Chaotic Wave Research Articles

Effect of horizontal temperature difference on Marangoni-thermocapillary convection in a shallow annular pool

The surface tension driven convection with the bidirectional temperature differences plays a very important role in many natural processes. However, most of the previous researches have focused only on the convection induced by a unidirectional temperature difference. In this paper, under the coexistence of bidirectional temperature differences, we conduct a series of numerical simulations to investigate the effect of horizontal temperature difference on the Marangoni-thermocapillary convection in a shallow annular pool. The critical values of bottom heat flux Qcri for transition from an axisymmetric steady flow to a three-dimensional unsteady flow at different values of Ma are determined. The result shows the horizontal temperature difference has a negative effect on the stability of Marangoni-thermocapillary convection. The simulation predicts two new state evolutions which do not appear in the convection with a unidirectional temperature difference. When Q is less than the Qcri value of 2.4×10-3, the Marangoni convection without horizontal temperature difference is steady and axisymmetric. When a small horizontal temperature difference is imposed, the convection called basic flow keeps steady and axisymmetric. When the value of Ma exceeds a certain threshold value Macri, the convection becomes a three-dimensional unsteady flow. After this unsteady flow happens, with the increase of Ma, the surface temperature fluctuation evolves from a punctate wave to a hydrothermal wave, and finally to a chaotic wave. Accordingly, the temperature oscillation with time is a periodically regular oscillation at first, then turns into a chaotic mess. When Q is larger than the corresponding Qcri value of 2.4×10-3, without a horizontal difference, the convection is unsteady and no basic flow exists in the variation process of Ma. With the increase of Ma, the surface temperature fluctuation evolves from a double hydrothermal wave to a single hydrothermal wave, and finally to a chaotic wave. The vertical heat transfer and horizontal temperature difference have different effects on the fluid, and their separate roles in driving fluid are determined. The bottom heat flux causes the surface fluid to flow in two opposite radial directions as the highest surface temperature is located in the middle region, while the horizontal temperature difference induces the surface fluid to flow in a single radial direction as the highest surface temperature appears at the hot wall. The combined action of these two forces generates different flows. The increase of horizontal temperature difference leads to the highest surface temperature, which originally appears in the middle region due to the bottom heat flux, and moves toward the hot wall. In this process, the horizontal temperature difference has a positive effect on the enhancement of flow near inner wall but it has a negative effect on the flow near outer wall.

Split-step approach to electromagnetic propagation through atmospheric turbulence using the modified von Karman spectrum and planar apertures

The impact of atmospheric phase turbulence on Gaussian beam propagation along propagation paths of varying lengths is examined using multiple random phase screens. The work is motivated by research involving generation and encryption of acousto-optic chaos, and the interest in examining propagation of such chaotic waves through atmospheric turbulence. A phase screen technique is used to simulate perturbations to the refractive index of the medium through the propagation path. A power spectral density based on the modified von Karman spectrum model for turbulence is used to describe the random phase behavior of the medium. In recent work, results for the numerical simulation of phase turbulence over a narrow region of space implemented by placing a planar aperture representing a (narrow) random phase screen were presented. Results are presented pertinent to extended phase screens (via multiple random-phase apertures) through which an incident Gaussian beam propagates incrementally via alternate phase transmission and diffraction along the propagation path. Additionally, for profiled electromagnetic waves (such as Gaussian), the scintillation index is evaluated for extended phase turbulence, and finally, fringe visibility due to the interference of double-Gaussian beams passing through extended turbulence is examined.

Rogue wave spectra of the Sasa–Satsuma equation

We analyze the rogue wave spectra of the Sasa–Satsuma equation and their appearance in the spectra of chaotic wave fields produced through modulation instability. Chaotic wave fields occasionally produce high peaks that result in a wide triangular spectrum, which could be used for rogue wave detection.

Collective motion of macroscopic spheres floating on capillary ripples: dynamic heterogeneity and dynamic criticality.

