Showing papers on "Bifurcation diagram published in 1988"

From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis

Abstract: Rather than enjoying a good PDF next a cup of coffee in the afternoon, instead they juggled in the same way as some harmful virus inside their computer. from equilibrium to chaos practical bifurcation and stability analysis is genial in our digital library an online admission to it is set as public so you can download it instantly. Our digital library saves in multipart countries, allowing you to get the most less latency epoch to download any of our books past this one. Merely said, the from equilibrium to chaos practical bifurcation and stability analysis is universally compatible afterward any devices to read.

A simple method for the calculation of postcritical branches

Abstract: The practical behaviour of problems exhibiting bifurcation with secondary branches cannot be studied in general by using standard path‐following methods such as arc‐length schemes. Special algorithms have to be employed for the detection of bifurcation and limit points and furthermore for branch‐switching. Simple methods for this purpose are given by inspection of the determinant of the tangent stiffness matrix or the calculation of the current stiffness parameter. Near stability points, the associated eigenvalue problem has to be solved in order to calculate the number of existing branches. The associated eigenvectors are used for a perturbation of the solution at bifurcation points. This perturbation is performed by adding the scaled eigenvector to the deformed configuration in an appropriate way. Several examples of beam and shell problems show the performance of the method.

Bifurcation Analysis of a Predator-Prey System Involving Group Defence

Abstract: A class of ODEs of generalized Gause type modeling predator-prey interaction is considered. The prey are assumed to exhibit a phenomenon called group defence, that is, predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase.Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a Hopf bifurcation. Conditions (that hold even in the case of no group defence) that ensure a subcritical Hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair (one stable and one unstable) of periodic orbits are given.Ecological ramifications are considered. Unlike the classical model, sufficient enrichment of the environment combined with group defence leads to extinction of the predator (deterministically) for almost all initial conditions, providing strong support fo...

A group-theoretic approach to computational bifurcation problems with symmetry

Abstract: Bifurcation of solution branches in static or steady problems of nonlinear mechanics is often associated with the underlying symmetry of the physical system. The use of group-theoretic methods in local bifurcation theory for problems with symmetry is well known. In this paper it is shown that the exploitation of symmetry via group invariance also yields an efficient computational approach to global bifurcation problems. These techniques are illustrated in the analysis of a lattice-dome structure with hexagonal symmetry. The methodology leads to a drastic reduction in numerical effort in the determination of several global solution branches, and enables the accurate computation of numerous singular points.

Bifurcations in finite element models with a non-associated flow law

Abstract: A numerical approach to bifurcation problems in soil mechanics is described. After locating the bifurcation point by a combination of an incremental-iterative loading procedure and an eigenvalue analysis of the tangent stiffness matrix, the solution is continued on the localization path by a suitable combination of the fundamental solution and the eigenvector belonging to the lowest eigenvalue. The procedures are applied in a bifurcation analysis of a cohesionless soil in a biaxial testing device. The results suggest that a diffuse bifurcation mode with a short wavelength is encountered whereupon a shear band gradually develops. The inclination angle of the shear band compares well with analytical formulae and with empirical data.

Phase Transitions and Their Bifurcation Analysis in a Large Population of Active Rotators with Mean-Field Coupling

Abstract: Bifurcation structures associated with the onset of macroscopic rhythms in a large population of active rotators are analyzed theoretically and numerically. The phase transition involving hyster· esis found numerically in a previous work are investigated in greater detail with particular attention to codimension·two bifurcation points. As a result, unexpectedly rich bifurcation structures are revealed. '

Noise and bifurcations

Abstract: The influence of while noise on bifurcating dynamical systems is investigated using both Fokker-Planck and functional integral methods. Noise leads to fuzzy bifurcations where physically relevant quantities become smooth functions of the bifurcation parameters. We study dynamical and probabilistic quantities, such as invariant measures, Liapunov exponents, correlation functions, and exit times. The behavior of these quantities near the deterministic bifurcation point changes for distinct values of the control parameter. Therefore the very concept of bifurcation point becomes meaningless and must be replaced by the notion of bifurcation region.

