Showing papers on "Bifurcation diagram published in 1995"

Asymmetry and Hopf bifurcation in spherical Couette flow

Abstract: Spherical Couette flow is studied with a view to elucidating the transitions between various axisymmetric steady‐state flow configurations. A stable, equatorially asymmetric state discovered by Buhler [Acta Mech. 81, 3 (1990)] consists of two Taylor vortices, one slightly larger than the other and straddling the equator. By adapting a pseudospectral time‐stepping formulation to enable stable and unstable steady states to be computed (by Newton’s method) and linear stability analysis to be conducted (by Arnoldi’s method), the bifurcation‐theoretic genesis of the asymmetric state is analyzed. It is found that the asymmetric branch originates from a pitchfork bifurcation; its stabilization, however, occurs via a subsequent subcritical Hopf bifurcation.

Remarks on food chain dynamics

Abstract: The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold and transcritical bifurcation curves of equilibria as well as flip, fold and transcritical bifurcation curves of limit cycles. The appearance of chaos in the model is proved to be related to a Hopf bifurcation and a degenerate homoclinic bifurcation in the prey-predator subsystem. The boundary of the chaotic region is shown to have a very peculiar structure.

Bifurcation for flow past a cylinder between parallel planes

Abstract: Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

On the bifurcation set of complex polynomial with isolated singularities at infinity

Abstract: © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Bifurcation, chaos, and voltage collapse in power systems

Abstract: A model of a power system with load dynamics is studied by investigating qualitative changes in its behavior as the reactive power demand at a load bus is increased. In addition to the saddle node bilification often associated with voltage collapse, the power system exhibits sub- and supercritical Hopf bifurcations, cyclic fold bifurcation, and period doubling bifurcation. Cascades of period doubling bifurcation terminate in chaotic invariant sets. The presence of these new bifurcations motivates a reexamination of the saddle-node bifurcation as the boundary of the feasible set of power injections.

Bifurcation and predictability analysis of a low-order atmospheric circulation model

Abstract: A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation The principal mechanisms leading to the appearance of complex dynamics around this bifurcation are described and various routes to chaotic behavior are identified, such as the transition through the period doubling cascade, the breakdown of an invariant torus and homoclinic bifurcations of a saddle-focus Non-trivial limit sets in the form of a chaotic attractor or a chaotic repeller are found in some parameter ranges Their presence implies an enhanced unpredictability of the system for parameter values corresponding to the winter season

Bifurcations of Attracting Cycles from Time-Delayed Chua's Circuit

Abstract: We study the bifurcations of attracting cycles for a three-segment (bimodal) piecewise-linear continuous one-dimensional map. Exact formulas for the regions of periodicity of any rational rotation number (Arnold’s tongues) are obtained in the associated three-dimensional parameter space. It is shown that the destruction of any Arnold’s tongue is a result of a border-collision bifurcation, and is followed by the appearance of a cycle of intervals with the same rotation number, whose dynamics is determined by a skew tent map. Finally, for the interval cycle the merging bifurcation corresponds to a homoclinic bifurcation of some point cycle.

Bifurcation analysis of two coupled periodically driven Duffing oscillators

Abstract: Bifurcation diagrams and phase diagrams of two coupled periodically driven identical Duffing oscillators are presented. It is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single, periodically driven, strictly dissipative oscillators The subpattern itself, however, is different from that of a single Duffing oscillator due, in particular, to Hopf bifurcations that are newly added to the bifurcation scenario.

Smallest chemical reaction system with Hopf bifurcation

Abstract: The smallest at most bimolecular chemical reaction system with Hopf bifurcation is presented. First the notion smallest reaction system is explained. Since the lowest number of intermediates has the highest priority in this characterization and since it has already been shown that three-component systems can have a Hopf bifurcation [1], the smallest reaction system must contain three intermediates. On the basis of a sufficient condition for a Hopf bifurcation in three-dimensional systems it is possible to find one reaction system which is according to the given characterization, the smallest one. In the first part of this paper it is shortly pictured and in the second part a more extensive proof that this system is really the searched smallest one is given.

A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: Application to a two-well oscillator

Abstract: An experimental study of local and global bifurcations in a driven two-well magneto-mechanical oscillator is presented. A detailed picture of the local bifurcation structure of the system is obtained using an automated bifurcation data acquisition system. Basins of attractions for the system are obtained using a new experimental technique: an ensemble of initial conditions is generated by switching between stochastic and deterministic excitation. Using this stochastic interrogation method, we observe the evolution of basins of attraction in the nonlinear oscillator as the forcing amplitude is increased, and find evidence for homoclinic bifurcation before the onset of chaos. Since the entire transient is collected for each initial condition, the same data can be used to obtain pictures of the flow of points in phase space. Using Liouville's Theorem, we obtain damping estimates by calculating the contraction of volumes under the action of the Poincare map, and show that they are in good agreement with the results of more conventional damping estimation methods. Finally, the stochastic interrogation data is used to estimate transition probability matrices for finite partitions of the Poincare section. Using these matrices, the evolution of probability densities can be studied.

Nonlocal continuum effects on bifurcation in the plane strain tension-compression test

Abstract: The paper examines nonlocal effects on bifurcation phenomena. A gradient plasticity model is used where a characteristic length is introduced in the yield criterion. Hill's well known framework of bifurcation theory is shown to hold in the presence of normality and a sufficient condition for uniqueness is given. Further, the regularizing effects of nonlocality are underlined. It is also shown that the underlying local continuum, obtained when the length scale goes to zero, always provides a lower bound for bifurcation stresses for the nonlocal continuum. Detailed analysis of bifurcation phenomena in the plane strain tension-compression test is carried out and compared to the results of Hill and Hutchinson for the local continuum. The results are qualitatively the same in the long wavelength domain while they differ markedly in the short wavelength domain. In this last case and in the elliptic regime, bifurcation modes disappear in tension while the corresponding stresses are significantly increased in the compressive regime.

