Showing papers on "Infinite-period bifurcation published in 2013"

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

Abstract: In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.

Subcritical bifurcation and bistability in thermoacoustic systems

Abstract: This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Abstract: Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

Bifurcation control of a congestion control model via state feedback

Abstract: This paper proposes to use a state feedback method to control the Hopf bifurcation for a novel congestion control model, i.e. the exponential random early detection (RED) algorithm with a single link and a single source. The gain parameter of the congestion control model is chosen as the bifurcation parameter. The analysis shows that in the absence of the state feedback controller, the model loses stability via the Hopf bifurcation early, and can maintain a stationary sending rate only in a certain domain of the gain parameter. When applying the state feedback controller to the model, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the model possesses a stable sending rate in a larger parameter range. Furthermore, explicit formulae to determine the properties of the Hopf bifurcation are obtained. Numerical simulations are given to justify the validity of the state feedback controller in bifurcation control.

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Abstract: Given a continuous family of C2 functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.

Bifurcation\instability forms of high speed railway vehicles

Abstract: The China high speed railway vehicles of type CRH2 and type CRH3, modeled on Japanese high speed Electric Multiple Units (EMU) E2 series and Euro high speed EMU ICE3 series possess different stability behaviors due to the different matching relations between bogie parameters and wheel profiles. It is known from the field tests and roller rig tests that, the former has a higher critical speed while large limit cycle oscillation appears if instability occurs, and the latter has lower critical speed while small limit cycle appears if instability occurs. The dynamic model of the vehicle system including a semi-carbody and a bogie is established in this paper. The bifurcation diagrams of the two types of high speed vehicles are extensively studied. By using the method of normal form of Hopf bifurcation, it is found that the subcritical and supercritical bifurcations exist in the two types of vehicle systems. The influence of parameter variation on the exported function Rec 1(0) in Hopf normal form is studied and numerical shooting method is also used for mutual verification. Furthermore, the bifurcation situation, subcritical or supercritical, is also discussed. The study shows that the sign of Re(λ) determinates the stability of linear system, and the sign of Rec 1(0) determines the property of Hopf bifurcation with Rec 1(0)>0 for supercritical and Rec 1(0)<0 for subcritical.

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

Abstract: In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.

Bifurcation analysis of an endogenous growth model

Abstract: This paper analyzes the dynamics of a variant of Jones (2002) semi-endogenous growth model within the feasible parameter space. We derive the long-run growth rate of the economy and do a detailed bifurcation analysis of the equilibrium. We show the existence of codimension-1 bifurcations (Hopf, Branch Point, Limit Point of Cycles, and Period Doubling) and codimension-2 (Bogdanov–Takens and Generalized Hopf) bifurcations within the feasible parameter range of the model. It is important to recognize that bifurcation boundaries do not necessarily separate stable from unstable solution domains. Bifurcation boundaries can separate one kind of unstable dynamics domain from another kind of unstable dynamics domain, or one kind of stable dynamics domain from another kind (called soft bifurcation), such as bifurcation from monotonic stability to damped periodic stability or from damped periodic to damped multiperiodic stability. There are not only an infinite number of kinds of unstable dynamics, some very close to stability in appearance, but also an infinite number of kinds of stable dynamics. Hence subjective prior views on whether the economy is or is not stable provide little guidance without mathematical analysis of model dynamics. When a bifurcation boundary crosses the parameter estimatesʼ confidence region, robustness of dynamical inferences from policy simulations are compromised, when conducted, in the usual manner, only at the parametersʼ point estimates.

Bautin bifurcation in a delay differential equation modeling leukemia

Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with l 1 = 0 and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation.

The number of limit cycles in a z2-equivariant liénard system

Abstract: In this paper, we consider the problem of limit cycle bifurcation near center points and a Z2-equivariant compound cycle in a polynomial Lienard system. Using the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we found some new and better lower bounds of the maximal number of limit cycles for this system.

The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems Without Sliding

Abstract: Planar discontinuous piecewise linear systems with two linearity zones, one of them being of focus type, are considered. By using an adequate canonical form under certain hypotheses, the bifurcation of a limit cycle, when the focus changes its stability after becoming a linear center, is completely characterized. Analytic expressions for the amplitude, period and characteristic multiplier of the bifurcating limit cycle are provided. The studied bifurcation appears in real world applications, as shown with the analysis of an electronic Wien bridge oscillator without symmetry.

Bifurcation of limit cycles in a quintic system with ten parameters

Abstract: In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of the computer algebra system MATHEMATICA, the first 11 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 11 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems. The results of Jiang et al. (Int. J. Bifurcation Chaos 19:2107–2113, 2009) are improved.

Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

Abstract: We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of A s = 3.19 and Re = 2.47 × 10 5 , where the saddle–node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio A s is caused by the saddle–node bifurcation.

On damping in the vicinity of critical points.

Abstract: The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added Oscillators with saddle–node, pitchfork and transcritical bifurcations are shown analytically to exhibit several interesting characteristics in the free decay response near a bifurcation A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results A transcritical bifurcation was selected because it may be used to represent generic bifurcation behaviour It is shown that the damping ratio can be used to predict a change in the stability with respect to changing system parameters

Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

Abstract: Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism

Abstract: In this paper, a predator-prey system with sex-structure and sexual favoritism is considered. Firstly, the impact of the sexual favoritism coefficient on the stability of the ordinary differential equation (ODE) model is studied. By choosing sexual favoritism coefficient as a bifurcation parameter, it is shown that a Hopf bifurcation can occur as it passes some critical value, and the stability of the bifurcation is also considered by using an analytical method. Secondly, the impact of the time delay on the stability of the delay differential equation (DDE) model is investigated, where time delay is regarded as a bifurcation parameter. It is found that a Hopf bifurcation can occur as the time delay passes some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived. Numerical simulations are performed to support theoretical results and some complex dynamic behaviors are observed, including period-halving bifurcations, period-doubling bifurcations, high-order periodic oscillations, chaotic oscillation, fast-slow oscillation, even unbounded oscillation. Finally, a brief conclusion is given. MSC:34K13, 34K18, 34K60, 37D45, 37N25, 92D25.

Dynamical behaviors of the periodic parameter-switching system

Abstract: In this paper, a periodic parameter-switching system about Lorenz oscillators is established. To investigate the bifurcation behavior of this system, Poincare mapping of the whole system is defined by suitable local sections and local mappings. The location of the fixed point and the parameter values of local bifurcations are calculated by the shooting method and Runge–Kutta method. Then based on the Floquent theory, we conclude that the period-doubling and saddle-node bifurcations play an important role in the generation of various periodic solutions and chaos. Meanwhile, upon the analysis of the equilibrium points of the subsystems, we explore the mechanisms of different periodic switching oscillations.

Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting

Abstract: The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.

Limit cycle bifurcations in a class of quintic Z 2-equivariant polynomial systems

Abstract: In this paper, we study a class of cubic Z 2-equivariant polynomial Hamiltonian systems under the perturbation of Z 2-equivariant polynomial of degree 5. First, we consider the unperturbed system and obtain necessary and sufficient conditions for the critical point (0,1) to be a nilpotent saddle, center, or cusp. We show that it can have 14 different phase portraits. Using the methods of Hopf and homoclinic bifurcation theory, we study the bifurcation problem of the perturbed system and prove that there exist 12 limit cycles.

Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems

Abstract: Using the Andronov-Hopf bifurcation theorem and the Poincare-Bendixson Theorem, this paper explores robust cyclical possibilities in a generalized Kolmogorov-Lotka-Volterra class of models with positive intraspecific cooperation in the prey population. This additional feedback effect introduces nonlinearities which modify the cyclical outcomes of the model. Using an economic example, the paper proposes an algorithm to symbolically construct the topological normal form of Andronov-Hopf bifurcation. In case the limit cycle turns out to be unstable, the possibilities of the dynamics converging to another limit cycle is explored.

A Critical Eigenvalues Tracing Method for the Small Signal Stability Analysis of Power Systems

Abstract: The continuation power flow method combined with the Jacobi-Davidson method is presented to trace the critical eigenvalues for power system small signal stability analysis. The continuation power flow based on a predictor- corrector technique is applied to evaluate a continuum of steady state power flow solutions as system parameters change; meanwhile, the critical eigenvalues are found by the Jacobi-Davidson method, and thereby the trajectories of the critical eigenvalues, Hopf bifurcation and saddle node bifurcation points can also be found by the proposed method. The numerical simulations are studied in the IEEE 30-bus test system.

Bifurcation behaviors of the two-state variable friction law of a rock mass system

Abstract: Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincare maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.

Existence and uniqueness and stability of limit cycle for a class of planar nonlinear systems

Abstract: In this paper, a class of planar nonlinear differential systems x = y(1+sin2mx), y = -x+δy+axy+bx3+cx2y+λx4ex2y is studied. By the formal series method based on Poincare ideas, the center and the focus are judged, and by the Dulac function, the non-existence of closed orbits is discussed. Meantime, by the Hopf bifurcation theory, some sufficient conditions for the existence of limit cycles which bifurcate from the equilibrium point are analyzed, then by some proper transforms, and by the theorem of L.A.Cherkas and L.I.Zheilevych, some sufficient conditions for the uniqueness and stability of limit cycles for such systems are established. Finally, one example is given for illustration.