Showing papers on "Infinite-period bifurcation published in 2002"
TL;DR: In this paper, the Hopf bifurcation behavior of a symmetric rotor/seal system was investigated using Muszynska's non-linear seal fluid dynamic force model.
64 citations
07 Aug 2002
TL;DR: In this article, the dynamic voltage stability of a power system is analyzed and compared in terms of the reduced and unreduced Jacobian matrix of the system, whose eigenstructure matches well with the reduced one; and thus can be used for bifurcation analysis.
Abstract: The dynamic of a large class of power systems can be represented by parameter dependent differential-algebraic models of the form x/spl dot/ = f (x, y, p) and 0 = g(x, y, p). When the parameter p of the system (such as load of the system) changes, the stable equilibrium points may lose their dynamic stability at local bifurcation points. The systems will lose its stability at the feasibility boundary, which is caused by one of three different local bifurcations: the singularity induced bifurcation, saddle-node and Hopf bifurcation. In this paper, the dynamic voltage stability of a power system is introduced and analyzed. Both the reduced and unreduced Jacobian matrix of the system are studied and compared. It is shown that the unreduced Jacobian matrix, whose eigenstructure matches well with the reduced one; and thus can be used for bifurcation analysis. In addition, the analysis avoids the singularity induced infinity problem, which may happen at reduced Jacobian matrix analysis, and is more computationally attractive.
50 citations
TL;DR: In this paper, a method for analysis of stochastic processes is presented which makes feasible the extraction of deterministic and random components of process dynamics directly from data, applied to time series from metal cutting.
39 citations
01 Feb 2002
TL;DR: In this paper, the qualitative behavior of a class of ratio-dependent predator-prey systems with delay at the equilibrium in the interior of the first quadrant is studied, and it is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions.
Abstract: Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions.AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14
37 citations
TL;DR: In this article, slow varying, conservative, one degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation, where a homoclinic orbit connects to a nonhyperbolic saddle point.
31 citations
TL;DR: In this paper, the existence of Hopf bifurcation points for a non-linear Hamiltonian system with hysteresis was studied and sufficient conditions for their existence were given.
Abstract: We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl--Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity.
27 citations
TL;DR: In this article, the bifurcation structure of a simple power-system model with respect to changes to both the real and reactive loads is investigated, with a Bogdanov-Takens point and hence homoclinic orbits; these orbits can be of Sil'nikov type with many coexisting periodic solutions.
Abstract: The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reactive loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov-Takens bifurcation point and hence homoclinic orbits; these orbits can be of Sil'nikov type with many coexisting periodic solutions. We may use the bifurcation calculations to divide the two-parameter plane into a number of regions, for which there are qualitatively different dynamics. We classify and further investigate the dynamical behavior in each of these regions, using a Monte Carlo method to investigate basins of attraction of various stable states. We then show how this classification can be used to denote each regions as either safe or unsafe with respect to the likelihood of voltage collapse.
26 citations
TL;DR: In this article, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homocalic orbit and 1-periodic orbit near Γ were studied.
Abstract: In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points.
25 citations
TL;DR: In this article, the authors derived a system of bifurcation equations for a potential system expressing local behavior at a hilltop point and derived the probability density function of critical loads that follows a Weibull-like distribution under the assumption that initial imperfections are subject to a multi-variate normal distribution.
16 citations
TL;DR: In this paper, the response of two nonlinearly coupled van der Poloscillators to a principal parametric excitation in the presence of one-to-one internal resonance is investigated.
Abstract: The response of a system of two nonlinearly coupled van der Poloscillators to a principal parametric excitation in the presence ofone-to-one internal resonance is investigated. The asymptoticperturbation method is applied to derive the slow flow equationsgoverning the modulation of the amplitudes and the phases of the twooscillators. These equations are used to determine steady-stateresponses, corresponding to a periodic motion for the starting system(synchronisation), and parametric excitation-response andfrequency-response curves. Energy considerations are used to studyexistence and characteristics of limit cycles of the slow flowequations. A limit cycle corresponds to a two-period amplitude- andphase-modulated motion for the van der Pol oscillators. Two-periodmodulated motion is also possible for very low values of the parametricexcitation and an approximate analytic solution is constructed for thiscase. If the parametric excitation increases, the oscillation period ofthe modulations becomes infinite and an infinite-period bifurcationsoccur. Analytical results are checked with numerical simulations.
