Showing papers on "Bifurcation diagram published in 1981"
Book•
01 Jan 1981
TL;DR: The Hopf Bifurcation Theorum has been used in many applications, such as Differential Difference and Integro-differential Equations (by hand).
Abstract: 1. The Hopf Bifurcation Theorum 2. Applications: Ordinary Differential Equations (by hand) 3. Numerical Evaluation of Hopf Bifurcation Formulae 4. Applications: Differential-Difference and Integro-differential Equations (by hand) 5. Applications: Partial Differential Equations (by hand).
2,090 citations
TL;DR: In this paper, the bifurcation of steady-state solutions of a reaction-diffusion equation in one space variable was studied, and it was shown that S is never critical for any cubic polynomial.
251 citations
TL;DR: In this paper, the authors generalized Hill's theory of bifurcation and stability in solids obeying normality to include a non-associated flow law, and a one-parameter family of linear comparison solids has been found that admits a potential and has the property that if uniqueness is certain for the comparison solid, then instability is precluded for the underlying elastic-plastic solid.
Abstract: In the present paper, Hill's theory of bifurcation and stability in solids obeying normality is generalized to include a non-associated flow law. A one-parameter family of linear comparison solids has been found that admits a potential and has the property that if uniqueness is certain for the comparison solid then bifurcation and instability are precluded for the underlying elastic-plastic solid. The uniqueness criterion derived may be used as a device to determine lower bounds to the magnitudes of primary bifurcation and instability stresses which are ordinarily unknown. A second linear solid is introduced whose constitutive relations have the same form as the elastic-plastic solid “in loading”. The first eigenstate of this solid gives an upper bound to the primary bifurcation state of the underlying elastic-plastic solid. The search for the genuine primary bifurcation state is therefore replaced by a search for upper and lower bounds in the situation when normality fails to hold. The theory is applied to problems of homogeneous stress states.
149 citations
01 Jan 1981
133 citations
118 citations
TL;DR: In this paper, the authors present numerical procedures for computing the Hopf bifurcation formulas which can determine the stability and location of the oscillation without integrating the parabolic partial differential equations.
90 citations
TL;DR: In this article, the steady-state bifurcations from the trivial solution of the reaction-diffusion equations associated to a model chemical reaction, the so-called Brusselator, are analyzed.
Abstract: In this paper we analyze the steady-state bifurcations from the trivial solution of the reaction-diffusion equations associated to a model chemical reaction, the so-called Brusselator. The present analysis concentrates on the case when the first bifurcation is from a double eigenvalue. The dependence of the bifurcation diagrams on various parameters and perturbations is analyzed. The results of reference [2] are invoked to show that further complications in the model would not lead to new behavior.
65 citations
TL;DR: In this paper, a branch of periodic solutions which exhibit the alternatives of the global Hopf bifurcation theorem is calculated for two general systems of differential equations, i.e., systems of the Hopf type.
Abstract: Branches of periodic solutions which exhibit the alternatives of the global Hopf bifurcation theorem are calculated for two general systems of differential equations. In the first, a branch of solu...
65 citations
TL;DR: In this article, the Davidenko-IVP is integrated by a self-correcting predictor-corrector method, which does not evaluate the Jacobians of H explicitly, instead, a quasi-Newton method of C. Broyden [Math. Comp., 19 (1965), pp. 577-593] is used in the corrector phase.
Abstract: An algorithm is presented which traces an implicitly defined curve ($H(x) = 0 $ for $H:\mathbb{R}^{N + 1} \to \mathbb{R}^N$). The Davidenko-IVP is integrated by a self-correcting predictor-corrector method which does not evaluate the Jacobians of H explicitly. Instead, a quasi-Newton method of C. Broyden [Math. Comp., 19 (1965), pp. 577–593] is used in the corrector phase. It is then shown how the algorithm may be used to locate bifurcation points and trace the bifurcating branches by introducing local perturbations of H in the sense of H. Jurgens, H. O. Peitgen, and D. Saupe [in Analysis and Computation of Fixed Points, S. M. Robinson, ed., Academic Press, New York]. Numerical results are reported on a difficult test problem and on the bifurcation of periodic solutions of a differential delay equation.
59 citations
TL;DR: In this paper, the bifurcation diagram of a laser with saturable absorber in the low and medium intensity regimes was investigated and the linear stability of the stationary solutions corresponding to these regimes was studied.
Abstract: We investigate the bifurcation diagram of a laser with saturable absorber in the low and medium intensity regimes The linear stability of the stationary solutions corresponding to these regimes is studied In the low intensity domain, a Hopf bifurcation point is determined from which a time-periodic solution emerges This solution is contructed and its stability is analyzed in the vicinity of the bifurcation point It is shown that this time-periodic solution is stable in a finite domain of the parameter space
39 citations
TL;DR: In this paper, a self-similar bifurcation structure is studied both analytically and numerically concerning their previous piecewise linear mapping which explained successfully the global bifurlcation structure found experimentally by Hudson et al. in the Belousov-Zhabotinsky reaction.