When a densely packed monolayer of macroscopic spheres floats on chaotic capillary Faraday waves, a coexistence of large scale convective motion and caging dynamics typical for glassy systems is observed. We subtract the convective mean flow using a coarse graining (homogenization) method and reveal subdiffusion for the caging time scales followed by a diffusive regime at later times. We apply the methods developed to study dynamic heterogeneity and show that the typical time and length scales of the fluctuations due to rearrangements of observed particle groups significantly increase when the system approaches its largest experimentally accessible packing concentration. To connect the system to the dynamic criticality literature, we fit power laws to our results. The resultant critical exponents are consistent with those found in densely packed suspensions of colloids.

Rogue waves of the Sasa-Satsuma equation in a chaotic wave field.

We study the properties of the chaotic wave fields generated in the frame of the Sasa-Satsuma equation (SSE). Modulation instability results in a chaotic pattern of small-scale filaments with a free parameter-the propagation constant k. The average velocity of the filaments is approximately given by the group velocity calculated from the dispersion relation for the plane-wave solution. Remarkably, our results reveal the reason for the skewed profile of the exact SSE rogue-wave solutions, which was one of their distinctive unexplained features. We have also calculated the probability density functions for various values of the propagation constant k, showing that probability of appearance of rogue waves depends on k.

Some aspects of image processing using foams

Abstract We have explored some concepts of chaotic dynamics and wave light transport in foams. Using some experiments, we have obtained the main features of light intensity distribution through foams. We are proposing a model for this phenomenon, based on the combination of two processes: a diffusive process and another one derived from chaotic dynamics. We have presented a short outline of the chaotic dynamics involving light scattering in foams. We also have studied the existence of caustics from scattering of light from foams, with typical patterns observed in the light diffraction in transparent films. The nonlinear geometry of the foam structure was explored in order to create optical elements, such as hyperbolic prisms and filters.

Synchronization and Waves in a Ring of Diffusively Coupled Memristor-Based Chua’s Circuits

In the present paper we report numerical observations of the spontaneous dynamics of N identical memristor-based Chua's circuits bidirectionally coupled in a ring geometry. Two different initial configurations are studied by varying N and the coupling strength: in the first configuration we consider only one circuit with non-zero initial conditions, in the second one all circuits have uniform random initial conditions. We observed both chaotic and non-chaotic synchronization. In the chaotic synchronized regime we identified emerging chaotic steady waves, characterized by an almost constant frequency and chaotic amplitude. Depending on the initial conditions, in the pseudo-sinusoidal oscillations regime both macroscopic quasi-periodic steady and traveling waves were observed.

Deciphering Dynamics of Epidemic Spread: The Case of Influenza Virus

In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns selected by the reaction–diffusion system under zero flux boundary conditions have been explored.Numerical simulations show that contact rate, β which is related to the reproduction number [Formula: see text], plays an important role in spatial pattern formation. It was found that diffusion has appreciable influence on spatial spread of epidemics. The wave of chaos appears to be a dominant mode of disease dispersal. This suggests a bidirectional spread for influenza epidemics. The epidemic propagates in the form of nonchaotic and chaotic waves as observed in H1N1 incidence data of positive tests in 2009 in the United States. We have conducted numerical simulations to confirm the analytic work and observed interesting behaviors. This suggests that influenza has a complex dynamics of spatial spread which evolves with time.

Ballistic and diffusive dynamics in a two-dimensional ideal gas of macroscopic chaotic Faraday waves.

We have constructed a macroscopic driven system of chaotic Faraday waves whose statistical mechanics, we find, are surprisingly simple, mimicking those of a thermal gas. We use real-time tracking of a single floating probe, energy equipartition, and the Stokes-Einstein relation to define and measure a pseudotemperature and diffusion constant and then self-consistently determine a coefficient of viscous friction for a test particle in this pseudothermal gas. Because of its simplicity, this system can serve as a model for direct experimental investigation of nonequilibrium statistical mechanics, much as the ideal gas epitomizes equilibrium statistical mechanics.