16 citations
TL;DR: In this paper, the authors investigated the Hopf bifurcation, chaos and asynchronous operation of a simple power system and showed that the chaos attractor breaks up into asynchronous operation when the resistance becomes larger.
Abstract: In this paper, the nonlinear phenomenon known as Hopf bifurcation, chaos and asynchronous operation of a simple power system are explored. Firstly, taking into account the nonlinearity of the generator shaft and the interaction of mechanics and electrics in the generator sets, the authors obtain a transient model by combining Park equations and mechanics equations. Then the Hopf bifurcation, period-doubling bifurcation and chaos caused by too large a line resistance are investigated with nonlinear mode and Floquet theory. The bifurcation figure of the system is also given. Further study shows that the chaos attractor breaks up into asynchronous operation when the resistance becomes larger. This way of loss-of-stability is different from that caused by loss-of-excitation.
Journal Article•
TL;DR: In this article, limit cycles in a cubic system were found and their distributions were studied by using the methods of bifurcation theory and qualitative analysis, and four limit cycles were identified.
Abstract: This paper concerns with limit cycles in a cubic system. Four limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis.
01 Jan 2002
TL;DR: In this article, the authors discuss discrete mechanical structures that when deformed may come into frictionless unilateral contact with rigid obstacles, and the presence of contact constraints is shown to introduce additional types of critical points, which they may label as non-smooth bifurcation points, corner limit points and end points.
Abstract: The present lecture notes discuss discrete mechanical structures that when deformed may come into frictionless unilateral contact with rigid obstacles. Since arbitrarily large displacements are considered, the structures may buckle, i.e. exhibit instabilities. Classically, for a structure not subjected to unilateral contact, critical points (where the stability behaviour of the structure may change character or bifurcation may occur) are divided into limit points and (smooth) bifurcation points. The presence of contact constraints is shown to introduce additional types of critical points, which we may label as non-smooth bifurcation points, corner limit points and end points.
TL;DR: In this article, the weakly nonlinear stability characteristics of short porous journal-bearing systems are presented on the basis of the Brinkman model, by applying the Hopf bifurcation theory.
TL;DR: In this paper, the authors studied the bifurcation of limit cycles in one-parameter unfoldings of quadractic dif- ferential systems in the plane having a degenerate critical point at the center.
Abstract: We investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic dif- ferential systems in the plane having a degenerate critical point at innit y. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from innit y together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then ex- panded in a Puiseux series with respect to the small parameter and its coefcients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous centerS3. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.
TL;DR: In this paper, a stability analysis of a non-linear prey-predator system under the influence of one dimensional diffusion has been investigated to determine the nature of the bifurcation point of the system.
Abstract: A stability analysis of a non-linear prey-predator system under the influence of one dimensional diffusion has been investigated to determine the nature of the bifurcation point of the system. The non-linear bifurcation analysis determining the steady state solution beyond the critical point enables us to determine characteristic features of the spatial inhomogeneous pattern arising out of the bifurcation of the state of the system.
10 Dec 2002
TL;DR: In this article, the authors provided an analytical condition under which bifurcation occurs, based on characteristics and position of the stable and unstable manifolds, and the exact shape of the region of attraction is provided.
Abstract: Bifurcation of the region of attraction for planar systems with one stable and one unstable pole under saturated linear state feedback is considered The boundary of the region of attraction can either possess an unbounded hyperbolic shape or be a bounded limit cycle The main contribution of this paper is to provide an analytical condition under which bifurcation occurs This condition is based on characteristics and position of the stable and unstable manifolds Furthermore, the exact shape of the region of attraction is provided
TL;DR: In this article, a closed-form parametrization of these branching solutions is given and the saddle-center bifurcation is studied in detail, along with two families of reversible orbits that are homoclinic to each one.
Abstract: In this paper, we study circular orbits of the J2 problem that are confined to constant-z planes They correspond to fixed points of the dynamics in a meridian plane It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them
TL;DR: In this paper, the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined, and the parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case.
Abstract: In this brief the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined. For an excitation of sufficient amplitude or frequency a Hopf bifurcation to a steady-state limitcycle is shown to result, appearing as a precursor to instability. The parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case. For weakly damped systems, the Hopf bifurcation can additionally exhibit a dependence on initial conditions. The resulting case of competing point and periodic attractors is discussed.
TL;DR: In this paper, the effect of bifurcation on the semi-active optimal control problem with fixed final state by using the cell mapping method is studied. And the global changes in the optimal control solution for different targets are studied.
