Showing papers on "Infinite-period bifurcation published in 1998"
TL;DR: In this paper, it is shown that the dynamical behavior of the disturbed system remains the same for all parameter values, regardless of the intensity of the disturbance, and that for any parameter value all solutions converge to each other almost surely (uniformly in bounded sets).
Abstract: In the deterministic pitchfork bifurcation the dynamical behavior of the system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed points split the state space into two macroscopically distinct regions. It is shown here that this bifurcation of the dynamical behavior disappears as soon as additive white noise of arbitrarily small intensity is incorporated the model. The dynamical behavior of the disturbed system remains the same for all parameter values. In particular, the system has a (random) global attractor, and this attractor is a one-point set for all parameter values. For any parameter value all solutions converge to each other almost surely (uniformly in bounded sets). No splitting of the state space into distinct regions occurs, not even into random ones. This holds regardless of the intensity of the disturbance.
151 citations
TL;DR: In this paper, a set of results of detecting nonlinear phenomena appearing in a turbine generator power system with series-capacitor compensation was described based on the Floquet theory as well as the Hopf bifurcation theorem.
Abstract: This paper describes a set of results of detecting nonlinear phenomena appearing in a turbine generator power system with series-capacitor compensation. The analysis was based on the Floquet theory as well as the Hopf bifurcation theorem. After the first Hopf bifurcation, the stable limit cycle bifurcates to a stable torus and an unstable limit cycle which connects to a stable limit cycle by a supercritical torus bifurcation. The stable limit cycle joins with an unstable limit cycle at a cyclic fold bifurcation. This unstable limit cycle is connected to the second Hopf. It has been also numerically demonstrated that such a strange sequence of periodic orbits is created by a q-axis damper winding.
28 citations
TL;DR: In this paper, computer simulations of an electric power system have been carried out to determine the possible bifurcations which lead to voltage collapse, and it is confirmed that an unstable limit cycle causes these bifurbcations for almost all cases.
Abstract: Computer simulations of an electric power system have been carried out to determine the possible bifurcations which lead to voltage collapse. Depending on the value of a capacitor, three kinds of the bifurcations are observed; blue sky disappearance of a chaotic attractor, of a limit cycle (via cyclic fold), or of a stable equilibrium (via subcritical Hopf) leads to voltage collapse. It is confirmed that an unstable limit cycle causes these bifurcations for almost all cases. A variety of nonlinear phenomena, including chaotic attractors and multiple coexisting limit cycles, are observed. From the point of view of an electric power system, voltage collapse studied in this paper will be the most possible phenomenon as compared with previous studies because condition of the system before voltage collapse is closer to a normal operation.
28 citations
TL;DR: In this article, the authors present results of a numerical study of a free-interface problem modelling self-propagating high-temperature synthesis (solid combustion) in a one-dimensional infinite medium.
Abstract: This paper presents results of a numerical study of a free-interface problem modelling self-propagating high-temperature synthesis (solid combustion) in a one-dimensional infinite medium. Evolution of the free interface exhibits a remarkable range of dynamical scenarios such as finite and infinite sequences of period doubling; the latter leading to chaotic oscillations, reverse sequences and infinite period bifurcation that may replace the supercritical Hopf bifurcation for some interface kinetics. Solutions were verified by using different numerical methods, including reduction to an integral equation for which convergence to the solutions has been demonstrated rigorously. Therefore, the ability of the free-interface model to generate the dynamical scenarios observed previously in models with a distributed reaction rate should be regarded as firmly established.
28 citations
TL;DR: In this paper, a bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North Western American Power System.
Abstract: A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System The system has five mechanical and two electrical modes The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation As a result, the power system oscillates subsynchronously with a small limit-cycle attractor As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe Consequently, there are no bounded motions The results show that adding damper windings may induce subsynchronous resonance
28 citations
TL;DR: In this article, two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models, are discussed.
Abstract: Considering a family of two-dimensional piecewise linear maps, we discuss two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models. It is shown that, in the case of so-called contact bifurcation of the 2 nd kind, the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor. In the case of so-called contact bifurcation of the 1 st kind, the reunion is a result of a contact of the attractor with its basin boundary which is fractal, including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary
26 citations
TL;DR: In this article, the existence of limit cycles in the Kaldor-Kalecki model is discussed by using the theory of dynamical systems and especially Hopf bifurcations.
