Chaotic Wave Research Articles

Analysis of two-dimensional combustion waves arising in the presence of a competitive endothermic reaction

We consider a system of reaction-diffusion equations describing combustion dynamics. The reaction is assumed to undergo two competitive reactions, one which is exothermic and one which is endothermic. The one-dimensional model has been shown to exhibit complex behaviour, from propagating combustion waves with a constant speed to period doubling cascades and the possibility of chaotic wave speeds. In this study, we extend the combustion model from one to two dimensions by exploring a model of an insulated strip with no heat loss and axially symmetric spread. In particular, we compare and contrast the behaviour of the systems in one and two dimensions. 
 
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Inelastic interactions, numerical breathers and chaotic wave fields for a focusing Kundu–Eckhaus equation in a nonlinear optical fiber

Investigation is made on a focusing Kundu–Eckhaus equation in nonlinear/quantum optics and fluid mechanics, for describing the optical properties of the femtosecond lasers and femtochemistry objects, and for examining the stability of the Stokes waves in weakly nonlinear dispersive media. We derive the one and two optical breather solutions via the Darboux transformation. Inelastic interactions between two single-hump optical breathers and between a single-hump optical breather and a double-hump optical breather are depicted. Numerical optical breathers are obtained via the split-step Fourier method: β affects the speeds of the numerical optical breathers and space energy distribution, while β has no effects on the total energy, where β is a real constant representing the quintic nonlinearity. Optical chaotic wave fields via three initial conditions, including the plane wave, one soliton and one breather, are derived. Among them, the optical chaotic wave field via the one breather is the most orderless via the probability density function. Total energy of the optical chaotic wave field via the one breather is the largest.

Optical breathers and multiple rogue waves via the modulation instability for a complex Ginzburg–Landau equation in the mode-locked laser operation

Investigation is made on a complex Ginzburg–Landau equation for the mode-locked laser operation. Optical breathers are numerically derived via the split-step Fourier method. Effects of the modulation instability on the dynamic behaviors of the optical breathers are investigated: when the growth rate of the modulation instability increases, the optical breather splits into two optical breathers earlier, and the two optical breathers separate faster. Via the energy distributions of the optical breathers, we obtain that when the fission phenomenon occurs, the energy decreases more than that of the fusion phenomenon. Multiple rogue waves in the chaotic wave field are obtained via the modulation instability. It is observed that multiple rogue waves occur earlier as the growth rate of the modulation instability becomes larger. Occurrence of the rogue wave can be predicted via the spectrum of the optical chaotic wave field.

Модель эволюции хаотических волновых процессов в сложных динамических системах на основе теории матричной декомпозиции

Модель эволюции хаотических волновых процессов в сложных динамических системах на основе теории матричной декомпозиции

Optical breathers and rogue waves via the modulation instability for a higher-order generalized nonlinear Schrödinger equation in an optical fiber transmission system

For describing the propagation of ultrashort pulses in a high-speed, long-distance optical fiber transmission system with the fourth-order dispersion, cubic–quintic nonlinearity, self-steepening and self-frequency shift, a higher-order generalized nonlinear Schrodinger equation is investigated. We get the rogue-wave solutions. Effects of the modulation instability on the optical rogue waves are studied: Increasing the growth rate of the modulation instability makes the existence time of the optical rogue wave shorter. We numerically derive the optical breathers in the chaotic wave fields via the modulation instability. Spectrum of the optical chaotic wave field can be used to indicate the appearance of the optical breather in the chaotic wave field. Optical rogue waves in the chaotic wave fields are also gotten via the modulation instability.

Transition of regular wave fronts to irregular wave fronts in gravity-driven thin films over topography

This article contributes to an understanding of the pathway from regular to chaotic traveling wave fronts over periodically undulated inclines in thin films. In order to investigate the transition from regular to chaotic waves, we used various undulation forms and varied the Reynolds number and the inclination angle in the measurements. Thereby, we revealed the first partially chaotic waves on a gravity-driven thin film channel flow. The wave is subdivided into: (i) the chaotic wave front and (ii) a regular wave tail. The area of the chaotic part can be increased by increasing the inertia of the system. Various phenomena on the flow were revealed: (a) bubble formation, (b) fingering, (c) splashes, and (d) pinch-offs. Our investigation leads to the conclusion that wave breaking over obstacles is a necessary condition for the transition from regular to chaotic wave fronts.

