The Dynamics and Bifurcation Control of a Singular Biological Economic Model

The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.

DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A FRACTIONAL ORDER SINGULAR LESLIE-GOWER PREY-PREDATOR MODEL

In this article,we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model,which describes the interaction bet ween populations of prey and predator,and takes into account the economic interest.We firstly obtain the solvability condition and the st ability of the model sys tem,and discuss the singularity induced bifurcation phenomenon.Next,we introduce a st ate feedback controller to elimina te the singularity induced bifurcation phenomenon,and discuss the optimal control problems.Finally,numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.

Dynamical Behavior and Singularities of a Single-machine Infinite-bus Power System

This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.

A SINGULAR BIOECONOMIC MODEL WITH DIFFUSION AND TIME DELAY

This paper studies a prey-predator singular bioeconomic system with time delay and diffusion, which is described by differential-algebraic equations. For this system without diffusion, there exist three bifurcation phenomena： Transcritical bifurcation, singularity induced bifurcation, and Hopf bifurcation. Compared with other biological systems described by differential equations, singularity induced bifurcation only occurs in singular system and usually links with the expansion of population. When the diffusion is present, it is shown that the positive equilibrium point loses its stability at some critical values of diffusion rate and periodic oscillations occur due to the increase of time delay. Furthermore, numerical simulations illustrate the effectiveness of results and the related biological implications are discussed.

Association of SIB Points with the Non-Degenerate Equilibria of the Extended DAE System

The stability of differential-algebraic equations (DAEs) was analyzed using singularity induced bifurcation (SIB) with one parameter. This kind of bifurcation arises in parameter-dependent DAEs having the form x·=f, 0=g. Extended DAE system reduction is introduced as a convenient method to compute the SIB points. Non-degeneracy conditions on the function g are needed. Aften verifying these conditions, the extended DAE system can be solved as an ODE by applying the implicit function theorem near the equilibrium point of the extended DAE system. These equilibrium points in turn include the SIB points of the original DAEs. The study of SIB points enables analysis of power system stability problems.