Showing papers on "Bifurcation diagram published in 1983"

Computational Methods in Bifurcation Theory and Dissipative Structures

Abstract: 1. Introduction.- 1.1 General Introduction.- 1.2 Dissipative Structures in Physical, Chemical, and Biological Systems.- 1.2.1 The problems of elastic stability.- 1.2.2 Bifurcations to divergence and flutter in flow-induced oscillations.- 1.2.3 Wheelset nonlinear hunting problem.- 1.2.4 Buckling of a shallow elastic arch.- 1.2.5 Dissipative structures in fluid mechanics.- 1.2.6 Biological systems.- 1.2.7 Reaction-diffusion problems.- 1.3 Basic Concepts and Properties of Nonlinear Systems.- 1.3.1 Steady-state solutions.- 1.3.2 Stability of solutions.- 1.3.3 Evolution systems.- 1.4 Examples.- 2. Multiplicity and Stability in Lumped-Parameter Systems (LPS).- 2.1 Steady-State Solutions.- 2.2 Dependence of Steady-State Solutions on a Parameter-Solution Diagram.- 2.3 Stability of Steady-State Solutions.- 2.4 Branch Points-Real Bifurcation.- 2.4.1 Evaluation of limit and bifurcation points.- 2.4.2 Direction of branches at a bifurcation point.- 2.4.2.1 Selecting starting points for the continuation algorithm.- 2.4.2.2 Illustrative examples.- 2.4.2.3 Bifurcation points with higher degeneration.- 2.4.3 Occurrence of isolas, isola formation.- 2.5 Branch Points-Complex Bifurcations.- 2.5.1 The Hopf bifurcation theorem.- 2.5.2 Direct decomposition technique for location of the complex bifurcation point.- 2.5.3 Direct iteration techniques.- 2.6 Bifurcation Diagram.- 2.7 Transient Behavior of LPS-Numerical Methods.- 2.7.1 Runge-Kutta methods.- 2.7.2 Multistep methods.- 2.7.3 Integration along the solution arc.- 2.7.4 Integration of phase trajectories for autonomous systems.- 2.7.5 Numerical methods for stiff systems of ODE.- 2.7.6 Systems of differential and algebraic equations.- 2.7.7 Integration of differential equations with time delay.- 2.8 Computation of Periodic Solutions.- 2.8.1 Transformation into an initial-value problem-the shooting method..- 2.8.2 Stability of periodic solutions.- 2.8.3 Continuation of periodic solutions.- 2.8.4 Bifurcation of periodic solutions.- 2.9 Chaotic Attractors.- 2.9.1 Characterization of chaotic attractors.- 2.9.2 Liapunov exponents.- 2.9.3 Power spectra.- 2.9.4 The Poincare map.- 3. Multiplicity and Stability in Distributed-Parameter Systems (DPS).- 3.1 Steady-State Solutions-Methods for Solving Nonlinear Boundary-Value Problems.- 3.1.1 Finite-difference methods.- 3.1.2 Quasi-linearization.- 3.1.3 Shooting methods.- 3.2 Dependence of Steady-State Solutions on a Parameter.- 3.3 Branch Points-Methods for Evaluating Real and Complex Bifurcation Points.- 3.3.1 Primary bifurcation.- 3.3.2 Secondary real bifurcation.- 3.3.3 Secondary complex bifurcation.- 3.4 Methods for Transient Simulation of Parabolic Equations-Finite-Difference Methods.- 3.4.1 Nonlinearity approximation.- 3.4.2 Automatic control of time step k.- 3.4.3 Automatic control of spatial step size h, equidistant net.- 3.4.4 Adaptive nonequidistant net.- 4. Development of Quasi-stationary Patterns with Changing Parameter.- 4.1 Quasi-stationary Behavior in LPS-Examples.- 4.2 Quasi-stationary Behavior in DPS-Examples.- 5. Perspectives.- Appendix A DERPAR-A Continuation Algorithm.- Appendix B SHOOT-An Algorithm for Solving Nonlinear Boundary-Value Problems by the Shooting Method.- Appendix C Bifurcation and Stability Theory.- C. 1 Invariant Manifolds and the Center-Manifold Theorem (Reduction of Dimension).- C.2 Normal Forms.- C.3 Bifurcation of Singular Points of Vector Fields.- C.4 Codimension of a Vector Field. Unfolding of a Vector Field.- C.5 Construction of a Versal Deformation.- C.6 Bifurcations of Codimension 2.- C.7 Bifurcations from Limit Cycles.- References.

Fractal Basin Boundaries, Long-Lived Chaotic Transients, And Unstable-Unstable Pair Bifurcation

Abstract: A new type of bifurcation to chaos is pointed out and discussed. In this bifurcation two unstable fixed points or periodic orbits are created simultaneously with a strange attractor which has a fractal basin boundary. Chaotic transients associated with the coalescence of the unstable-unstable pair are shown to be extraordinarily long-lived.

