Amplitude Equations Research Articles

Patterns dynamics in a diffusive prey–predator system with ratio-dependent functional response in a planar domain

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out.

Amplitude equation for a diffusion–reaction system in presence of complexing reaction with the activator species: the Brusselator model

Amplitude equation for a diffusion–reaction system in presence of complexing reaction with the activator species: the Brusselator model

Pattern Control of Neural Networks with Two-Dimensional Diffusion and Mixed Delays

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns.

Investigation of soliton solutions for the NWHS model with temperature distribution in an infinitely long and thin rod

This study proposed a modified [Formula: see text]-expansion method to seek the new exact traveling wave solutions of the Newell–White–Head–Segel (NWHS) Model. This is an amplitude equation utilized for distributing temperature within a rod that is infinitely thin and long, or determining the flow velocity of a fluid through a pipe that is infinitely long but has a small diameter. The modified [Formula: see text]-expansion method is used to extract the new exact solutions. The solutions of this model are categorized in hyperbolic, trigonometric, and rational forms. Moreover, we compare our results with the new auxiliary equation method. The 3D, line and corresponding contour representation of these solutions are depicted by choosing the different values of parameters.

AMPLITUDE EQUATIONS AND ORDER PARAMETERS OF HUMAN SARS-COV-2 INFECTIONS AND IMMUNE REACTIONS: A MODEL-BASED APPROACH

Several recent modeling studies have attempted to understand the human immune reaction to SARS-CoV-2 infections. Such a model-based understanding provides a sound basis for fighting COVID-19 on the level of individual patients. However, in this context, a worked-out nonlinear physics analysis providing insights into underlying amplitude equations and potential COVID-19 order parameters has not been conducted so far. In order to conduct such an analysis, a three-variable virus dynamics model is considered that can account for the human immune reaction to SARS-CoV-2 infections. The model amplitudes equations are derived and the relevant order parameter is determined. In line with theoretical reasoning, it is demonstrated that the order parameter and its amplitude determine the initial stage of SARS-CoV-2 infections, in general, and the initial dynamics of immune reactions, in particular. Explicitly, this finding is demonstrated for data from four COVID-19 patients. For those patients it is also demonstrated that the remnant of the order parameter determines the final disease decline phase. In this context, a time-resolved eigenvalue analysis is conducted that reveals that the transition from the initial stage to the decline stage which is associated with a switch of the leading eigenvalue from a positive to a negative value. It is argued that the immune reaction essentially contributes to this switch. From a medical-physics point of view, this observation suggests that the immune reaction of COVID-19 patients can stabilize the virus-free fixed point of affected sites.

Spatiotemporal dynamics analysis and parameter optimization of a network epidemic-like propagation model based on neural network method

In this paper, a reaction-diffusion model is established to study the dynamic behavior of rumor propagation. Firstly, we consider the existence of the positive equilibrium points. Then, we perform a stability analysis to study the conditions for the occurrence of Turing instability. Secondly, we use multiscale analysis to derive the expression of the amplitude equation. In the process of numerical simulation, the reality is considered. It shows that controlling the spread rate of rumor and the number of new Internet users have a great effect on curbing the spread of online rumor. Furthermore, it is proved that the analysis of amplitude equation plays a decisive role in the formation of Turing patterns. We also discuss the phenomenon of Turing patterns when the network structure changes and verify the rationality of the model by Monte Carlo method. Finally, we consider two methods based on statistical principle and convolutional neural network severally to identify the parameters of the reaction-diffusion system with Turing instability by using stable patterns. The statistical principle-based method offers superior accuracy, whereas the convolutional neural network-based approach significantly reduces recognition time and cuts down time costs.

Research on Pattern Dynamics Behavior of a Fractional Vegetation-Water Model in Arid Flat Environment

In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α, and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β, changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.

Complex dynamic analysis of a reaction–diffusion predator–prey model in the network and non-network environment

This paper considers an improved two-dimensional reaction–diffusion predator–prey model with time delay. Firstly, the distribution of the equilibrium points of the system and the existence conditions are discussed. Secondly, under the assumption of the existence of equilibrium points, the linear approximations of the system in both network and non-network settings are derived. Thirdly, the Turing instability conditions for both non-delay and delay systems are investigated, including cases of diffusion-induced and delay-induced instabilities. Fourthly, amplitude equations are derived based on the non-delay system. Finally, extensive numerical simulations are conducted to validate and illustrate the theoretical results, and to analyze and explain their practical significance. The above results effectively demonstrate that the theoretical findings, simulations, and natural reality are consistent. The numerical modeling conducted on the network structure based on the Monte Carlo method exhibits a more diverse range of dynamic behaviors in the parameter of the Holling III functional response function.