23 citations
TL;DR: In this paper, a numerical study of natural convection in a two-dimensional container of unity aspect ratio with unstable temperature distributions on the side walls and adiabatic top and bottom walls is discussed for a Boussinesq fluid with unity Prandtl number.
23 citations
TL;DR: It is shown how more than one eigenvalue can diverge in an analogous manner, with applications in electrical power systems, as well as proving singularity-induced bifurcation.
Abstract: It is well known that an equilibrium of a semi-explicit, index-1 differential-algebraic equation under a parameter variation may encounter the singularity manifold. It is a generic property of this encounter that one eigenvalue of the linear stability mapping associated with the equilibrium will pass from one half of the complex plane to the other without passing through the imaginary axis. This is known as singularity-induced bifurcation and an equivalent result is proven in this paper. While this property is generic, it is shown how more than one eigenvalue can diverge in an analogous manner, with applications in electrical power systems.
22 citations
TL;DR: In this paper, a bifurcation analysis is used to investigate the complex dynamics of a heavily loaded turbine-generator system connected to an infinite busbar through a series capacitor-compensated transmission line.
20 citations
TL;DR: In this paper, the bifurcation of limit cycles in Lienard systems is studied and a sequence of algebraic approximations are obtained in the parameter space, each algebraic approximation represents an exact lower bound to the BIFURcation set.
TL;DR: In this article, the transient characteristics of a nonequilibrium phase transition in a model of abiased van der Pol oscillator were investigated, and the state-independent driving term which triggers the bifurcation from limit cycle to fixed point was treated as a randomly fluctuating quantity.
Abstract: The transient characteristics of a nonequilibrium phase transition is investigated in a model of abiased van der Pol oscillator. The state-independent driving term which triggers the bifurcation from limit cycle to fixed point is treated as a randomly fluctuating quantity. The advancement of the Hopf bifurcation is explained as a result of noise-induced periodicity found in this model system. The phase boundary separating the two attractors is determined numerically and is interpreted as stochastic bifurcation locus in parameter space. The phenomenon of critical slowing down occurring on the fixed point side is found to be similar to that which occurs in a deterministic system. The relevent critical exponent is estimated to have the mean field value of unity, irrespective of how the stochastic bifurcation points are approached in a two-dimensional parameter space.
TL;DR: In this article, the experimental bifurcation analysis of the electrochemical steady state and oscillatory behavior of the Fe 14.8 M H 3 PO 4 system was studied by using as control parameters the applied potential, the Fe-disc rotation speed and an external resistance.
TL;DR: In this paper, the authors studied dynamic bifurcations in deformation from a uniform state of rapid extension for a rectangular block and showed that the long wavelength modes are suppressed because their rate of formation is too slow compared to the background deformation.
Abstract: Dynamic bifurcations in deformation from a uniform state of rapid extension are studied for a rectangular block. The block may be viewed as a segment of a thin shell which is one period of a deformation which is periodic along the circumference. The material is taken to be incompressible and characterized by an incrementally linear, rate-independent elastic-plastic constitutive law. The stress-state in the block prior to bifurcation is approximately uniform and the effect of bifurcation mode inertia is taken into account. The bifurcation problem is formulated as a two-variable problem in terms of a constant stress and a variable representing the rate of growth of a bifurcation mode, here called the localization speed. A qualitative lower limit on growth of a diffuse mode in terms of the localization speed and a limit on the background motion due to the requirement of subsonic deformation are suggested. Bifurcation stress-localization speed distributions are computed and important changes from the quasistatic bifurcation behavior responsible for the phenomenon of multiple necking are shown to exist. In particular, it is found that the long wavelength modes are suppressed because their rate of formation is too slow compared to the background deformation. The main focus is on diffuse bifurcations; however, other bifurcation phenomena are touched upon by way of incremental wave propagation arguments.
TL;DR: In this article, the authors studied the bifurcation in which a cycle branches off from an equilibrium in a continuous one-parameter family of equilibria, as the parameter passes through a critical value.
TL;DR: In this paper, an asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2bifurcations in terms of the system control parameters.
Abstract: Oscillations described by autonomous three-dimensional differential equation systems display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario, the key instability is usually an initial bifurcation from a single period oscillation to its subharmonic of period two, or the reverse. An asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2 bifurcations in terms of the system control parameters.
TL;DR: In this article, the dynamics of weakly non-linear, single-degree-of-freedom, harmonically excited Hamiltonian systems as a bifurcation parameter varies slowly across the points of instability.
