Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semianalytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.

Stability analysis of the projectile based on random center manifold reduction

The center manifold method has been widely used in the field of stochastic dynamics as a dimensionality reduction method.This paper studied the angular motion stability of a projectile system under random disturbances.The random bifurcation of the projectile is studied using the idea of the Routh-Hurwitz stability criterion,the center manifold reduction,and the polar coordinates transformation.Then,an approximate analytical presentation for the stationary probability density function is found from the related Fokker–Planck equation.From the results,the random dynamical system of projectile generates three different dynamical behaviors with the changes of the bifurcation parameter and the noise strength,which can be a reference for projectile design.

Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.

DYNAMICAL BEHAVIOR OF AN INNOVATION DIFFUSION MODEL WITH INTRA-SPECIFIC COMPETITION BETWEEN COMPETING ADOPTERS

In this paper,we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation(product)in a particular region.The model exhibits two equilibria,namely,the adopter-free and an interior equilibrium.The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number(BIN)R_(A).It is investigated that the adopter free steady-state is stable if R_(A)<1.By consideringτ(the adoption experience of the adopters)as the bifurcation parameter,we have been able to obtain the critical value ofτresponsible for the periodic solutions due to Hopf bifurcation.The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem.Exhaustive numerical simulations in the support of analytical results have been presented.

STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage （time delay） is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period （time delay, T） passes through a critical value. Applying the normal form theory and the center manifold argument, we de- rive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination

In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.

Singularly perturbed bifurcation subsystem and its application in power systems

The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.

Dynamic bifurcation of the n-dimensional complex Swift-Hohenberg equation

This paper is concerned with the bifurcation of a complex Swift-Hohenberg equation. The attractor bifurcation of the complex Swift-Hohenberg equation on a one- dimensional domain （0, L） is investigated. It is shown that the n-dimensional complex Swift-Hohenberg equation bifurcates from the trivial solution to an attractor under the Dirichlet boundary condition on a general domain and under a periodic boundary condition when the bifurcation parameter crosses some critical values. The stability property of the bifurcation attractor is analyzed.

Asymptotic Bifurcation Solutions for Perturbed Kuramoto-Sivashinsky Equation

Stability and dynamic bifurcation in the perturbed Kuramoto-Sivashinsky （KS） equation with Dirichlet boundary condition are investigated by using central manifold reduction procedure. The result shows, as the bifurcation parameter crosses a critical value, the system undergoes a pitchfork bifurcation to produce two asymptotically stable solutions. Furthermore, when the distance from bifurcation is of comparable order ε2 （｜ε｜≤1）, the first two terms in e-expansions for the new asymptotic bifurcation solutions are derived by multiscale expansion method. Such information is useful to the bifurcation control.

Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model

Motivated by an animal territoriality model,we consider a centroidal Voronoi tessellation algorithm from a dynamical systems perspective.In doing so,we discuss the stability of an aligned equilibrium configuration for a rectangular domain that exhibits interesting symmetry properties.We also demonstrate the procedure for performing a center manifold reduction on the system to extract a set of coordinates which capture the long term dynamics when the system is close to a bifurcation.Bifurcations of the system restricted to the center manifold are then classified and compared to numerical results.Although we analyze a specific set-up,these methods can in principle be applied to any bifurcation point of any equilibrium for any domain.

Traveling Waves for an Epidemic Model with Spatio-Temporal Delays

This paper is concerned with traveling waves for an epidemic model with spatio-temporal delays. It is shown that the traveling wave solutions are still persistently existing in the model when the non-locality is caused by small average time delays via the geometric singular perturbation theory and the center manifold theorems.

A Schistosomiasis Model with Diffusion Effects

In this paper, we propose a schistosomiasis model in which two human groups share the water contaminated by schistosomiasis and migrate each other. The dynamical behavior of the model is studied. By calculation, the threshold value is given, which determines whether the disease will be extinct or not. The existence and global stability of the parasite-free equilibrium and the locally stability of the endemic equilibrium are discussed. Numerical simulations indicate that the diffusion from the mild endemic village to severe endemic village is benefit to control schistosomiasis transmission;otherwise it is bad for the disease control.