Coupled circularly polarized electromagnetic soliton states in magnetized plasmas

The interaction between two co-propagating electromagnetic pulses in a magnetized plasma is considered, from first principles, relying on a fluid-Maxwell model. Two circularly polarized wavepackets by same group velocities are considered, characterized by opposite circular polarization, to be identified as left-hand- or right hand circularly polarized (i.e. LCP or RCP, respectively). A multiscale perturbative technique is adopted, leading to a pair of coupled nonlinear Schrödinger-type (NLS) equations for the modulated amplitudes of the respective vector potentials associated with the two pulses. Systematic analysis reveals the existence, in certain frequency bands, of three different types of vector soliton modes: an LCP-bright/RCP-bright coupled soliton pair state, an LCP-bright/RCP-dark soliton pair, and an LCP-dark/RCP-bright soliton pair. The value of the magnetic field plays a critical role since it determines the type of vector solitons that may occur in certain frequency bands and, on the other hand, it affects the width of those frequency bands that are characterized by a specific type of vector soliton (type). The magnetic field (strength) thus arises as an order parameter, affecting the existence conditions of each type of solution (in the form of an envelope soliton pair). An exhaustive parametric investigation is presented in terms of frequency bands and in a wide range of magnetic field (strength) values, leading to results that may be applicable in beam-plasma interaction scenarios as well as in space plasmas and in the ionosphere.

Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems

The paper discusses three-dimensional models of the propagation of stochastic internal waves in hydrodynamic systems: ’half-space - half-space’, ’half-space - layer with rigid lid’, and ’layer with solid bottom - layer with rigid lid’. In constructing the models, the layers are considered to be ideal fluids separated by a contact surface. The main objective of the modeling is to obtain a dynamic equation for the stochastic amplitude of surface waves. A comparative analysis of the obtained results has been conducted. In order to control the contribution of nonlinear terms, a dimensionless non-numerical parameter has been introduced. The models are distinguished by boundary conditions that determine the general form of solutions. As a result, a dynamic equation for the stochastic amplitude of internal waves has been derived. After ensemble averaging of the amplitudes, the dynamic equation is formulated in integral form using Fourier-Stieltjes integrals. The dynamic equation reveals two-wave and three-wave interactions, as well as the contribution of dispersion to wave dynamics. An investigation of the boundary case of the transition of internal waves in the ’half-space - half-space’ system to surface waves in the absence of an upper liquid layer confirms the validity of the results.

Discontinuity waves in temperature and diffusion models

We analyse shock wave behaviour in a hyperbolic diffusion system with a general forcing term which is qualitatively not dissimilar to a logistic growth term. The amplitude behaviour is interesting and depends critically on a parameter in the forcing term. We also develop a fully nonlinear acceleration wave analysis for a hyperbolic theory of diffusion coupled to temperature evolution. We consider a rigid body and we show that for three-dimensional waves there is a fast wave and a slow wave. The amplitude equation is derived exactly for a one-dimensional (plane) wave and the amplitude is found for a wave moving into a region of constant temperature and solute concentration. This analysis is generalized to allow for forcing terms of Selkov–Schnakenberg, or Al Ghoul-Eu cubic reaction type. We briefly consider a nonlinear acceleration wave in a heat conduction theory with two solutes present, resulting in a model with equations for temperature and each of two solute concentrations. Here it is shown that three waves may propagate.

Stability, modulation instability and explicit-analytical solutions for the Hamiltonian amplitude equation

Stability, modulation instability and explicit-analytical solutions for the Hamiltonian amplitude equation

Effect of Angular Speed Variations on the Nonlinear Vibrations of a Rotational Spring-Mass System

A rotating spring-mass system is considered using polar coordinates. The system contains a cubic nonlinear spring with damping. The angular velocity harmonically fluctuates about a mean velocity. The dimensionless equations of motion are derived first. The velocity fluctuations resulted in a direct and parametric forcing terms. Approximate analytical solutions are sought using the Method of Multiple Scales, a perturbation technique. The primary resonance and the principal parametric resonance cases are investigated. The amplitude and frequency modulation equations are derived for each case. By considering the steady state solutions, the frequency response relations are derived. The bifurcation points are discussed for the problems. It is found that speed fluctuations may have substantial effects on the dynamics of the problem and the fluctuation frequency and amplitude can be adjusted as passive control parameters to maintain the desired responses.

Pattern formation for a charge transfer model with cross-diffusion

This paper is devoted to a charge transfer model with cross-diffusion under Neumann boundary conditions. By applying the linear stability theory, sufficient conditions for the occurrence of Hopf bifurcation and Turing instability are successfully established. Moreover, the amplitude equations are derived near the critical value of Turing instability according to the standard multiple-scale analysis. Our research indicates that the entire pattern generated by this model is controlled by the cross-diffusion parameter d21. Dynamic pattern formation is described theoretically and numerically, such as changing control parameters for spot pattern, strip pattern, and spot-strip pattern.

