Value Of Bifurcation Parameter Research Articles

The flow curvature method applied to canard explosion

The aim of this work is to establish that the bifurcation parameter value leading to a canard explosion in dimension 2 obtained by the so-called geometric singular perturbation method can be found according to the flow curvature method. This result will be then exemplified with the classical Van der Pol oscillator.

Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian

The Jacobian elliptic functions are generalized and applied to bifurcation problems associated with p-Laplacian. The values of bifurcation parameter and the corresponding solutions are represented in terms of common parameters, and a complete description of the bifurcation diagram and a closed form representation of the corresponding solutions are obtained. As a by-product of the representation, it turns out that a kind of solution is also a solution of an eigenvalue problem of p / 2 -Laplacian.

Effects of Colored Noise on Periodic Orbits in a One-Dimensional Map

Noise can induce inverse period-doubling transition and chaos. The effects of the colored noise on periodic orbits, of the different periodic sequences in the logistic map, are investigated. It is found that the dynamical behaviors of the orbits, induced by an exponentially correlated colored noise, are different in the mergence of transition, and the effects of the noise intensity on their dynamical behaviors are different from the effects of the correlation time of noise. Remarkably, the noise can induce new periodic orbits, namely, two new orbits emerge in the period-four sequence at the bifurcation parameter value μ = 3.5, four new orbits in the period-eight sequence at μ = 3.55, and three new orbits in the period-six sequence at μ = 3.846, respectively. Moreover, the dynamical behaviors of the new orbits clearly show the resonancelike response to the colored noise.

PREDATOR-PREY IN PATCHY SPACE WITH DIFFUSION

In this paper we formulate a predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other one. Numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, i. e. the stable constant steady state loses its stability and spatially non-constant stationary solutions, a pattern emerge.

The nature of Hopf bifurcation for the Gompertz model with delays

In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays τ ̄ = τ 1 / τ 2 . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of τ ̄ . The function describing stability of HB is periodic with respect to τ ̄ . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data.

Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

In this paper, we investigate the generalization of the Camassa–Holm equation u t + K ( u m ) x − ( u n ) x x t = [ ( ( u n ) x ) 2 2 + u n ( u n ) x x ] x , where K is a positive constant and m , n ∈ N . The bifurcation and some explicit expressions of peakons and periodic cusp wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. Further, in the process of obtaining the bifurcation of phase portraits, we show that K = m + n 1 + n c n − m + 1 n is the peakon bifurcation parameter value for the equation. From the bifurcation theory, in general, the peakons can be obtained by taking the limit of the corresponding periodic cusp waves. However, we find that in the cases of n ≥ 2 , m = n + 1 , when K tends to the corresponding bifurcation parameter value, the periodic cusp waves will no longer converge to the peakons, instead, they will still be the periodic cusp waves. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature. By further studying the cause of this phenomenon, we show that this planar system has some different characters from the previous Camassa–Holm systems. What is more, we obtain some periodic cusp wave solutions in the form of polynomial functions, which are different from those in the form of exponential functions. Some previous results are extended.

Switching Exponent Scaling near Bifurcation Points for Non-Gaussian Noise

We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.

A hyperbolic Lindstedt–Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators

A hyperbolic Lindstedt–Poincaré method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Liénard oscillator is studied in detail, and the present method’s predictions are compared with those of Runge– Kutta method to illustrate its accuracy.

Decoherence of qubits interacting with a nonlinear dissipative classical or quantum system

Quantum trajectories given by the quantum state diffusion theory are used to study the dynamics of a pair of qubits interacting with the quantized Duffing oscillator. The latter is a nonlinear dissipative and driven quantum system that in the semiclassical limit displays typical bifurcations and chaotic behavior of the classical model. It is shown that the oscillator part of the qubits-oscillator system in the semiclassical limit is well described by the classical model despite the oscillator-qubit interaction. On the other hand, exact quantum dynamics of the qubits pair is completely different from the classical model as long as there is non-negligible qubit-oscillator interaction. Dynamics of the interqubits entanglement and the qubits pair von Neumann entropy is studied in the deeply quantum and various degrees of semiclassical regimes and for different values of the oscillator's bifurcation parameter. It is concluded that the decoherence of the qubits pair by the dissipative nonlinear oscillator is more effective when the oscillator is in the more classical regime, and if the semiclassical oscillator dynamics is chaotic rather than regular.

An operator method for studying regular bifurcations in multiparameter systems

In this paper, we present the basic postulates of a new method for studying a large class of bifurcation problems with multidimensional degenerations. The method makes it possible to detect bifurcation parameter values; it leads to an iteration procedure for approximately studying problems depending on many parameters. Applications to the theory dynamical systems are also discussed.

Bifurcations of a class of singular biological economic models

This paper studies systematically a prey–predator singular biological economic model with time delay. It shows that this model exhibits two bifurcation phenomena when the economic profit is zero. One is transcritical bifurcation which changes the stability of the system, and the other is singular induced bifurcation which indicates that zero economic profit brings impulse, i.e., rapid expansion of the population in biological explanation. On the other hand, if the economic profit is positive, at a critical value of bifurcation parameter, the system undergoes a Hopf bifurcation, i.e., the increase of delay destabilizes the system and bifurcates into small amplitude periodic solution. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here. In addition, we study numerically that the system undergoes a saddle-node bifurcation when the bifurcation parameter goes through critical value of positive economic profit.

