Unstable limit cycles in an electric power system and basin boundary of voltage collapse

It has been reported that a saddle node bifurcation or a blue sky bifurcation causes voltage collapse in an electric power system. In these references, computer simulations are carried out with the voltage magnitude of the generator bus terminal held constant. The generator model described by differential equations of internal flux linkages allows the voltage magnitude of the generator bus terminal to change. By using this model, we have carried out computer simulations of the power system to determine the cause of voltage collapse. It is a cyclic fold bifurcation of the stable limit cycle caused by an unstable limit cycle that leads to the voltage collapse. The involvement of complicated sequences of unstable limit cycles with cyclic fold bifurcations is confirmed, and the voltage collapse which is caused by perturbation for steady states is related to these unstable limit cycles. This is very interesting from the point of view of a nonlinear problem. From the point of view of a power system, the power system will fluctuate in practice even in normal operation, and may sometimes operate beyond the limit of its stability in recent year. It is very important in this situation that we clarify bifurcations of limit cycles on the power system.     

Bifurcation and chaos in a Monod type food chain chemostat with pulsed input and washout

In this paper, we introduce and study a model of a Monod type food chain chemostat with pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat

In this paper, we introduce and study a model of a predator–prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Computing a closest bifurcation instability in multidimensional parameter space

Summary Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ₀, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ₀ to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ_* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ₀. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ₀ of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.     

Bifurcation control of subsynchronous resonance using TCSC

This paper presents the use of Thyristor-Controlled Series Capacitor (TCSC) to control bifurcations of subsynchronous resonance (SSR) in multimachine power system. The modified second system of the IEEE second benchmark model of subsynchronous resonance is considered. The dynamics of the damper windings, automatic voltage regulator (AVR), and power system stabilizer (PSS) on SSR in power system are included. In the case of neglecting TCSC, the results show that the operating point of the system loses stability via subcritical Hopf bifurcation. When we add TCSC to the system, all bifurcations are eliminated. Therefore, the Thyristor-Controlled compensated system never loses stability at any realistic firing angle.     

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Catastrophic bifurcations in a second-order dynamical system with application to acid rain and forest collapse

In this paper we present a second-order nonlinear dynamical system modelling the interactions of trees and damaging insects in a forest. With this model we discuss the influence of acidic deposition, an increase of which can cause sudden insect infestations and the collapse of the forest ecosystem. The analysis is carried out by finding the bifurcations of the system and by proving that under suitable conditions, such bifurcations can be catastrophic. The conditions for bifurcation can be explicitly given, and this facilitates the biological interpretation of the results.     

Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays

In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations.     

一类具有时滞和Holling Ⅲ型功能性反应的捕食模型的稳定性和Hopf分支

研究一类具有时滞和Holling Ⅲ型功能性反应的捕食模型的稳定性和Hopf分支.以滞量为参数,得到了系统正平衡点的稳定性和Hopf分支存在的充分条件,给出了确定Hopf分支方向和分支周期解的稳定性的计算公式.     

A Five-Dimensional System of Navier-Stokes Equation for a Two-Dimensional Incompressible Fluid on a Torus

A five-mode truncation of Navier-Stokes equation for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented, the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system are revealed.     

Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Stability and bifurcation analysis for a discrete-time model of Lotka-Volterra type with delay

A discrete model of Lotka-Volterra type with delay is considered, and a bifurcation analysis is undertaken for the model. We derive the precise conditions ensuring the asymptotic stability of the positive equilibrium, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark-Sacker bifurcations occur, but codimension 2 (fold-Neimark-Sacker, double Neimark-Sacker and resonance 1:1) bifurcations may also be present. The direction and the stability of the Neimark-Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.     

The dynamics of Bertrand model with bounded rationality

The paper considers a Bertrand model with bounded rational. A duopoly game is modelled by two nonlinear difference equations. By using the theory of bifurcations of dynamical systems, the existence and stability for the equilibria of this system are obtained. Numerical simulations used to show bifurcations diagrams, phase portraits for various parameters and sensitive dependence on initial conditions. We observe that an increase of the speed of adjustment of bounded rational player may change the stability of Nash equilibrium point and cause bifurcation and chaos to occur. The analysis and results in this paper are interesting in mathematics and economics.     

Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response

We perform a bifurcation analysis of a discrete predator-prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F₁,F₂ and F₃ and collect complete information on this in a single scheme. In the case of F₂ we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM. Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.     

Effects of Hard Limits on Bifurcation, Chaos and Stability

An SMIB model in the power systems,especially that concering the effects of hard limits onbifurcations,chaos and stability is studied.Parameter conditions for bifurcations and chaos in the absence ofhard limits are compared with those in the presence of hard limits.It has been proved that hard limits can affectsystem stability.We find that (1) hard limits can change unstable equilibrium into stable one;(2) hard limits canchange stability of limit cycles induced by Hopf bifurcation;(3) persistence of hard limits can stabilize divergenttrajectory to a stable equilibrium or limit cycle;(4) Hopf bifurcation occurs before SN bifurcation,so the systemcollapse can be controlled before Hopf bifurcation occurs.We also find that suitable limiting values of hard limitscan enlarge the feasibility region.These results are based on theoretical analysis and numerical simulations,such as condition for SNB and Hopf bifurcation,bifurcation diagram,trajectories,Lyapunov exponent,Floquetmultipliers,dimension of attractor and so on.     

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.     

Two-parameter bifurcations in a network of two neurons with multiple delays

We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.