Complex Dynamical Behavior of Holling–Tanner Predator-Prey Model with Cross-Diffusion

Spatial predator-prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator-prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black-eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross-diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross-diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.

Investigation Of Dual-Pump Fiber Optical Parametric Amplifier Performance Driven by Energy Transfer Between Four-Wave Mixing Process

Fiber optical parametric amplifier (FOPA) is operated based on energy transfer from pump waves to signal wave and at the end of the fiber, an idler wave is generated. This process is called four-wave mixing (FWM). Even though effects of higher-order dispersion coefficients, fiber length, fiber nonlinearity, fiber attenuation, pump powers, pump wavelength separation and distance of central pump wavelength with ZDW on gain profiles have been examined by previous researchers, but on different fiber or numerically studied using the Optisys system, analytical model or different amplitude equations. Thus, in this study, the above-mentioned parameters on the gain performance of dual pump fiber optical parametric amplifier (FOPA) using highly nonlinear shifted fiber (HNL-DSF) as a medium will be numerically investigated using ode45 function in Matlab. The gain at a certain wavelength can be obtained by solving 4 coupled amplitude equations with fiber loss and pump depletion that govern the four-wave mixing (FWM) process of pumps, signal and idler waves. Simulations results indicate positive gives poor or no gain, meanwhile, an addition of to negative widens the bandwidth, but there is no significant effect with the addition of . Besides, an increase of fiber length, nonlinearity and pump powers improve gain performance, but an increase of fiber loss decays the gain amplitude. Increment of pump separation will enhance flatness of gain at wavelength far from central wavelength but results in an increase of gain reduction at the central wavelength. Lastly, must be positive, not too small and not bigger than 1.125nm to get a high, broader and lesser ripples gain.

Periodic, Quasi-Periodic and Phase-Locked Oscillations and Stability in the Fiscal Dynamical Model with Tax Collection and Decision-Making Delays

In this paper, we consider the complex dynamics of a fiscal dynamical model, which was improved from Wolfstetter classical growth cycle model by Sportelli et al. The main work of the present paper is to study the impact of fiscal policy delays on the national income adjustment processes using a dynamical method, such as double Hopf bifurcation analysis. We first use DDE-BIFTOOL to find the double Hopf bifurcation points of the system, and draw the bifurcation diagrams with two bifurcation parameters, i.e. the tax collection delay [Formula: see text] and the public expenditure decision-making delay [Formula: see text]. Then we employ the method of multiple scales to obtain two amplitude equations. By analyzing these amplitude equations, we derive the classification and unfolding of these double Hopf bifurcation points. And three types of double Hopf bifurcations are found. Finally, we verify the results by numerical simulations. We find complex dynamic behaviors of the system via the analytical method, such as stable equilibrium, stable periodic, quasi-periodic and phase-locked solutions in respective regions. The dynamical phenomena can help policy makers to choose a proper range of the delays so that they could effectively formulate fiscal policies to stabilize the economy.

Patterns and Their Large-Scale Distortions in Marangoni Convection with Insoluble Surfactant

Nonlinear dynamics of patterns near the threshold of long-wave monotonic Marangoni instability of conductive state in a heated thin layer of liquid covered by insoluble surfactant is considered. Pattern selection between roll and square planforms is analyzed. The dependence of pattern stability on the heat transfer from the free surface of the liquid characterized by Biot number and the gravity described by Galileo number at different surfactant concentrations is studied. Using weakly nonlinear analysis, we derive a set of amplitude equations governing the large-scale roll distortions in the presence of the surface deformation and the surfactant redistribution. These equations are used for the linear analysis of modulational instability of stationary rolls.

Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

Analysis of Spatially Extended Excitable Izhikevich Neuron Model near Instability

The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs , we demonstrate various results of spatiotemporal architecture ( PDEs ) using a suitable parameter regime. We analytically perform the saddle node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different types of firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method and then expansion in powers of a small control parameter. The instabilities and Turing bifurcation are established using the theoretical analysis and numerical simulations. The spatial system has potential effects on the deterministic system as a result of the diffusive matrices with various couplings and the coupled oscillators with this nearest neighbor coupling show synchronization measured by the synchronization factor analysis. Our results qualitatively reproduce different phenomena of the extended excitable system based with an efficient analytical scheme.

A Genetic-Based Hybrid Algorithm Harmonic Minimization Method for Cascaded Multilevel Inverters with ANFIS Implementation

Selective harmonic elimination pulse-width modulation (SHEPWM) is a widely adopted method to eliminate harmonics in multilevel inverters, yet solving harmonic amplitude equations is both time consuming and not accurate. This method is applied here for a 7-level cascaded multilevel inverter (CMLI) with erroneous DC sources. To meet the seven harmonic amplitude equations, two notches are applied with the use of higher switching frequency than nominal. These notches can be placed in six different positions in the voltage wave, and each was assessed in a separate manner. In order to solve the equations, a hybrid algorithm composed of genetic algorithm (GA) and Newton–Raphson (N-R) algorithm is applied to achieve faster convergence and maintain the accuracy of stochastic methods. At each step of the modulation index (M), different positions for the notches are compared based on the distortion factor (DF2%) benchmark, and the position with lowest DF2% is selected to train an artificial neural fuzzy interface system (ANFIS). ANFIS will receive the DC sources’ voltages together with required M and will produce one output; thus, eight ANFISs are applied to produce seven firing angles, and the remaining one is to determine which one of the notches’ positions should be used. Software simulations and experimental results confirm the validity of this proposed method. The proposed method achieves THD 8.45% when M is equal to 0.8 and is capable of effectively eliminating all harmonics up to the 19th order.

COVID-19 outbreaks follow narrow paths: A computational phase portrait approach based on nonlinear physics and synergetics

From the perspective of mathematical epidemiology, COVID-19 epidemics emerge due to instabilities in epidemiological systems. It is shown that the COVID-19 outbreaks follow highly specified paths in epidemiological state spaces. These paths are described by phase portraits that can be readily computed from epidemiological models defined in terms of nonlinear dynamical systems. The paths are predicted by order parameters and amplitude equations that are well known in nonlinear physics and synergetics to exist at instability points. The approach is illustrated for SIR, SEIR and SEIAR models and epidemic outbreaks in China, Italy and West Africa. Identifying such COVID-19 order parameters may help in forecasting COVID-19 epidemics and predicting the impacts of intervention measures.