Branch switching in bifurcation problems for ordinary differential equations

Abstract: The problem of switching branches in boundary-value problems of ordinary differential equations is considered. Three non-local methods for calculating emanating solutions near a nontrivial bifurcation point are proposed. These methods calculate one solution on an emanating branch (without a priori exact knowledge of the bifurcation point). Other solutions on the branch can be obtained by global continuation. The methods are convenient as they consist in solving boundary-value problems by standard software. The construction of an initial approximation of the emanating solution is outlined. A characteristic feature of the proposed methods is that they can be easily automated; the user can avoid nearly all preparatory work. The methods are tested on several examples arising in different application areas.

Introduction to Dynamic Bifurcation.

Abstract: : Dynamic bifurcation theory in differential equations is concerned with the changes that occur in the structure of the limit sets of solutions as parameters in the vector field are varied. For example, if the vector field is the gradient of a function with a finite number of critical points, then the omega-limit set of each orbit is an equilibrium point. Thus, one must be concerned with how the number of equilibrium points changes with the parameters (this is usually called static bifurcation theory), how the stability properties of the equilibrium points change and the manner in which the equilibrium points are connected to each other by orbits. If the vector field is not the gradient of a function, then other types of limiting motions can occur; for example, periodic orbits, invariant tori, homoclinic and heteroclinic orbits. The purpose of these notes is to give an introduction to the methods used in determining how these more complicated limit sets change as parameters vary. (Author)

Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics

Abstract: A local two parameter bifurcation theorem concerning the bifurcation from steady states of time periodic solutions of a nonlinear system of partial, integro-differential equations is proved. A Hopf bifurcation theorem is derived as a corollary. By means of independent and dependent variable changes this theorem is applicable to the general McKendrick equations governing the growth of an age-structured population (with the added feature here of a possible gestation period). The theorem is based on a Fredholm theory developed in the paper for the associated linear equations. An application is given to an age-structured population whose fecundity is density and age dependent and it is shown that for a sufficiently narrow age-specific “reproductive and resource consumption window” steady state instabilities, accompanied by sustained time periodic oscillations, occur when the birth modulus surpasses a critical value.

An explicit example of Hopf bifurcation in fluid mechanics

Abstract: It is observed that a complete and explicit example of Hopf bifurcation appears not to be known in fluid mechanics. Such an example is presented for the rotating Benard problem with free boundary conditions on the upper and lower faces, and horizontally periodic solutions. Normal modes are found for the linearization, and the Veronis computation of the wave numbers is modified to take into account the imposed horizontal periodicity. An invariant subspace of the phase space is found in which the hypotheses of the Joseph-Sattinger theorem are verified, thus demonstrating the Hopf bifurcation. The criticality calculations are carried through to demonstrate rigorously, that the bifurcation is subcritical for certain cases, and to demonstrate numerically that it is subcritical for all the cases in the paper.

Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory

Abstract: In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches. Introduction. In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is assumed to be perfectly flexible and is suspended between fixed supports. The distance between the supports is assumed to be greater than the natural length of the wire and therefore the wire will always be in tension. The magnetic field is assumed to be constant and is directed parallel to the line between the supports. We show that this problem can be solved exactly. The set of solutions exhibits the classic bifurcation phenomenon. The bifurcation parameter is X = IB where I is the current in the wire and B the strength of the magnetic field. For all values of X > 0 there is a trivial solution in which the wire remains straight. However at the eigenvalues of the problem obtained by linearizing the equilibrium equation about the trivial solution, bifurcation occurs and we obtain branches of nontrivial solutions. The situation is quite similar to the classical problem of the buckling of a beam. Actually in our case the branches of solutions are two-dimensional manifolds because the problem is invariant under rotation about the axis of the supports. This work was motivated by the discussion in [1] (see ?4.7, especially Figure 29 and the reference quoted therein). There are two cases to consider as the wire is (elastically) homogeneous or not. Of course it is of interest to study the stability of the equilibria with respect to the (hyperbolic) equations of motion. This is in general a very difficult problem and in the present case it is complicated even further by the multiplicity of equilibria arising from the rotational invariance. In the case of the homogeneous wire we can study the operator obtained by linearization about a nontrivial solution. We are able to locate Received by the editors June 16, 1982. 1980 Mathematics Subject Classification. Primary 73C50; Secondary 58E07. (D1983 American Matheniatical Society 0002-9947/82/0000-1374/$03.50 377 This content downloaded from 157.55.39.180 on Mon, 25 Apr 2016 06:09:45 UTC All use subject to http://about.jstor.org/terms

Homoclinic bifurcation of perturbed reversible systems

Abstract: In this contribution certain types of semilinear second order equations are studied in a Hilbert space. The vector field has a quasi accretive linearization, i. e. it is strictly accretive up to a finite dimensional part. These equations include elliptic boundary value problems in cyclindrical domains as well as reaction-diffusion systems. Bounded solutions are sought, in particular those which join two saddle nodes (heteroclinic solutions) or a saddle node with itself (homoclinic solution). The reversible case is studied by using a center manifold reduction. Global existence and persistence under nonreversible perturbations is shown for homoclinic solutions.