TL;DR: In this paper, the saddle-node bifurcation with the coalescence of two unstable periodic orbits is studied using the codimension-two bifurbcation, which is a common phenomenon in nonlinear systems.
Abstract: A saddle-node bifurcation with the coalescence of a stable periodic orbit and an unstable periodic orbit is a common phenomenon in nonlinear systems. This study investigates the mechanism of producing another saddle-node bifurcation with the coalescence of two unstable periodic orbits. The saddle-node bifurcation results from a codimension-two bifurcation that a period doubling bifurcation line tangentially intersects a saddle-node bifurcation line in a parameter plane. Based on the bifurcation theory, the saddle-node bifurcation with the coalescence of two unstable periodic orbits is studied using the codimension-two bifurcation.
TL;DR: A bifurcation scenario that corresponds to disappearance of a saddle-node cycle on the Klein bottle filled with homoclinic trajectories of this cycle is constructed and a classification of bIfurcation scenarios that shows which of them can be realized in the unimodal case is given.
Abstract: Nonlocal bifurcations of vector fields on the Klein bottle are studied. The problem is to construct a bifurcation scenario that corresponds to disappearance of a saddle-node cycle on the Klein bottle filled with homoclinic trajectories of this cycle. For the global Poincare map specified by a unimodal function, a complete description of bifurcation scenarios is obtained. The bifurcation scenario corresponding to an arbitrary unimodal function is written out. Also, a classification of bifurcation scenarios that shows which of them can be realized in the unimodal case is given.
Journal Article•
TL;DR: In this article, the authors consider a general approach to the analysis of the behavior of dynamical systems depending on a parameter the variation of which eventually leads to the loss of stability or disappearance of a steady-state mode.
Abstract: We consider a general approach to the analysis of the behavior of dynamical systems depending on a parameter the variation of which eventually leads to the loss of stability or disappearance of a steady-state mode. Quantitative characteristics of such effects are calculated for a self-sustained oscillation system with dry friction. Along with the cases of instantaneous and quasistatic change of the parameter, we consider a scenario referred to as diagnostic. This scenario involves additional specific perturbations of the state variables, which enables one to assess the closeness of the system to a bifurcation state and to formulate a bifurcation prediction criterion convenient for practical utilization.
Journal Article•
TL;DR: In this paper, the authors applied implicit theory to convert 2D problem into 1D one, and then computed Taylor expansion of the bifurcation function and analyzed its zero poins.
Abstract: First by applying implicit theory to convert 2 dimension problem into 1 dimension one, then by computing Taylor expansion of the bifurcation function and analyzing its zero poins, certain conditions were given under which the period doubling bifurcation happens. At the end, by analyzing the eigenvalue of map f 2 at 2 periodic points the stability of then were obtained.
TL;DR: In this article, a time-dependent bifurcation model and its control problem are studied and an important mechanism for dynamical hysteresis and pulsing oscillation in parametric control systems is revealed.
Abstract: A time-dependent bifurcation model and its control problem are studied. Firstly, delayed bifurcating transition phenomena with memory effects of the model with time-dependent parameters varying in either the positive or the negative direction are analysed. Secondly, a parametric control problem with feedback for the time-dependent model is investigated. The existence and stability of dynamical hysteresis cycles are obtained by qualitative analysis of bifurcation and stability. Finally, an important mechanism for dynamical hysteresis and pulsing oscillation in parametric control systems is revealed as the result of delayed bifurcating transitions when the bifurcation parameter varies periodically across the steady bifurcation value.
TL;DR: In this article, an averaging method for nonlinear systems with time delay has been proposed to analyze the Hopf bifurcation curve and homoclinic bifurbation curve.
Abstract: This paper studies bifurcation set of a nonlinear system with time delay. It is known that it is difficult to investigate this system by analytical methods, because this system is described by a difference-differential equation which is usually difficult to solve. To analyze this kind of a system, we have already introduced an averaging method for the functional differential equation. The previous work showed that this averaging method is effective for nonlinear system with time delay. Applying this averaging method, this paper studies bifurcation set of this kind of a system with a fundamental harmonic response. The result shows that this system have a simple harmonic motion, when a delay is small and an angular frequency of external force is large. Furthermore, it is shown that this system have a derivative point from which the Hopf bifurcation curve, a saddle-node bifurcation curve and homoclinic bifurcation curve are derived.