The Dynamical Properties of a Class of Discrete Smith Diffusion Model

In this paper, the dynamical properties of Smith type diffusion model with Dirichlet boundary conditions are studied. The properties of hyperbolic fixed points and non-hyperbolic fixed points of the model are analyzed. By using the central manifold theorem, the bifurcation phenomenon of the model is studied. The results show that flip, transcritical, pitchfork and Fold-flip bifurcations exist at non-hyperbolic fixed points.

Generalized Hopf Bifurcation in a Delay Model of Neutrophil Cells Model

The DDE-Biftool software is applied to solve the dynamical stability and bifurcation problem of the neutrophil cells model. Based on Hopf point finding with the stability property of the equilibrium solution loss, the continuation of the bifurcating periodical solution starting from Hopf point is exploited. The generalized Hopf point is tracked by seeking for the critical value of free parameter of the switching phenomena of the open loop, which describes the lineup of bifurcating periodical solutions from Hopf point. The normal form near the generalized Hopf point is computed by Lyapunov-Schimdt reduction scheme combined with the center manifold analytical technique. The near dynamics is classified by geometrically different topological phase portraits.

The Classification to Stationary Process of Tidal Motion Observed at the Time of Kuroshio’s Meandering

The tide level displays information about the state of the sea current and the tidal motion. The tide level of the southern coast of Japan Island is affected strongly by Kuroshio Current flowing in the western part of North Pacific Ocean. When Kuroshio takes the straight path and flow along the Japan Islands, the tide level increases, and it is calculated from two tide level data observed at Kushimoto and Uragami in the southern part of Kii Peninsula. In contrast, the tide level decreases at the time when Kuroshio leaves from the Japan Islands. In this paper, the hourly tidal data are analyzed using the Autocorrelation Function (ACF) and the Mutual Information (MI) and the phase trajectories at first. We classify the results into 5 types of tidal motion. Each categorized type is investigated and characterized precisely using the mean tide level and the unit root test (ADF test) next. The frequency of the type having unstable tidal motion increases when the Kuroshio Current is non-meandering or in a transition state or the tide level is high, and the type shows a non-stationary process. On the other hand, when the Kuroshio Current meanders, the tidal motion tends to take a periodical and stable state and the motion is a stationary process. Though it is not frequent, we also discover a type of stationary and irregular tidal motion.

Bifurcation Analysis of a Neutrophil Periodic Oscillation Model with State Feedback Control

The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.

Periodical Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model with Feedback Control

The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.

Two-Dimensional Manifolds with Computation V-Function of ODE Systems

The computation of stable or unstable manifold of two-dimensional is developed, which is an efficient method in studying stable structure analysis of system character geometrically. The Lorentz stable manifold is computed by the fixed arclength method and the hyperbolic equilibrium is a saddle. The two-dimensional stable structure of Lorentz manifold is significant in people’s usual view. We also introduce the V-function to compute the V-manifold correspondingly. The defined V-function is smooth in the unstable direction of the manifold. Especially, the routh to period-doubling attractor on manifold surface is discussed too.

Turing Instability of Gray-Scott Reaction-Diffusion Model with Time Delay Effects

The reaction diffusion Gray-Scott model with time delay is put forward with the assumption of Neumann boundary condition is satisfied. Based on the Turing bifurcation condition, the Turing curves on two parameter plane are discussed without time delay. The normal form is computed via applying Lyapunov-Schmidt reduction method in system of PDE, and the bifurcating direction of pitchfork bifurcation underlying codimension-1 singularity of Turing point is computed. The continuation of Pitchfork bifurcation is simulated with varying free parameter continuously near the turing point, which is in coincidence with the theoritical analysis results. The wave pattern formation in the case of turing instability is also simulated which discover Turing oscillation phenomena from periodicity to irregularity.

Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises

In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.

Asymptotic Behavior of Stochastic Strongly Damped Wave Equation with Multiplicative Noise

In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.

Random Attractor of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation

In this paper, we consider the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise. We first deal with random term by using Ornstein-Uhlenbeck process and establish the wellness of the solution, then the existence of global random attractor are proved.

Global Attractor and Dimension Estimation for a 2D Generalized Anisotropy Kuramoto-Sivashinsky Equation

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the global attractor. Finally, we get the upper bound estimation of the Haus-dorff and fractal dimension of attractor.

Deriving the Exact Percentage of Dark Energy Using a Transfinite Version of Nottale’s Scale Relativity

In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.

Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term

In this article, we prove the existence of exponential attractors of the nonclassical diffusion equation with critical nonlinearity and lower regular forcing term. As an additional product, we show that the fractal dimension of the global attractors of this problem is finite.

The Inertial Manifold for a Class of Nonlinear Higher-Order Coupled Kirchhoff Equations with Strong Linear Damping

This paper considers the long-time behavior for a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of the inertial manifold while such equations satisfy the spectral interval condition.

Voltage Control of a 12/8 Pole Switched Reluctance Generator Using Fuzzy Logic

This paper presents a voltage controller for a 12/8 switched reluctance generator (SRG). SR machines have a robust with simple structure because they have no windings or permanent magnets on the rotor, so they are capable of working at high speed application and at rough condition. However, due to nonlinear nature, doubly salient poles, time variant parameters, and non-ideal current waveform, The SRG output voltage inherently contains ripples. The proposed voltage controller is based on fuzzy logic that is very efficient over uncertainty and parameter variations. The effectiveness of the proposed control method is verified by comparing the obtained result with that of a PI-controller.

State Reconstruction for Complex Dynamical Networks with Noises

The state reconstruction problem is addressed for complex dynamical networks coupled with states and outputs respectively, in a noisy transmission channel. By using Lyapunov stability theory and H∞ performance, two schemes of state reconstruction are proposed for the complex dynamical networks with the nodes coupled by states and outputs respectively, and the estimation errors are convergent to zeros with H∞ performance index. A numerical simulation demonstrates the effectiveness of the proposed observers.

Fixed Point Theorems of Hegedus Contraction Mapping in Some Types of Distance Spaces

In the present paper, we prove some fixed point theorems of Hegedus contraction in some types of distance spaces, dislocated metric space, left dislocated metric space, right dislocated metric space and dislocated quasi-metric metric space which are generalized metrics spaces where self-distances are not necessarily zero.

The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays

In this paper, we study the global and pullback attractors for a strongly damped wave equation with delays when the force term belongs to different space. The results following from the solution generate a compact set.