Temporal flooding of regular islands by chaotic wave packets

We investigate the time evolution of wave packets in systems with a mixed phase space where regular islands and chaotic motion coexist. For wave packets started in the chaotic sea on average the weight on a quantized torus of the regular island increases due to dynamical tunneling. This flooding weight initially increases linearly and saturates to a value which varies from torus to torus. We demonstrate for the asymptotic flooding weight universal scaling with an effective tunneling coupling for quantum maps and the mushroom billiard. This universality is reproduced by a suitable random matrix model.

Chaotic Wave Solutions in a Nonlocal Model for Media With Vibrating Inclusions

The present paper is devoted to the investigation of wave solutions of a mathematical model of a complex medium that takes into account the nonlocal interaction of structural elements of the main medium and the vibration dynamics of an additional continuum (inclusions). By the methods of qualitative and numerical analysis, it is shown that an autonomous dynamic system with four-dimensional phase space has periodic, multiperiodic, quasiperiodic, and chaotic solutions. We construct bifurcation diagrams of the development of vibrations depending on the parameters of nonlocality of the dynamic equation of state of the medium

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point \((N^*,P^*,Z^*)\) via numerical simulation. And the sequence cold spots \(\rightarrow \) stripe–spots mixtures \(\rightarrow \) stripes \(\rightarrow \) hot stripe–spots mixtures \(\rightarrow \) hot spots \(\rightarrow \) chaotic wave patterns controlled by parameters \(a_1\) or \(c_1\) in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.

Remarks on nodal volume statistics for regular and chaotic wave functions in various dimensions

We discuss the statistical properties of the volume of the nodal set of wave functions for two paradigmatic model systems which we consider in arbitrary dimension s≥2: the cuboid as a paradigm for a regular shape with separable wave functions and planar random waves as an established model for chaotic wave functions in irregular shapes. We give explicit results for the mean and variance of the nodal volume in the arbitrary dimension, and for their limiting distribution. For the mean nodal volume, we calculate the effect of the boundary of the cuboid where Dirichlet boundary conditions reduce the nodal volume compared with the bulk. Boundary effects for chaotic wave functions are calculated using random waves which satisfy a Dirichlet boundary condition on a hyperplane. We put forward several conjectures on what properties of cuboids generalize to general regular shapes with separable wave functions and what properties of random waves can be expected for general irregular shapes. These universal features clearly distinguish between the two cases.

Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion

Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point(u2*,v2*,w2*), and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameterc1, the sequence “holes→holes-stripe mixtures→stripes→spots-stripe mixtures→spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point(u1*,v1*,0), and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.

SHORELINE CHANGES IN THUA THIEN-HUE LAGOON SYSTEM

In southern Thua Thien Hue province, a long and narrow sandy coast (barrier islands) splits water inside lagoons from outside, Gulf of Tonkin. Shoreline is impacted directly by waves, so it often causes robust changes in northeast monsoon, which could influence local people and economy significantly. In this study, the results showed that Thuan An inlet moved toward northwest, the extreme (weather) phenomena made the inlet open/closure repeatedly, and so on. From the results, we can classify into two basic types of shoreline changes: one is movement of Thuan An inlet; the other is topographic change in barrier islands. Their lytho-hydrodynamic processes have opposite directions, contrasting magnitudes on power and time. The impact of two characteristics in northeast monsoon was analyzed particularly as reasons: one was chaotic waves in front of Gulf of Tonkin; the other was wave field from enhanced NE monsoon spells. Comparing the results of two cases near shore, angle of wave ray deflected oppositely, magnitudes of wave power and time were different. Mechanism of wave impact in the extreme case (opening/closure inlet), unstable dynamic status of barrier islands were studied as well.

Prenatal corticosteroids modify glutamatergic and GABAergic synapse genomic fabric: insights from a novel animal model of infantile spasms.