Homoclinic and heteroclinic bifurcations of Vector fields

Abstract: We study a bifurcation of homoclinic and heteroclinic orbits in a two or more parameter family of autonomous ODEs, where the unperturbed system has two heteroclinic orbits joined at a common saddle point Under some assumptions on eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system, we can show that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system A bifurcation diagram is given for such families Some homoclinic bifurcations are also treated including the one producing a twice-rounding homoclinic orbit

Iterates of maps with symmetry

Abstract: In this paper the elementary aspects of bifurcation of fixed points, period doubling, and Hopf bifurcation for iterates of equivariant mappings are discussed. The most interesting of these is an algebraic formulation of the hypotheses of Ruelle’s theorem (D. Ruelle [1973], “Bifurcations in the presence of a symmetry group,” Arch. Rational Mech. Anal., 51, pp. 136–152) on Hopf bifurcation in the presence of symmetry.In the last sections this result is used to show that Hopf bifurcation from standing waves in a system of ordinary differential equations with $O(2)$ symmetry can lead directly to motion on an invariant 3-torus; indeed, depending on the exact symmetry of the standing waves, one might expect to see three invariant 3-tori emanating from such a bifurcation. The unexpected third frequency comes from drift along the torus of standing waves whose existence is forced by the $O(2)$ symmetry.

Generalized Hopf Bifurcation and Its Dual Generalized Homoclinic Bifurcation

Abstract: We show the duality between the generalized Hopf bifurcation (GHB) and the generalized homoclinic bifurcation (GHB*). This duality is twofold: (1) the Poincare normal forms at a weak focus and at a weak saddle, (2) the bifurcation diagrams of the GHB and the GHB*. Since the GHB is well known, the GHB* may be considered as the heart of the article.

K-theoretic methods in bifurcation theory

Abstract: We introduce to a new approach to bifurcation using the index bundle of the linearization at the trivial branch

On degenerate Hopf bifurcation with broken O(2) symmetry

Abstract: A symmetry-breaking Hopf bifurcation in an O(2)-equivariant system generally produces a branch of standing waves and two branches of oppositely propagating travelling waves. This generic bifurcation assumes three non-degeneracy conditions on the cubic terms of the Poincare-Birkhoff normal form. When these conditions fail more complicated behaviour accompanies the bifurcation; in particular one finds secondary bifurcations of quasiperiodic waves. For these degenerate bifurcations, the effects of perturbations which break the reflection symmetry are considered. The perturbed system retains a residual SO(2) symmetry. Qualitatively these perturbations have three effects: (1) they split the double multiplicity eigenvalues to that the travelling waves bifurcate separately, (2) they perturb the primary standing wave branches to secondary branches of modulated waves and (3) they produce new steady-state bifurcations along the modulated wave branches.

Organization of periodic orbits in the driven Duffing oscillator.

Abstract: The global organization of the periodic orbits in the periodically driven Duffing oscillator is described in terms of the relative rotation rates of pairs of orbits We compare this organization with that determined by the second return of a horseshoe map and find a complete agreement, up to the global torsion, for all orbit pairs and all the regions of the bifurcation diagram studied In addition, we characterize topologically different flows in the Duffing family by their global torsion

Hopf bifurcation and the stability of non-linear age-dependent population models

Abstract: The possibility of Hopf bifurcation into stable orbits is considered for the Gurtin-MacCamy model of age-dependent population dynamics, in which the mortality function depends only on the population while fertility is a fairly general function of age as well as population. In addition an algorithm is produced which provides a necessary condition for Hopf bifurcation to occur for arbitrary systems of ordinary differential equations.

Multiparameter bifurcation diagrams in predator-prey models with time lag

Abstract: A predator-prey model is considered in which prey is limited by the carrying capacity of the environment, and predator growth rate depends on past quantities of prey. Conditions for stability of an equilibrium, and its bifurcation are established taking into account all the parameters.

Experiments on bifurcation of periodic states into tori for a periodically forced chemical oscillator

Abstract: We study experimentally continuous transitions from quasiperiodic to periodic states for a time‐periodically forced chemical oscillator. The chemical reaction is the hydration of 2,3‐epoxy‐1‐propanol, and is carried out in a continuous stirred tank reactor (CSTR). Periodic oscillatory states are observed to arise in the autonomous system through supercritical Hopf bifurcations as either the total flow rate or the cooling coil temperature is changed. Under conditions of oscillation for the autonomous system, small‐amplitude periodic variation of the total flow rate generates an attracting two‐torus from the stable limit cycle. From the experiments we determine the structure of the toroidal flow, stroboscopic phase portraits, and circle maps as a function of the forcing amplitude and period. A continuous transition from the quasiperiodic to a periodic state, in which the two‐torus contracts to a closed curve (Neimark–Sacker torus bifurcation), is observed as the forcing amplitude is increased at a constant forcing period, or as the forcing period is changed at a constant moderate forcing amplitude. Qualitative theoretical predictions compare well with the experimental observations. This paper presents the first experimental observation of a Neimark–Sacker torus bifurcation in a forced chemical oscillator system, and relates the bifurcation diagram of the unforced system to that of the forced system.