Distributed temperature measurement with millimeter-level high spatial resolution based on chaotic laser

The high-precision structural health monitoring of large civil structures and materials are increasingly demanded with widely using the distributed fiber sensors. A Brillouin optical correlation domain analysis for millimeter-levelhigh spatial resolution sensing using broadband chaotic laser is proposed and demonstrated. Through the analysis of the influence of polarization state and feedback strength on the chaotic laser, we experimentally achieve a broadband chaotic laser with a spectrum over 7.5 GHz in –3 dB which means that the theoretical spatial resolution is 3 mm, and we also successfully measure the distribution of fiber Brillouin gain spectrum with a temperature over 300 m measurement range with 7.05 mm spatial resolution, which is the first time that the sensor system based on chaotic laser has achieved the measurement with millimeter-level. However, there is still a difference in spatial resolution between the experimental and theoretical values. We can find that the chaotic laser has a time-delay feature; besides, with the broadening of chaotic laser, the threshold of stimulated Brillouin scattering in optical fibers increases while the Brillouin gain will weaken if the pump power is not enough here, and the cross-correlation peak of chaotic laser will narrow. All these problems cause the Brillouin gain signal to be easily submerged by noise, so the performance of the chaotic Brillouin optical correlation domain analysis system will decrease ultimately. Therefore, we also propose an optimization of Brillouin optical correlation domain analysis system by introducing the time-gated scheme into pump branch. It is obvious that the peak power of the pump wave is heightened by more than 9.5 dB after being amplitude-modulated by a square pulse with a pulse width of greater than acoustic phonon lifetime, and the signal-to-back ground noise ratio of the gain spectrum is improved effectively in theory; the cross correlation between chaotic pump wave and probe waveis locked within a pulse duration time, and the residual stimulated Brillouin scattering interactions existing outside the central correlation peak can be largely inhibited. In this optimized setup, the performance of the distributed temperature sensing is improved to 3.12 mm spatial resolution, which corresponds well to the theoretical value. The improved chaotic Brillouin optical correlation domain analysis technology will have a great potential application in high-precision structural health monitoring of large civil structures.

Stimulated Brillouin scattering of terahertz electromagnetic pulses in paraelectrics

The nonlinear acoustoelectromagnetic phenomena in the terahertz (THz) range in the crystalline paraelectric SrTiO3 are investigated theoretically. The goal is to investigate an appearance of short THz electromagnetic (EM) pulses. The moderate cooling to the temperature T ≈ 77 K is considered. The resonant three-wave interaction of two counterpropagating EM THz waves with the longitudinal acoustic wave of the difference frequency is under investigation that is called the stimulated Brillouin scattering. This resonant interaction is due to the quadratic nonlinearity called electrostriction. The appearance of short EM pulses is possible when the input amplitudes of the signal EM wave are comparable with the pump amplitude and the duration of the input pulses is quite long. In the temporal scale associated with the EM wave propagation within a crystal, the cubic EM nonlinearity and the EM wave dispersion affect this three-wave resonant interaction. Under the development of the resonant nonlinear interaction, the modulation instability occurs that results in the complex and even chaotic wave modulation.

Chaotic Wave Motions and Chaotic Dynamic Responses of Piezoelectric Laminated Composite Rectangular Thin Plate Under Combined Transverse and In-Plane Excitations

In the present work, the chaotic wave motions and the chaotic dynamic responses are investigated for a four-edge simply supported piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Based on the reductive perturbation method, the complicated partial differential nonlinear governing equation of motion for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations is transformed into an equivalent and soluble nonlinear wave equation. The heteroclinic orbit and resonant torus are obtained for the unperturbed nonlinear wave equation. The topological structures of the unperturbed and the perturbed nonlinear wave equations are investigated on the fast and the slow manifolds. The persistence of the heteroclinic orbit is studied for the perturbed nonlinear wave equation through the Melnikov method. The geometric analysis is utilized to prove that the heteroclinic orbit goes back to the stable manifold of the saddle point on the slow manifold under the perturbations. The existence of the homoclinic orbit is conformed for the perturbed nonlinear wave equation by the first and the second measures. When the homoclinic orbit is broken, the chaotic motions occur in the Smale horseshoe sense for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Numerical simulations are finished to study the influence of the damping coefficient on the propagation properties of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Both theoretical study and numerical simulation results indicate the existence of the chaotic wave motions and the chaotic dynamic responses of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations.

Universal spectral correlations in the chaotic wave function and the development of quantum chaos

We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix $\rho_A$ of a spatial subsystem A. We find that the spectrum of $\rho_A$ is described by a so-called Wishart random matrix, which exhibits universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings. We use these universal spectral characteristics of $\rho_A$ as a definition of chaos in the wave function. A simple and precise characterization of such correlations is a segment of linear growth at sufficiently long times, recently called the "ramp", of the spectral form factor. Specifically, numerical results for the spectral form factor of $\rho_A$ of generic non-integrable systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit an universal ramp identical to that appearing for a random pure state, in which $\rho_A$ is the Wishart random matrix. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ramp set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of $\rho_A$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ramp in the spectral form factor of $\rho_A$, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized chaotic regime.