Simple models for bifurcations creating horseshoes

Abstract: We present a class of simple models for global bifurcations creating horseshoes Some properties known for Henon mappings are easily obtained for these models such as, eg, the existence of nontrivial hyperbolic sets Kneading sequences techniques allow us to exhibit explicit differences with the global bifurcation diagram for maps of the interval Explicit examples displaying wild hyperbolic sets and infinitely many sinks are also given as an illustration of the simplicity of these models

Hopf bifurcation via Volterra series

Abstract: The Hopf bifurcation theorem gives a method of predicting oscillations which appear in a nonlinear system when a parameter is varied. There are many different ways of proving the theorem and of using its results, but the way which is probably the most useful, to control and system theorists, uses Nyquist loci in much the same way as the describing function method does. The main advantages of this method are dimensionality reduction, which eases the calculation, and the ability to cope with higher-order approximations than are used in the original Hopf theorem. This paper shows how such an approach to the Hopf bifurcation follows naturally and easily from Volterra series methods. Such use of Volterra series in nonlinear oscillations appears to be new. In many problems, the calculations involved are simplified when the Volterra series approach is taken, so the approach has practical merits as well as theoretical ones.

Bifurcation and chaos in a simple feedback control system

Abstract: We discuss local and global bifurcations exhibited by a second order oscillatory system with nonlinear restoring forces, when subject to first order linear feedback control. The resulting third order system exhibits codimension two bifurcations in which multiple limit cycles and homoclinic orbits are created. Moreover, when subject to periodic desired position feedback, chaotic responses are possible.

Splitting of Steady Multiple Eigenvalues May Lead to Periodic Cascading Bifurcation

Abstract: A general bifurcation problem is considered that depends on two parameters in addition to the bifurcation parameter $\lambda $. It is assumed that all primary bifurcation states correspond to steady solutions and that they branch supercritically. Then it is shown that for a range of system parameters and near a triple primary bifurcation point the following cascade of bifurcations from the minimum primary bifurcation state is possible. As $\lambda $ increases there is secondary and then tertiary bifurcation to steady states and finally Hopf bifurcation at a quarternary bifurcation point. Related transitions have been observed experimentally in thermal convection and other hydrodynamic stability problems. In addition, we show that Hopf bifurcation near a double primary bifurcation point is not possible when both primary states near the double point bifurcate supercritically. However, it is possible near such a double bifurcation point if imperfections are included in the formulation, as we demonstrate.

Numerical computation of hopf-bifurcation points for parabolic diffusion-reaction differential-equations

Abstract: A numerical method for the direct computation of Hopf bifurcation points for a system of parabolic “diffusion-reaction” partial differential equations is described.A finite-difference discretization in space is used to approximate these equations by a system of first-order ordinary differential equations. The Hopf bifurcation of the latter system is computed by using Kubicek’s method. The results are improved by Richardson extrapolation. For approximation of the space operator, a standard three-point difference formula as well as the Stormer–Numerov technique are used. The method proposed is illustrated on an example describing heat and mass transfer in a porous catalyst particle.

Pinning-free soliton lattices and bifurcation in a discrete double-well model: exact results

Abstract: It is shown that the chain of coupled particles in the double-well potential introduced by Schmidt (1979) is completely integrable in the static limit. The chaotic behaviour and the associated infinite series of bifurcations found in the related discrete phi 4 theory are absent in the model. The solutions are generally unpinned soliton lattices. The model exhibits a bifurcation where a hyperbolic fixed point becomes elliptic and splits into two hyperbolic fixed points. The bifurcation does not lead to chaos.

Global Bifurcation Diagram in Nonlinear Diffusion Systems

Abstract: Publisher Summary This chapter discusses the global phenomena of pattern formation in the systems of reaction–diffusion equations. The system is assumed to possess Turing's diffusion-induced instability, which appears typically in mathematical biology. A key in this chapter is the discovery of singular branches that possess both boundary and interior transition layers and of singular limit points as its consequence. The structure of solutions at the singular-shadow edge seems to play the pole of the organizing centre of the whole global structure.

An organizing center for optical bistability and self-pulsing

Abstract: The problem of self-pulsing in optically bistable systems is discussed within the framework of imperfect bifurcation theory. The joint appearance of a hysteresis cycle in the cw-transmission curve and of transitions to self-pulsing is described as an interaction between steady-state and Hopf bifurcations induced by varying the incident field intensity. The bifurcation equations for the most degenerate case are shown to be determined by a corank-two and codimension-four polynomial normal form. This form can be extracted from analytical and numerical studies on the Maxwell-Bloch equations, and acts as an organizing center for bistable switching and the self-pulsing mechanism. The structurally stable unfolded bifurcation diagrams are analyzed. Besides describing correctly and in a comprehensive way all bifurcations to self-pulsing that have so far been observed, a number of new generic transitions are predicted. These include self-pulsing from the low transmission branch and transitions leading to the formation of islands with self-pulsing behavior.