*Forced Oscillations of a Self-Oscillating Bimolecular Surface Reaction Model

Abstract: The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

Modulated rotating waves in O(2) mode interactions

Abstract: The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.

On the stationary distribution of self-sustained oscillators around bifurcation points

Abstract: A double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones Systems undergoing Hopf bifurcations belong to this class As an illustrative example, the generalized van der Pol oscillator is considered around its bifurcation point A calculation is carried out up to third order in both the noise intensity and the bifurcation parameter (damping coefficient)

Systematics of the Periodic Windows in the Lorenz Model and its Relation with the Antisymmetric Cubic Map

Abstract: The Lorenz model and the antisymmetric cubic map enjoy the same discrete symmetry. A careful study of a one-dimensional bifurcation diagram obtained numerically from the Lorenz model reveals that the systematics of periodic windows is closely related to that of the cubic map. In addition, we determined the drift of the fundamental frequency and thus proposed a nomenclature for the periodic windows by associating an absolute period to each window, which in turn agrees with the corresponding word made of three letters for most of the observed periods.

A global index for bifurcation of fixed points.

Abstract: 3~ is called the set of trivial fixed points. We are interested in connected branches of fixed points of / bifurcating from y. Let ^--={(λ, x) e 0\\y\\f(^ x) = x} be the set of nontrivial fixed points of /. 28 -— F n y is the set of bifurcation points. If J^ is compact we define an index BI(/) which lies in the (k — l)-stem π^_ χ =co fc_1(pt); ω^ the stable homotopy theory (see G. W. Whitehead [25], Chapter 12). For small values of k these groups are well known. For example, o>0(pt)^Z, co1(pt)^Z2, co2(pt)^Z2. If ΒΙ(/)ΦΟ then a connected subset £f of y exists which bifurcates from y and is not contained in any compact subset of &. There is a striking analogy to the fixed point index (or the Brouwer-Leray-Schauder degree), where the fixed points correspond to the bifurcation points. In particular, BI (/) is homotopy invariant and additive.

Bifurcation of periodic orbits for non-positive definite hamiltonian systems

Abstract: We consider the bifurcations of periodic solutions of a family of non–positive definite Hamiltonian systems of n degrees of freedom near the origin as the family passes through a semisimple resonance, We begin with a smooth Hamiltonian H with a general semisimple quadratic part H2 and then construct a normal form of H with respect to Hg up to fourth order terms and make a versal deformation. We apply the Liapunov–Schmidt reduction in the presence of symmetry and further reduce the resulting bifurcation equation to a gradient system. Thus, the study of periodic solutions of the original system is reduced to finding critical points of a real-valued function. As an application, we consider a system with two degrees of freedom in 1:−1 semisimple resonance by using suitable choices of the parameters to study the bifurcation as the eigenvalues split along the imaginary axis or across it and we obtain complete bifurcation patterns of periodic orbits on each energy level

Topological equivalence of bifurcation problems

Abstract: A number of models have been developed, using singularity theory, for analysing the bifurcation of solutions to nonlinear problems as parameters vary. In all such models, there appear moduli, which parametrise families with continuously varying bifurcation behaviour. However, if one investigates the qualitative behaviour of the bifurcation branching, then, with the exception of certain special values, such families are qualitatively the same and the moduli effectively disappear. Such a qualitative investigation can be carried out by considering the 'topological equivalence' of bifurcation problems. The author considers such topological equivalence and derives sufficient conditions that: (1) bifurcation problems are topologically equivalent (and so describe the same qualitative branching behaviour for solutions); (2) bifurcation problems are topologically determined by a particular part of their Taylor expansions; and (3) moduli parameters are topologically redundant (and can be ignored) in 'universal models', which are models describing all possible bifurcation phenomena in a given problem.