Finite state machines and recurrent neural networks—automata and dynamical systems approaches

Abstract: We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine ℳ the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and ℳ. is established in the context of the algebraic theory of automata. In the second approach, the RNN is viewed as a set of discrete-time dynamical systems associated with input symbols of ℳ. In particular, issues concerning network representation of loops and cycles in the state transition diagram of ℳ. are shown to provide a basis for the interpretation of learning process from the point of view of bifurcation analysis. The circumstances under which a loop corresponding to an input symbol x is represented by an attractive fixed point of the underlying dynamical system associated with x axe investigated. For the case of two recurrent neurons, under some assumptions on weight values, bifurcations can be understood in the geometrical context of intersection of increasing and decreasing parts of curves defining fixed points. The most typical bifurcation responsible for the creation of a new fixed point is the saddle node bifurcation.

Nonlinear analysis of the stick-slip bifurcation in the creep-controlled regime of dry friction.

Abstract: We perform an experimental study of the amplitude and frequency shift of the stick-slip (SS) oscillations of a paper-on-paper block and spring system close to the SS--steady-sliding Hopf bifurcation. The confrontation of the experimental results with a weakly nonlinear analytical analysis of the dynamics and with direct numerical calculations yields very satisfactory agreement with the model of creep-controlled dry friction proposed by Heslot et al. [Phys. Rev. E 49, 4973 (1994)].

New methods for computing a saddle-node bifurcation point for voltage stability analysis

Abstract: This paper proposes methods for calculating saddle-node bifurcation points of power system power flow equations. The first method calculates a saddle-node bifurcation point along a given ray in the parameter space of power flow equations. By exploiting the special structure of power flow equations, the method calculates a saddle-node bifurcation point along a given ray as a solution to a constrained optimization problem. The constrained optimization can be solved efficiently with standard optimization methods. The second method calculates a locally closest saddle-node bifurcation point with respect to the operating point. This method uses an iterative process of computing a saddle-node bifurcation point along a ray, and then updating the direction of the ray for calculating a closer saddle-node bifurcation point. The method is a quasi-Newton method that updates the direction of a ray based on the first-order derivatives and the approximations to the second-order derivatives of the distance between saddle-node bifurcation points and the operating point. The paper compares the proposed methods with other methods on two test examples. >

Periodic orbits, bifurcation diagrams and the spectroscopy of C2H2 system

Abstract: The principal families of periodic orbits that emerge from the stationary points of the six‐dimensional potential energy surface of the C2H2 molecular system, as well as periodic orbits from saddle‐node bifurcations, have been located and propagated for an energy range up to 36 500 cm−1 above the absolute minimum of the potential. The bifurcation diagrams of these periodic orbits reveal the regions of phase space where the dynamics are regular or chaotic (with soft or hard chaos) for acetylene, vinylidene, and the region over these two isomers. An association of the structure of phase space with spectroscopic findings is made by calculating Gutzwiller’s semiclassical trace formula and classical survival probability functions.

Structure in the bifurcation diagram of the Duffing oscillator.

Abstract: We identify four levels of structure in the bifurcation diagram of the two-well periodically driven Duffing oscillator, plotted as a function of increasing control parameter T, the period of the driving term. The superstructure, or bifurcation peninsula, repeats periodically as T increases by \ensuremath{\sim}2\ensuremath{\pi}, beginning and ending with symmetric period-one orbits whose local torsions differ by 2. Within each bifurcation peninsula there is a systematic window structure. The primary window structure is due to Newhouse and Newhouse-like orbits. Fine structure is due to a Farey sequence of well-ordered orbits between the primary windows. Hyperfine structure consists of very narrow windows associated with non-well-ordered orbits. We construct a template for the Duffing oscillator, a two-dimensional return map, and a one-dimensional return map which describes the systematics of orbit creation and annihilation. All structures are identified by topological indices. Our predictions are based on, and compatible with, numerical computations.

Investigation of the bifurcation character of the H-mode in ASDEX Upgrade

Abstract: The so-called 'dithering cycles', ie periodic L-H-L transitions, are studied in the ASDEX Upgrade tokamak The cycle repetition frequency is approximately 2 kHz; high-resolution ECE measurements show that Tc changes during the cycle consistent with the assumption of the periodic build-up and destruction of the edge transport barrier characteristic of the H-mode A study of the conditions under which the cycles occur reveals that with the unfavourable Del B drift direction, the cycles are suppressed These observations are interpreted in a schematic bifurcation diagram; we find that the width of the bifurcation varies with the reversal of the Del B drift

The centres in the reduced Kukles system

Abstract: In this paper we consider the family of the cubic systems of Kukles (1944) with the condition that one of the parameters a7 is zero. Under this restriction the centre conditions were given by Kukles. The study of this family exhibits properties and issues which are important in the problem of the full classification of cubic systems with a centre. The family is formed of four strata, one of which is made up of quadratic systems and was studied before by Schlomiuk (1993). If we consider the three strata formed by truly cubic system we have a first (second) stratum consisting of systems symmetric with respect to the x-axis (y-axis) and a third stratum consisting of systems with two invariant straight lines and having an elementary first integral obtained by the Darboux method. Systems in either one of the symmetric strata do not possess elementary first integrals generically. The first stratum is formed by integrable systems having a Liouvillian first integral. We show that systems in the second stratum have no Liouvillian first integral. We give the full bifurcation diagram of each stratum of truly cubic systems.