Complex pattern evolution of a two-dimensional space diffusion model of malware spread

In order to investigate the propagation mechanism of malware in cyber-physical systems (CPSs), the cross-diffusion in two-dimensional space is attempted to be introduced into a class of susceptible-infected (SI) malware propagation model depicted by partial differential equations (PDEs). Most of the traditional reaction-diffusion models of malware propagation only take into account the self-diffusion in one-dimensional space, but take less consideration of the cross-diffusion in two-dimensional space. This paper investigates the spatial diffusion behaviour of malware nodes spreading through physical devices. The formations of Turing patterns after homogeneous stationary instability triggered by Turing bifurcation are investigated by linear stability analysis and multiscale analysis methods. The conditions under the occurence of Hopf bifurcation and Turing bifurcation in the malware model are obtained. The amplitude equations are derived in the vicinity of the bifurcation point to explore the conditions for the formation of Turing patterns in two-dimensional space. And the corresponding patterns are obtained by varying the control parameters. It is shown that malicious virus nodes spread in different forms including hexagons, stripes and a mixture of the two. This paper will extend a new direction for the study of system security theory.

Stochastic dynamics of a nonlinear vibration energy harvester subjected to a combined parametric and external random excitation: The distinct cases of Itô and Stratonovich stochastic integration

The Stratonovich and Itô interpretations of a stochastic integral are mathematically consistent, distinct representations that yield contrasting response characteristics for stochastic dynamical systems subjected to multiplicative noise, such as parametric random excitation. The contrasting dynamics from the two interpretations currently remain unexplored in vibration energy harvester dynamics. We analytically investigate the dynamics of a generic piezoelectric, nonlinear vibration energy harvester simultaneously subjected to stochastic damping and external random excitation, focusing on the contrasting dynamics engendered by the two interpretations. Numerically solving the Stratonovich and Itô stochastic differential equations in the nonequilibrium regime of harvester dynamics, we find that the Stratonovich version yields significantly higher root mean square values of the harvester voltage output, for lower values of the mechanical damping coefficient. Furthermore, we find positive values for the leading Lyapunov exponent indicating instabilities in the harvester response unique to the Stratonovich interpretation. Additionally, comparing the results for the harvester electrical output between the cases of external excitation represented by the white noise and the Ornstein–Uhlenbeck noise processes, we find lower root mean square values of the voltage output in the latter case. Moreover, the results indicate that the averaged voltage output becomes progressively lower with increasing correlation time of the Ornstein–Uhlenbeck process. Studying the equilibrium dynamics by deriving Fokker–Planck equations for the response amplitude of the harvester using the harmonic balance method and stochastic averaging techniques, we solve for the stationary probability densities to find that the harvester is more likely to attain a well-defined equilibrium state under the Itô interpretation. In summary, the results contribute to the theoretical understanding of the fundamental contrast between the Stratonovich and Itô interpretations in vibration energy harvester dynamics and are expected to motivate advances in harvester design for various applications.

Estimating maximum initial wave amplitude of subaerial landslide tsunamis: A three-dimensional modelling approach

Landslide tsunamis, responsible for thousands of deaths and significant damage in recent years, necessitate the allocation of sufficient time and resources for studying these extreme natural hazards. This study offers a step change in the field by conducting a large number of three-dimensional numerical experiments, validated by physical tests, to develop a predictive equation for the maximum initial amplitude of tsunamis generated by subaerial landslides. We first conducted a few 3D physical experiments in a wave basin which were then applied for the validation of a 3D numerical model based on the Flow3D-HYDRO package. Consequently, we delivered 100 simulations using the validated model by varying parameters such as landslide volume, water depth, slope angle and travel distance. This large database was subsequently employed to develop a predictive equation for the maximum initial tsunami amplitude. For the first time, we considered travel distance as an independent parameter for developing the predictive equation, which can significantly improve the predication accuracy. The predictive equation was tested for the case of the 2018 Anak Krakatau subaerial landslide tsunami and produced satisfactory results.

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

Analysis of Self-Gravitating Fluid Instabilities from the Post-Newtonian Boltzmann Equation.

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy-momentum tensor calculated from a post-Newtonian Maxwell-Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory.

The high-order approximation of SPDEs with multiplicative noise via amplitude equations

The aim of this paper is to investigate the high-order approximation of a class of SPDEs with cubic nonlinearity driven by multiplicative noise with the help of the amplitude equations. The highlight of our work is the provision of approximate solutions with enhanced accuracy. Precisely, previous researches primarily concentrated on deriving approximate solutions via the first-order amplitude equations. However, this paper constructs approximate solutions by utilizing both first-order and second-order amplitude equations. And, we rigorously prove that such approximate solutions enjoy improved convergence property. To further illustrate our demonstration intuitively, we apply our main theorem to stochastic Allen–Cahn equation and present a numerical analysis.