Different Types Of Bifurcations Research Articles

High-order amplitude equation for steps on the creep curve

We consider a model proposed by one of the authors for a type of plastic instability found in creep experiments which reproduces a number of experimentally observed features. The model consists of three coupled non-linear differential equations describing the evolution of three types of dislocations. The transition to the instability has been shown to be via Hopf bifurcation leading to limit cycle solutions with respect to physically relevant drive parameters. Here we use reductive perturbative method to extract an amplitude equation of up to seventh order to obtain an approximate analytic expression for the order parameter. The analysis also enables us to obtain the bifurcation (phase) diagram of the instability. We find that while supercritical bifurcation dominates the major part of the instability region, subcritical bifurcation gradually takes over at one end of the region. These results are compared with the known experimental results. Approximate analytic expressions for the limit cycles for different types of bifurcations are shown to agree with their corresponding numerical solutions of the equations describing the model. The analysis also shows that high order nonlinearities are important in the problem. This approach further allows us to map the theoretical parameters to the experimentally observed macroscopic quantities.

Characterization of blowout bifurcation by unstable periodic orbits

Blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor, lying in some invariant subspace, becomes transversely unstable. We establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor. We argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits . There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. @S1063-651X~97!50902-0# PACS number~s!: 05.45.1b Recently, a novel type of bifurcation has been discovered in chaotic dynamical systems @1,2#. This is the so-called ‘‘blowout bifurcation’’ that occurs in dynamical systems with a symmetric invariant subspace. Let S be the invariant subspace in which there is a chaotic attractor. Since S is invariant, initial conditions in S result in trajectories that remain in S forever. Whether the chaotic attractor in S is also an attractor in the full phase space depends on the sign of the largest Lyapunov exponent L’ computed for trajectories in S with respect to perturbations in the subspace T which is transverse to S. When L’ is negative, S attracts trajectories transversely in the vicinity of S and, hence, the chaotic attractor in S is an attractor in the full phase space. If L’ is positive, trajectories in the neighborhood of S are repelled away from it and, consequently, the attractor in S is transversely unstable and it is hence not an attractor in the full phase space. Blowout bifurcation occurs when L’ changes from negative to positive values. There are distinct physical phenomena associated with the blowout bifurcation. For example, near the bifurcation point where L ’ is negative, if there are other attractors in the phase space, then typically, the basin of the chaotic attractor in S is riddled @3#. When L ’ is slightly positive, if there are no other attractors in the phase space, the dynamics in the transverse subspace T exhibits an extreme type of temporally intermittent bursting behavior, the on-off intermittency @4,5#. Recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems @6#. In the study of chaos theory, it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics @7‐9#. Periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems. For example, the period-doubling bifurcation @10# and the saddlenode bifurcation are bifurcations of periodic orbits. Catastrophic events in chaotic systems such as crises @11# and basin boundary metamorphoses @12# are triggered by collision of periodic orbits, usually of low period, embedded in different dynamical invariant sets. The birth of Wada basin boundaries, meaning common boundaries of more than two basins of attraction, is caused by a saddle-node bifurcation on the basin boundary @13#. More recent study indicates that the riddling bifurcation, bifurcation that gives birth to a riddled basin, is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in S @14#. In view of the role of periodic orbits played in these major bifurcations, it is desirable to study the blowout bifurcation by periodic orbits. In this regard, Ashwin, Buescu, and Stewart have noticed that as a system parameter changes towards the blowout bifurcation point, more and more atypical invariant measures become transversely unstable @2#. At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable. In this paper, we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace S .I n particular, we argue that near the bifurcation, there exist two groups of periodic orbits S s and S u , each having an infinite number of members, one transversely stable and another transversely unstable, respectively. The sign of the largest transverse Lyapunov exponent L’ is determined by the relative weights of S s and S u : L’ is negative ~positive! when S s ~S u! weighs over S u ~S s!. ~A precise definition of the ‘‘weights’’ will be described in the sequel. ! At the bifurcation, the weights of S s and S u are balanced. In contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits @10‐14#, blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits . The num