Prenatal exposure to corticosteroids has long-term postnatal somatic and neurodevelopmental consequences. Animal studies indicate that corticosteroid exposure-associated alterations in the nervous system include hypothalamic function. Infants with infantile spasms, a devastating epileptic syndrome of infancy with characteristic spastic seizures, chaotic irregular waves on interictal electroencephalogram (hypsarhythmia) and mental deterioration, have decreased concentrations of adrenocorticotrophic hormone (ACTH) and cortisol in cerebrospinal fluid, strongly suggesting hypothalamic dysfunction. We have exploited this feature to develop a model of human infantile spasms by using repeated prenatal exposure to betamethasone and a postnatal trigger of developmentally relevant spasms with NMDA. The spasms triggered in prenatally primed rats are more severe compared to prenatally saline-injected ones and respond to ACTH, a treatment of choice for infantile spasms in humans. Using autoradiography and immunohistochemistry, we have identified a link between the spasms in our model and the hypothalamus, especially the arcuate nucleus. Transcriptomic analysis of the arcuate nucleus after prenatal priming with betamethasone but before trigger of spasms indicates that prenatal betamethasone exposure down-regulates genes encoding several important proteins participating in glutamatergic and GABAergic transmission. Interestingly, there were significant sex-specific alterations after prenatal betamethasone in synapse-related gene expression but no such sex differences were found in prenatally saline-injected controls. A pairwise relevance analysis revealed that, although the synapse gene expression in controls was independent of sex, these genes form topologically distinct gene fabrics in males and females and these fabrics are altered by betamethasone in a sex-specific manner. These findings may explain the sex differences with respect to both normal behaviour and the occurrence and severity of infantile spasms. Changes in transcript expression and their coordination may contribute to a molecular substrate of permanent neurodevelopmental changes (including infantile spasms) found after prenatal exposure to corticosteroids.

Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model

Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrödinger equation has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave field generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves.

Spatiotemporal dynamics in a delayed diffusive predator model

In this paper, we investigate the spatiotemporal dynamics of a delayed reaction–diffusion Leslie–Gower model. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: Pure Turing instability gives birth to spots, spots–stripes-mixture, stripes, stripes–holes-mixture and holes patterns, pure Hopf instability to spiral wave pattern, and Hopf–Turing instability to chaotic wave pattern. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion predator–prey model, and indicate that time delay play an important roles in pattern formation.

Non-intrusive detection of air–water surface roughness in self-aerated chute flows

Stepped spillways are known to enhance the energy dissipation potential when compared with common smooth invert chutes. In the skimming flow regime, the self-aerated flow becomes more chaotic and surface waves (referred to as air–water surface roughness) are enhanced when the discharge decreases. In this study, experiments on smooth invert and stepped chutes models with a slope of 1: 2 were conducted to characterize this surface roughness by the use of a high-speed camera and ultrasonic sensor. It was found that the amount of entrapped air – at the water level where the air concentration is 90% – is reduced when a stepped spillway is considered. With decrease in step height (and on smooth invert chutes), entrapped air became more relevant. Wave heights increased with increase in step height and exceeded significantly the characteristic water level with 90% air concentration which was detected by a conductivity probe. A wide range of wave frequencies indicated the turbulent structure of stepped spillway flows.

Model for Shock Wave Chaos

We propose the following model equation, u(t) + 1/2(u(2)-uu(s))x = f(x,u(s)) that predicts chaotic shock waves, similar to those in detonations in chemically reacting mixtures. The equation is given on the half line, x<0, and the shock is located at x = 0 for any t ≥ 0. Here, u(s)(t) is the shock state and the source term f is taken to mimic the chemical energy release in detonations. This equation retains the essential physics needed to reproduce many properties of detonations in gaseous reactive mixtures: steady traveling wave solutions, instability of such solutions, and the onset of chaos. Our model is the first (to our knowledge) to describe chaos in shock waves by a scalar first-order partial differential equation. The chaos arises in the equation thanks to an interplay between the nonlinearity of the inviscid Burgers equation and a novel forcing term that is nonlocal in nature and has deep physical roots in reactive Euler equations.