The Calculation Process of the Limit Cycle for Lorenz System

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

FOUR–DIMENSIONAL BRUSSELATOR MODEL WITH PERIODICAL SOLUTION

In the paper, a four-dimensional model of cyclic reactions of the type Prigogine's Brusselator is considered. It is shown that the corresponding dynamical system does not have a closed trajectory in the positive orthant that will make it inadequate with the main property of chemical reactions of Brusselator type. Therefore, a new modified Brusselator model is proposed in the form of a four-dimensional dynamic system. Also, the existence of a closed trajectory is proved by the DN-tracking method for a certain value of the parameter which expresses the rate of addition one of the reagents to the reaction from an external source.

Double Hopf Bifurcation of a Hematopoietic System with State Feedback Control

The dynamics of a system composed of hematopoietic stem cells and its relationship with neutrophils is ubiquitous due to periodic oscillating behavior induce cyclical neutropenia. Underlying the methodology of state feedback control with two time delays, double Hopf bifurcation occurs as varying time delay to reach its threshold value. By applying center manifold theory, the analytical analysis of system exposed the different dynamical feature in the classified regimes near double Hopf point. The novel dynamics as periodical solution and quasi-periodical attractor coexistence phenomena are explored and verified by numerical simulation.

Application of Extended Geometrical Criterion to Population Model with Two Time Delays

Geometrical criterion is a flexible method to be applied to a type of delay differential equations with delay dependent coefficient. The criterion is used to solve roots attribution of the related characteristic equation in complex plane effectively by introducing a new parameter skillfully. An extended geometrical criterion is developed to compute the stability of DDEs with two time delays. It is found that stability switching phenomena arise while equilibrium solution loses its stability and becomes unstable, then retrieve its stability again. Hopf bifurcation and the bifurcating periodic solution is analyzed by applying central manifold reduction method. The novel dynamical behaviors such as periodical solution bifurcating to chaos are discovered by using numerical simulation method.

A reactor model with delay because of convection

In this article a model of a nuclear reactor which is made of two differential equations with delay was analysed. There was made a linear analysis and defined the area of the asymptotic stability of the model and the area in which there appears a stable periodical solution of one frequency. The analytic form of the mentioned solution was received using the bifurcation theory.

New Student Laboratory Work about Pulsational Phenomena in Astronomy

The pulsation phenomenon is inherent to most types of object and it plays a great role at certain stages in the evolution of objects in the universe. That is why students must study this phenomenon in the framework of laboratory hours. Often the study of these phenomena is reduced to an analysis of some differential equations with variable coefficients. A class of these equations is connected with the stability problem of the oscillations of self-gravitating systems (S. Nuritdinov, Sov. Astron., 1985, 29, 293). In order to carry out this laboratory work every student is required to compose a computer program using the periodical solution stability method and the parameter resonance theory. The program will find the critical amplitude of the pulsation and some dependences between physical parameters.