Resonance in modulation instability from non-instantaneous nonlinearities

To explore resonance phenomena in the nonlinear region, we show by experimental measurements and theoretical analyses that resonance happens in modulation instability from non-instantaneous nonlinearities in photorefractive crystals. With a temporally periodic modulation in the external bias voltage, corresponding to a modulation in the nonlinear strength, an enhancement in the visibility of MI at resonant frequency is reported through spontaneous optical pattern formations. Theoretical curves obtained from a nonlinear non-instantaneous Schrödinger equation give good agreement to experimental data.

Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate

This paper presents the study on the chaotic wave and chaotic dynamics of the nonlinear wave equations for a simply supported truss core sandwich plate combined with the transverse and in-plane excitations. Based on the governing equation of motion for the simply supported sandwich plate with truss core, the reductive perturbation method is used to simplify the partial differential equation. According to the exact solution of the unperturbed equation, two different kinds of the topological structures are derived, which one structure is the resonant torus and another structure is the heteroclinic orbit. The characteristic of the singular points in the neighborhood of the resonant torus for the nonlinear wave equation is investigated. It is found that there exists the homoclinic orbit on the unperturbed slow manifold. The saddle-focus type of the singular point appears when the homoclinic orbit is broken under the perturbation. Additionally, the saddle-focus type of the singular point occurs when the resonant torus on the fast manifold is broken under the perturbation. It is known that the dynamic characteristics are well consistent on the fast and slow manifolds under the condition of the perturbation. The Melnikov method, which is called the first measure, is applied to study the persistence of the heteroclinic orbit in the perturbed equation. The geometric analysis, which is named the second measure, is used to guarantee that the heteroclinic orbit on the fast manifold comes back to the stable manifold of the saddle on the slow manifold under the perturbation. The theoretical analysis suggests that there is the chaos for the Smale horseshoe sense in the truss core sandwich plate. Numerical simulations are performed to further verify the existence of the chaotic wave and chaotic motions in the nonlinear wave equation. The damping coefficient is considered as the controlling parameter to study the effect on the propagation property of the nonlinear wave in the sandwich plate with truss core. The numerical results confirm the validity of the theoretical study.

Observing spatio-temporal dynamics of excitable media using reservoir computing.

We present a dynamical observer for two dimensional partial differential equation models describing excitable media, where the required cross prediction from observed time series to not measured state variables is provided by Echo State Networks receiving input from local regions in space, only. The efficacy of this approach is demonstrated for (noisy) data from a (cubic) Barkley model and the Bueno-Orovio-Cherry-Fenton model describing chaotic electrical wave propagation in cardiac tissue.

Reconfigurable dynamic all-optical chaotic logic operations in an optically injected VCSEL**Project supported by the National Natural Science Foundation of China (Grant No. 61475120) and the Innovative Projects in Guangdong Colleges and Universities, China (Grant No. 2015KTSCX146).

In a chaotic system of vertical cavity surface emitting laser (VCSEL) with external optical-injection, we propose a novel implementation scheme for reconfigurable dynamic all-optical chaotic logic operations (AOCLOs). Under different key parameters, such as the bias current, the injection strength and the frequency detuning of the injected light field and the VCSEL, we also explore the evolutions of the polarization-bistability with the amplitude of the injected light field when the output of VCSEL is chaotic wave. According to the dynamic evolutions, we find out the optimal value of the frequency detuning that is considered as a control logic signal, and further implement different AOCLOs, such as AND, NAND, OR, NOR, XOR, and XNOR, by controlling the logic operation of the control logic signal between two logic inputs. Moreover, the ability to reconstruct these logic operations is demonstrated under relatively low noise strength of the spontaneous emission.

Transformation of mode-2 internal solitary wave over a pseudo slope-shelf

Numerical simulations are performed to investigate the effect of wave amplitude in a numerical wave tank on the evolution of a convex mode-2 internal solitary wave (ISW) propagating over a pseudo slope-shelf. A finite volume method based on a Cartesian grid system is adopted to solve the Navier-Stokes equations using Improved Delayed Detached Eddy Simulation model for the turbulent closure. Numerical results reveal three types of waveform during wave generation on the flat bottom: (1) pseudo vortex shedding in the case of very large initial amplitude; (2) PacMan phenomenon in large amplitude; and (3) smooth mode-2 ISW for small amplitude. During wave propagation on the plateau, the first type of waveform induces a quasi-elevated mode-1 ISW; the second generates chaotic internal waves with significant reduction in amplitude; while the third renders a slightly deformed mode-2 ISW across the plateau. Moreover, the decrease in the magnitude of leading trough is more intense than that in the leading crest due to strong wave-obstacle interaction in the case of very large initial wave amplitude.