Abstract: Many non-linear dynamical systems are characterized by bifurcation parameters which vary with time This paper presents an analytical framework to analyze the dynamics of weakly non-linear, single-degree-of-freedom, harmonically excited Hamiltonian systems as a bifurcation parameter varies slowly across the bifurcation points, or points of instability Formal results concerning the slowly varying normal forms of such systems in the neighborhood of a bifurcation point, are derived The use of matched asymptotic expansions and non-linear boundary layers, in analyzing the slowly varying normal forms is briefly summarized The simplified boundary layer equations capture all the essential “local” dynamics of the original Hamiltonian system during transition across a bifurcation The developed theory is illustrated through detailed analyses of the dynamics of two classical Hamiltonian systems: the forced, undamped, non-linear Mathieu equation and the forced, undamped Duffing equation as the excitation frequency slowly varies ( non-stationary excitation ) across the points of instabilities (simple bifurcations) in these systems
TL;DR: In this paper, a new type of onset of chaos in a multidimensional dissipative system is reported that has been observed under a circumstance in which a delicate balance of order-formation and chaos is materialized.
TL;DR: In this article, the existence of infinitely many points of pitchfork bifurcation, and the global properties of the solution curves were studied for several classes of boundary value problems.
TL;DR: In this article, the effect of Gaussian white noise (type P noise) on low flow rate chaos in both systems far from and close to bifurcation points using the three-variables ODE of the Belousov-Zhabotinsky reaction developed by Gyorgyi and Field was investigated.
Abstract: We investigate systematically the effect of Gaussian white noise (type P noise) on the low flow rate chaos in both systems far from and close to bifurcation points using the three-variables ODE of the Belousov-Zhabotinsky reaction developed by Gyorgyi and Field When the noise is added to the chaos with the bifurcation parameter far from bifurcation points, the chaos trajectories are slightly scattered However, in the chaos having the bifurcation parameter near bifurcation points, it happens that topological entropy is constant but the Lyapunov exponent decreases We have found that this phenomenon, named “noise-induced order”, appears in intermittent chaos with the internal structure of m -periodic oscillation, and that “noise-induced order” is caused by an increase in the length of laminar region and the subsequent change of the invariant density
01 May 1998
TL;DR: In this article, the delay is regarded as the lag due to gestation and the criteria for existence of interior equilibrium and its global stability are derived. But the delay was not considered in this paper.
Abstract: In this study, we have considered a prey-predator model reflecting the predator interference with discrete time delay. This delay is regarded as the lag due to gestation. In absence of delay, the criteria for existence of interior equilibrium and its global stability are derived. By choosing the delay as a bifurcation parameter, we have shown that a Hopf bifurcation may occur when the delay passes its critical value. Finally, we have derived the criteria for stability switches and verified the results through computer simulation.
TL;DR: In this article, necessary and sufficient conditions for the occurrence of a bifurcation in the equilibrium path of a discrete structural system are established as a consequence of the degeneracy of the solution of the rate problem at a critical point.
Abstract: Necessary and sufficient conditions for the occurrence of a bifurcation in the equilibrium path of a discrete structural system are established as a consequence of the degeneracy of the solution of the rate problem at a critical point. Such result is based on the properties of the elastic-plastic rate problem formulated as a linear complementarity problem (LCP) in terms of plastic multipliers (the moduli of the plastic strain rate vectors) as basic unknowns. The conditions here given allow to distinguish, both theoretically and practically, among bounded bifurcations, unbounded bifurcations, limit points, and unloading points. All of the needed quantities depend either on the starting situation or on the actual known term increment; there is no need to compute eigenvalues or eigenvectors of stiffness matrices. The results obtained can be seen as a refinement, for the discrete elastic-plastic problem, of the uniqueness theory given by Hill. The refinement allows covering the case of vector-valued yield fun...
TL;DR: In this paper, a stochastic system with a vanishing diffusion at deterministic fixed points is analytically studied, where the system is locked at one of the deterministic stable or unstable fixed points with probabilities which depend on the noise intensity and an initial condition.
Abstract: A stochastic system with a vanishing diffusion at deterministic fixed points is analytically studied. Above the critical noise intensity, the noise stabilizes the deterministic unstable fixed points. The system is locked at one of the deterministic stable or unstable fixed points with probabilities which depend on the noise intensity and an initial condition. An example is studied numerically showing the bifurcation of locked states.