Bifurcations in nonlinear integral models of biological systems

A bifurcation analysis is suggested for nonlinear integral models of age-distributed biological populations. The analysis shows that the integral model of one-species population with intraspecies competition has zero and positive stationary states for some values of a bifurcation parameter. The nontrivial positive stationary state is initially stable and becomes unstable as the parameter grows. The obtained results are discussed and compared with the corresponding results in differential and difference models.

Investigation of subharmonic ferroresonant oscillations in power systems

Steady state bifurcation diagram based on continuation technique is a versatile tool for analyzing the behaviour of ferroresonance circuit, since it gives the visualization of multiple solutions. Stability analysis of the solutions obtained by steady state bifurcation diagram [K. Al-Anbarri, R. Ramanujam, T. Keerthiga, K. Kuppusamy, Analysis of non-linear phenomenon in MOV connected transformer, IEE Proc. Generation Transmission Distrib.148 (6) (2001) 562–566; K. Al-Anbarri, R. Ramanujam, R. Saravanaselvan, K. Kuppusamy, Effect of iron core loss nonlinearity on the chaotic ferroresonance in power transformers, Electric Power Syst. Res. 65 (2003) 1–12; K. Al-Anbarri, R. Ramanujam, Ch. Subba Rao, K. Kuppusamy, Effect of circuit configuration on chaotic ferroresonance in a power transformer, Electric Power Components Syst. 30 (2002) 1015–1031] demarcates stable fundamental ferroresonant solution, unstable fold and flip segments. Flip segment in steady state bifurcation diagram [K. Al-Anbarri, R. Ramanujam, T. Keerthiga, K. Kuppusamy, Analysis of nonlinear phenomenon in MOV connected transformer, IEE Proc. Generation Transmission Distrib.148 (6) (2001) 562–566; K. Al-Anbarri, R. Ramanujam, R. Saravanaselvan, K. Kuppusamy, Effect of iron core loss nonlinearity on the chaotic ferroresonance in power transformers, Electric Power Syst. Res. 65 (2003) 1–12; K. Al-Anbarri, R. Ramanujam, Ch. Subba Rao, K. Kuppusamy, Effect of circuit configuration on chaotic ferroresonance in a power transformer, Electric Power Components Syst. 30 (2002) 1015–1031] implies unstable fundamental solution, but does not reveal the range, order and stability of subharmonics. This paper proposes an approach to explore the unstable flip segment further and unearth the hidden subharmonics. As a consequence of this approach, continuum of stable solutions of well-defined domains can be identified. It is found that the existence of subharmonic solutions is sensitive to various factors such as transformer saturation index, non-linearity in the core loss and presence of an arrester. Exhaustive studies reveal that in the presence of an arrester, subharmonic mode exists for wider range of bifurcation parameter value at a high value of core saturation index, there by reducing the range of chaotic attractors.

Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics

Abstract A nonlinear stochastic dynamical model on a typical HAB algae diatom and dianoflagellate densities was created and presented in this paper. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. The singular boundary theory of diffusion process and the invariant measure theory were applied in analyzing the bifurcation of stability and the Hopf bifurcation of the stochastic system. The critical value of the stochastic Hopf bifurcation parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov–Schmidt reduction and the Crandall–Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.

Guaranteeing dynamic performance in nonlinear non-smooth power systems

Performance specifications place restrictions on the dynamic response of many systems, including power systems. It is often convenient to know parameter values that induce limiting behaviour, Le., force the system to just meet the specification. This limiting behaviour is a form of grazing bifurcation, and can be formulated as a boundary value problem. The paper develops a numerical shooting method for computing such bifurcation parameter values, and an optimization algorithm for finding bifurcation values that are closest to a given parameter set.

PEAKONS AND THEIR BIFURCATION IN A GENERALIZED CAMASSA–HOLM EQUATION

Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation [Formula: see text] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

Peakons and periodic cusp waves in a generalized Camassa–Holm equation

We study the peakons and the periodic cusp wave solutions of the following equation: u t+2ku x−u xxt+auu x=2u xu xx+uu xxx with a,k∈ R , which we will call the generalized Camassa–Holm equation, or simply the GCH equation, for when a=3 it was the so-called CH equation given by R. Camassa and D.D. Holm [Phys. Rev. Lett. 71 (11) (1993) 1661–1664]. They showed that the CH equation has a class of new solitary wave solutions called “peakons”. J.P. Boyd [Appl. Math. Comput. 81 (2–3) (1997) 173–187] studied another class of new periodic wave solutions called “coshoidal waves”. Using the bifurcation method of the phase plane, we first construct peakons and show that a=3 is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter a or the wave speed c varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.

Comparative analysis of approximate and precise mapping for a «bouncing ball»

Approximate and precise mappings for a bouncing ball are presented. The results of comparison of dynamics showed by these mappings are given: phase portraits, dynamical regimes topography charts, bifurcation parameter values etc. It is shown that the mapping of a bouncing ball in its traditional form occupies an intermediate position between the physically motivated and formal models, and in certain areas of parameter space it should be regarded as a formal model.

Bifurcation of switched nonlinear dynamical systems

This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincare mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we analyze the behavior of a two-dimensional nonlinear autonomous system whose state space is constrained on two half planes concerned with state dependent switching characteristics. From investigation of bifurcation diagrams, we conclude that the tangent and global bifurcations play an important role for generating various periodic solutions and chaos. Some theoretical results are confirmed by laboratory experiments.