Easy encoding and low bit‐error‐rate chaos communication system based on reverse‐time chaotic oscillator

A new chaos communication system based on reverse-time chaotic oscillator (RTCO) is proposed in this study. In the system, driven by bipolar sequence, RTCO can directly generate chaotic wave signals that can encode arbitrary binary information, which is much easier than that of the existing chaotic communication scheme in that needs the initial condition estimation. Then the analytical expression of matched filter for the basis function of RTCO is derived. The proposed matched filter is capable of decreasing the effect of noise in the channel. Next, the binary information can be obtained by detecting the summation of multi-sampling during the symbol period through a setting threshold over additive white Gaussian noise (AWGN) channel, which further decreases the influence of noise in decoding procedure. In addition, the binary information can also be obtained over the Rayleigh channel, and its bit error rate (BER) expression is derived. Finally, the feasibility and the validity of the proposed system are given with numerical simulations. It is shown that in the proposed communication system, the encoded chaotic signal is generated with a much simpler method. Furthermore, the BER is lower than that in over AWGN channel, and the performance of the proposed system over Rayleigh channel is better than that of differential-chaos-shift-keying.

Intermittency in the Hodgkin-Huxley model.

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.

Characterization of Brillouin dynamic grating based on chaotic laser

The Brillouin dynamic grating (BDG) based on chaotic laser has particular advantages over the conventional BDG, for example, the creation of single and permanent BDG. To gain insight into the chaotic BDG, we theoretically investigate the reflection and gain spectra characteristics of the chaotic BDG generated in the polarization maintaining fiber. We find that the reflection spectral width of the chaotic BDG is inversely proportional to the effective grating length and the variation in the gain spectral width is negligible with respect to the effective grating length. The widths of the reflection and gain spectra are not affected by the power of the chaotic pump wave. Besides, we analyze that the occurrence of the weak BDG in the generation process of the chaotic BDG leads to the side-lobe of the reflected pulse.

Thermal radiation induced ignition of multipoint turbulent explosions

Abstract The severity of vapour cloud explosions is typically correlated with the peak over-pressure or, more accurately, with the total impulse caused by the pressure waves. The flame speed arising from strong (e.g. quasi-stable) turbulent deflagrations is frequently used to provide an indication of potential damage. However, conventional flame propagation mechanisms can present difficulties in terms of explaining the resulting damage. For example, the over-pressure in the Buncefield vapour cloud explosion was much higher than that predicted by conventional models. Alternative propagation mechanisms include intermittent localised strong explosions or detonations potentially supported by forward thermal radiation causing multi-point ignition of dust particles ahead of the advancing flame front. Such mechanisms are here explored using particles coated with acetylene black as the radiation target due to their relationship with soot emissions. A continuous wave laser operating in the near infrared was used as the radiation source with experiments performed in a flame tube using fuel lean CH4/H2/Air mixtures. It is shown that ignition kernels caused by irradiated particles can successfully be entrained into the main flow and/or recirculation zones formed around obstacles and cause multipoint explosions. The resulting relationship between fuel consumption ahead of the advancing flame and the evolution of the strength of the explosion is shown to be complex and typically lead to increased explosion durations with reduced peak pressures. It is also shown that chaotic pressure wave interactions can substantially increase both the explosion duration and the peak pressure depending on the timing of the radiation induced ignition.

Combined Nonlinear Analysis of Atrial and Ventricular Series for Automated Screening of Atrial Fibrillation

Atrial fibrillation (AF) is the most common cardiac arrhythmia in clinical practice. It often starts with asymptomatic and short episodes, which are difficult to detect without the assistance of automatic monitoring tools. The vast majority of methods proposed for this purpose are based on quantifying the irregular ventricular response (i.e., RR series) during the arrhythmia. However, although AF totally alters the atrial activity (AA) reflected on the electrocardiogram (ECG), replacing stable P-waves by chaotic and time-variant fibrillatory waves, this information has still not been explored for automated screening of AF. Hence, a pioneering AF detector based on quantifying the variability over time of the AA morphological pattern is here proposed. Results from two public reference databases have proven that the proposed method outperforms current state-of-the-art algorithms, reporting accuracy higher than 90%. A less false positive rate in the presence of other arrhythmias different from AF was also noticed. Finally, the combination of this algorithm with the classical analysis of RR series variability also yielded a promising trade-off between AF accuracy and detection delay. Indeed, this combination provided similar accuracy than RR-based methods, but with a significantly shorter delay of 10 beats.