical intuition, the system minimizes work generation and dissipation when one component of momentum is extracted. However, when the sink absorbs all incoming momentum, the system configures itself such that one of the convection plumes aligns directly with the sink. This state is the most hydrodynamically stable, but it maximizes, rather than minimizes extracted mechanical work. Furthermore, in the case of only vertical momentum extraction, there are two attractors, with different stabilities. Numerical experiments involving slow variations of the horizontal momentum extraction show a clear history dependence. This hysteresis preserves information about the system's past states, and hence represents a primitive memory. The momentum sink can also be used to manipulate the horizontal position of the flow field, with potential applications in microfluidics and laminar convection systems. This simple system exhibits the phenomena of autocatalysis (during the initial growth of the convection plumes), negative feedback (the attractors are either fully or quasistable), memory, and elementary computation.

Chaotic Rayleigh-Bénard convection with finite sidewalls

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Spatiotemporal dynamics of the covariant Lyapunov vectors of chaotic convection

We explore the spatiotemporal dynamics of the spectrum of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection. We use the inverse participation ratio to quantify the amount of spatial localization of the covariant Lyapunov vectors. The covariant Lyapunov vectors are found to be spatially localized at times when the instantaneous covariant Lyapunov exponents are large. The spatial localization of the Lyapunov vectors often occurs near defect structures in the fluid flow field. There is an overall trend of decreasing spatial localization of the Lyapunov vectors with increasing index of the vector. The spatial localization of the covariant Lyapunov vectors with positive Lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed. We find that a weighted covariant Lyapunov vector is useful for the visualization and interpretation of the significant connections between the Lyapunov vectors and the flow field patterns.

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection

We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

Dynamics of Scroll Wave in a Three-Dimensional System with Changing Gradient

The dynamics of a scroll wave in an excitable medium with gradient excitability is studied in detail. Three parameter regimes can be distinguished by the degree of gradient. For a small gradient, the system reaches a simple rotating synchronization. In this regime, the rigid rotating velocity of spiral waves is maximal in the layers with the highest filament twist. As the excitability gradient increases, the scroll wave evolutes into a meandering synchronous state. This transition is accompanied by a variation in twisting rate. Filament twisting may prevent the breakup of spiral waves in the bottom layers with a low excitability with which a spiral breaks in a 2D medium. When the gradient is large enough, the twisted filament breaks up, which results in a semi-turbulent state where the lower part is turbulent while the upper part contains a scroll wave with a low twisting filament.

Short-term prediction of dynamical behavior of flame front instability induced by radiative heat loss

We apply nonlinear forecasting to the time series of the flame front instability induced by radiative heat loss to test for the short-term predictability and long-term unpredictability characteristic of deterministic chaos in flame front instability. Our results indicate that the flame front instability represents high-dimensional chaos generated via the period-doubling cascade process reported in our previous study [H. Gotoda, K. Michigami, K. Ikeda, and T. Miyano, Combust Theory Modell. 14, 479 (2010)], while its short-term behavior is predictable using a local nonlinear predictor based on the Sugihara-May method [H. Gotoda, H. Nikimoto, T. Miyano, and S. Tachibana, Chaos 20, 013124 (2011); G. Sugihara and R. M. May, Nature 344, 734 (1990)] as well as a generalized radial basis function network as a global nonlinear predictor. The feasibility of a new approach based on short-term prediction is also discussed in this work from the practical viewpoint of combustion systems.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Diagnosis of spatiotemporal chaos in wave envelopes of a nematic electroconvection pattern

In this paper we report and analyze complex spatiotemporal dynamics recorded in electroconvection in the nematic liquid crystal I52, driven by an ac voltage slightly above the onset value. The instability mechanism creating the pattern is an oscillatory (Hopf) instability, giving rise to two pairs of counterpropagating rolls traveling in oblique directions relative to the unperturbed director axis. If a system of nonlinear partial differential equations shows the same set of unstable modes, the pattern above the onset is represented in a weakly nonlinear analysis as a superposition of the traveling rolls in terms of wave envelopes varying slowly in space and time. Motivated by this procedure, we extract slowly varying envelopes from the space-time data of the pattern, using a four-wave demodulation based on Fourier analysis. In order to characterize the spatiotemporal dynamics, we apply a variety of diagnostic methods to the envelopes, including the calculation of mean intensities and correlation lengths, global and local Karhunen-Loève decompositions in Fourier space and physical space, the location of holes, the identification of coherent vertical structures, and estimates of Lyapunov exponents. The results of this analysis provide strong evidence that our pattern exhibits extensive spatiotemporal chaos. One of its main characteristics is the presence of coherent structures of low and high intensities extended in the vertical (parallel to the director) direction.

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.

Extensive chaos in Rayleigh-Bénard convection

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.

Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection

Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.