Analytic approach on geometric structure of invariant manifolds of the collinear Lagrange points

An analytical method is proposed to find geometric structures of stable,unstable and center manifolds of the collinear Lagrange points.In a transformed space,where the linearized equations are in Jordan canonical form,these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points.These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time.Thus the so-called geometric structure of these invariant manifolds is obtained.The stable,unstable and center manifolds are tangent to the stable,unstable and center eigenspaces,respectively.As an example of applicability,the invariant manifolds of L 1 point of the Sun-Earth system are considered.The stable and unstable manifolds are symmetric about the line from the Sun to the Earth,and they both reach near the Earth,so that the low energy transfer trajectory can be found based on the stable and unstable manifolds.The periodic or quasi-periodic orbits,which are chosen as nominal arrival orbits,can be obtained based on the center manifold.

Singularly Perturbed Renormalization Group Method and Its Significance in Dynamical Systems Theory

In this paper,we mainly investigate three topics on the renormalization group(RG)method to singularly perturbed problems:1)We will present an explicit strategy of RG procedure to get the approximate solution up to any order.2)We will refer that the RG procedure can,in fact,be used to get the normal form of differential dynamical systems.3)We will also present the approximating center manifolds of the perturbed systems,and investigate the long time asymptotic behavior by means of RG formula.

Attractor Bifurcation and Phase Transition for Liquid ~4He

Experiments show that the liquid helium-4 has either superfluid phase or solid phase when tem- perature decreases or the pressure of the system changes. In this paper, we discuss the equations which govern these phase transitions and derive the criterions for these phase changes. Meanwhile, we give related approxi- mate solutions and draw the phase diagram. In addition, we also prove that the liquid helium-4 bifurcates from the trivial solution to an attractor as parameters cross certain critical value. The topological structure of the bifurcated attractor is also illustrated.

Bogdanov–Takens Bifurcation in a Neutral Delayed Hopfield Neural Network with Bidirectional Connection

In this paper,a neutral Hopfield neural network with bidirectional connection is considered.In the first step,by choosing the connection weights as parameters bifurcation,the critical point at which a zero root of multiplicity two occurs in the characteristic equation associated with the linearized system.In the second step,we studied the zeros of a third degree exponential polynomial in order to make sure that except the double zero root,all the other roots of the characteristic equation have real parts that are negative.Moreover,we find the critical values to guarantee the existence of the Bogdanov–Takens bifurcation.In the third step,the normal form is obtained and its dynamical behaviors are studied after the use of the reduction on the center manifold and the theory of the normal form.Furthermore,for the demonstration of our results,we have given a numerical example.

QUALITATIVE STUDY FOR A MULTI-DRUG RESISTANT TB MODEL WITH EXOGENOUS REINFECTION AND RELAPSE

One tuberculosis transmission model is formulated by incorporating exogenous reinfec- tion, relapse, and two treatment stages of infectious TB cases. The global stability of the unique disease-free equilibrium is obtained by applying the comparison principle if the effective reproduction number for the full model is less than unity. The existence and stability of the boundary equilibria are given by introducing the invasion reproduction numbers. Furthermore, the existence and local stability of the endemic equilibrium are addressed under some conditions.

On the existence of traveling wave solutions and upper and lower bounds for some Fisher-iolmogorov type equations

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher-Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

ON COMPLEX DYNAMIC CONTROL SYSTEMS

This paper presents some recent works on the control of dynamic systems,which have certain complex properties caused by singularity of the nonlinear structures,structure^varyings, or evolution process etc. First, we consider the structure singularity of nonlinear control systems. It was revealed that the focus of researches on nonlinear control theory is shifting from regular systems to singular systems. The singularity of nonlinear systems causes certain complexity. Secondly, the switched systems are considered. For such systems the complexity is caused by the structure varying. We show that the switched systems have significant characteristics of complex systems. Finally, we investigate the evolution systems. The evolution structure makes complexity, and itself is a proper model for complex systems.