Parametric small signal amplification near pitchfork bifurcations

Abstract The nonlinear dielectric properties of ferroelectrics during polarization reversal are used to amplify small periodic perturbations. The ferroelectric serves as a nonlinear capacitor in a series-resonance circuit. The inductance is a linear one. Such a series-resonance circuit is a system which offers many features of nonlinear dynamical systems. It is possible to observe different types of bifurcations (pitchfork bifurcations, period-doubling cascades and chaotic behaviour) well known from the theory of nonlinear dynamical systems. Near such bifurcations the system can amplify suitable chosen periodic perturbations. These perturbations are coupled into the system by amplitude modulation of the driving voltage. Different types of bifurcations have been investigated concerning their ability of amplifying small signals.

The bifurcation of the unstable periodic orbits in bounded and unbounded three-disk billiards

The bifurcation of unstable periodic orbits (UPOS) in bounded and unbounded billiards are investigated. The billiard systems studied in this paper consist of three disks and have C3 nu symmetry. It is found numerically that these systems have essentially two different types of bifurcation for changing the structure of the UPOS. The first type of bifurcation is caused by the tangential collision of the trajectory with a convex boundary segment. The second type of bifurcation is caused by the collision of the trajectory with a vertex point, where two smooth boundary segments meet at a finite angle. The vertex points on the boundary play the central geometrical role of organizing the UPOS in these billiard systems.

Dynamic Stability and Chaos of System with Piecewise Linear Stiffness

The dynamic stability behavior of a single‐degree‐of‐freedom system with piecewise linear stiffness is considered. The analytical expression representing the divergence of perturbed trajectories is derived. The mechanism triggering dynamic instability of trajectories and the cause of chaotic behavior are then studied. Liapunov exponents are used as a quantitative measure of system stability. Numerical results including bifurcation diagrams and largest Liapunov exponents of a system with symmetric bilinear stiffness are presented. Several different types of bifurcation and nonlinear phenomena are identified, including pitchfork, fold, flip, boundary crisis, and intermittency of type 3. To illustrate the initial‐condition dependent nature of the problem, basins of attraction of multiple steady‐state responses are determined on the phase plane using the simple cell mapping method.

Local (Classical) And Global Bifurcations In Non-linear, Non-gradient Autonomous Dissipative Structural Systems

A general analysis for non-linear, non-gradient, multiple-parameter structural systems with or without damping described by autonomous ordinary differential equations is comprehensively presented. Attention is focused on bifurcational systems under follower loading exhibiting a trivial stable fundamental path from which a post-divergence or post-oscillatory instability may occur. Conditions for establishing regions of existence of adjacent equilibria, critical and stability conditions as well as different types of bifurcations (with zero, double zero and pure imaginary eigenvalues), for a smooth variation of the control parameter, are thoroughly explored and discussed. Global (dynamic) bifurcations (associated with stable limit cycles), irrelevant to any characteristic properties of the Jacobian eigenvalues, are discovered in a certain small neighborhood of a compound branching. It seems that such global bifurcations with trajectories passing through the saddles of the trivial path are due to the interaction of the first and second buckling modes which occur in the aforementioned small region of adjacent equilibria. Loading discontinuity phenomena for values of the control parameter defining the foregoing compound branching are detected. The effect of damping and other findings based on local (classical) analyses are compared with the results of this nonlinear analysis, and serious discrepancies are observed showing that a precise modelling must be generic; including both non-linearities and damping.The analysis is illustrated by using as a model Ziegler's classical model, the response of which is fully assessed.

Topological entropy and bifurcations of chaotic scattering.

Topological entropy is defined and its behavior investigated near bifurcations of chaotic scattering. The result shows that for different types of bifurcations the topological entropy exhibits distinct behavior.

On the solution of non-linear finite element equations

The paper deals with the basic requirements in the construction of a reliable continuation procedure. Adaptive step length determination and calculation of critical equilibrium states are discussed. For simple critical points an algorithm, which does not need classification between different types of bifurcations or even distinction between limit vs bifurcation point, is described. Situations where the extension of the parameter space could reveal vital information concerning the behaviour of the structure being analysed, are addressed.

Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the murray optimum

We have used X-ray microangiography to investigate the hypothesis that the potent endogenous vasodilator endothelium-derived relaxing factor (EDRF) contributes to the maintenance of "optimality" in vascular branching by modulating the diameters of the parent (D0) and daughter (D1 and D2) arteries at bifurcations. Five anatomically different types of bifurcation were studied in buffer-perfused rabbit ear preparations both under resting conditions and after pharmacological constriction by 5-hydroxytryptamine (5HT). A range of flow rates (1-5 ml min-1) was employed as release of EDRF from endothelial cells is stimulated by shear stress. Experimental data obtained in the presence and absence of EDRF activity were compared with theoretical predictions in three ways. (1) Junction exponents (x) were determined at each bifurcation from the equation Dx1 + Dx2 = Dx0, and their frequency distributions constructed. Murray (1926a, Proc. natn. Acad. Sci., U.S.A. 12, 207-214; 1926b, J. gen. Physiol. 9, 835-841.) proposed that x will be exactly 3 if power losses and intravascular volume are minimized simultaneously. In unconstricted preparations, either in the presence or absence of EDRF activity, and in preparations constricted by 0.1 microM 5HT in the presence of EDRF activity, the modes and medians of the frequency distributions of x were found to be close to 3 at all flow rates. In contrast, in 0.1 microM 5HT-constricted preparations in the absence of EDRF activity, no single mode common to all flow rates was apparent and medians were significantly larger at all flow rates. (2) Theoretically "optimal" branching angles were derived from experimental diameter measurements using four mathematical models which minimize respectively the total surface area, total volume, total drag (shear stress) and total power losses at bifurcations (Murray, 1926b). These calculated branching angles were then compared with actual branching angles. EDRF activity was found to be necessary for accurate prediction of branching angles by the minimum volume and power loss models in 5HT-constricted but not in resting preparations. (3) For each model or "minimization principle", there is an optimal mathematical relationship between the junction exponent, x, and the angle between daughter arteries, psi 12, at a bifurcation (Roy & Woldenberg, 1982, Bull. math. Biol. 44, 349-360.) Experimentally determined values of x and psi 12 agreed closely with those predicted both by the minimum volume and the minimum power loss principles, except again in 5HT-constricted preparations in the absence of EDRF activity.(ABSTRACT TRUNCATED AT 250 WORDS)

Bifurcations in chemical reactors via feedback system theory

By considering some examples of chemical reactors we present a study of different types of bifurcations (static, dynamic or Hopf and, mainly, their degeneracies) taking place in those systems when some distinguishable parameters are varied. The richness of the dynamics near those singular conditions is so varied that considerable work has been done to synthesize that particular behaviour in a general classification in the unfolding of the bifurcations. The present paper summarizes recent results and presents some new ones using an equivalent feedback configuration of the ordinary differential equation system and a graphical approach to study Hopf bifurcations. Moreover degenerate bifurcations of codimensions 2 and 3 are identified with this frequency domain approach.

A recursion formula for the number of stable orbits in the cubic map

A recursion formula for the number of stable cycles in the cubic map is given by analysing different types of bifurcations and investigating the stability of all periodic orbits.

Etude d’un modèle de rèpartition des débits dans le lit vasculaire

The authors try to precise the distribution of a Newtonian fluid at the level of a glass-tube bifurcation, a model grossly similar to blood vessel branches. Different types of bifurcations are obtained by means of translucent polyester resins. The length of the tubes is chosen in order not to alter the flow at the entrance and at the outlet (the length is approximately 90 times greater than the diameter). The fluid used is a mixture of water and glycerin; its viscosity is determined according to the temperature. The flow rates are measured by chronometry and the pressure with a pressure gauge made of capillary tubes. With certain hypotheses, the geometrical characteristics of the bifurcations permit the calculation of the induced charge losses; however, the shape of the area of influence and its evolution cannot be accurately precised.