TL;DR: In this article, a truncated fifth-order model has been solved numerically, in which there is a bifurcation from symmetrical to asymmetrical oscillations, leading to a period-doubling cascade which is followed immediately by a periodhalving cascade and a partitioning back to symmetry.
01 Jan 1981
TL;DR: In this paper, stable numerical methods for the approximation of the solutions of a nonlinear parameter-dependent equation near a non-trivial bifurcation point are discussed, and the problem of finding the bifurlcation point is reformulated as a well-posed equation of higher dimension.
Abstract: We discuss stable numerical methods for the approximation of the solutions of a nonlinear parameter - dependent equation near a non-trivial bifurcation point. The problem of finding the bifurcation point is reformulated as a well-posed equation of higher dimension. The nearby branches can be calculated in a stable manner after applying a certain transformation having its origin in the Lyapunov — Schmidt theory. We also treat the perturbed bifurcation problem and present numerical results.
TL;DR: In this article, the authors use Laplace's method of steepest descent to study bifurcation in the presence of small noise, and apply it to the study of the dynamics of noisy, constrained or implicitly defined dynamical systems.
01 Jan 1981
TL;DR: In this article, a unicomponent reaction-diffusion system with trigger type dynamics and combined boundary conditions is considered, where the boundary permeabilities and reservoir concentrations as well as the dimension of the system are the control parameters.
Abstract: Pattern formation in a unicomponent reaction-diffusion system with trigger type dynamics and combined boundary conditions is considered. The boundary permeabilities and reservoir concentrations as well as the dimension of the system are the control parameters. The whole assemblage of steady states, their bifurcations and changes under the variation of these parameters is described. Among all steady distributions possible for given values of the parameters, only the simplest ones prove to be asymptotically stable. The relation to catastrophe theory is discussed.
TL;DR: In this paper, a simplification of the bifurcation analysis is presented, illustrated by a discussion of two important special cases exhibiting secondary bifurbcation of periodic solutions.
TL;DR: The existence and first order asymptotic approximation of two one-parameter families of periodic orbits in the neighborhood of a 1st species-2nd species bifurcation orbit are established for small values of the mass ratio as discussed by the authors.
Abstract: The existence and first order asymptotic approximation of two one-parameter families of periodic orbits in the neighborhood of a 1st species–2nd species bifurcation orbit are established for small values of the mass ratio $\mu > 0$.
TL;DR: The BIFOR2 as mentioned in this paper analyzes Hopf bifurcation points in ODEs by finding critical value(s) of a user-specified parameter v such that a stationary (equilibrium) solution x"*(v) loses linear stability by virtue of a complex conjugate pair of eigenvalues.
TL;DR: In this article, convergence properties of projection methods for the computation of Hopf branches off simple eigenvalues for general operator equations and for the Hopf bifurcation branches for ordinary differential equations are established.
01 Dec 1981
TL;DR: In this article, the classical swing equation model of a synchronous generator is augmented to include the effects of variable damping, nonlinear frequency dependence of the input torque, and lossy transmission lines.
Abstract: The classical swing equation model of a synchronous generator is augmented to include the effects of variable damping, nonlinear frequency dependence of the input torque, and lossy transmission lines. In each case oscillations are shown to occur for certain parameter values. These oscillations are due to a Hopf bifurcation.
01 Jan 1981
TL;DR: In this article, a hierarchy of increasingly accurate boundary conditions for the truncated interval problem is determined, and numerical techniques for error estimation and the determination of an appropriate truncation point are discussed.
Abstract: Part I. "Asymptotic Boundary Conditions for Ordinary Differential Equations"
The numerical solution of two point boundary value problems on semi-infinite intervals is often obtained by truncating the interval at some finite point. In this thesis we determine a hierarchy of increasingly accurate boundary conditions for the truncated interval problem. Both linear and nonlinear problems are considered. Numerical techniques for error estimation and the determination of an appropriate truncation point are discussed.
A Fredholm theory for boundary value problems on semi-infinite intervals is developed, and used to prove the stability of our numerical methods.
Part II. "Numerical Hopf Bifurcation"
Several numerical methods for locating a Hopf bifurcation point of a system of o.d.e.'s or p.d.e.'s are discussed. A new technique for computing the Hopf bifurcation parameters is also presented. Finally, well-known numerical techniques for simple bifurcation problems are adapted for Hopf bifurcation problems. This provides numerical techniques for computing the bifurcating branch of periodic solutions, possibly including turning points and simple bifurcation points. The stability of the periodic solutions is also discussed.
TL;DR: In this article, the authors consider a class of simple examples and investigate the connection between the solutions of the nonlinear wave equation and the solution of the corresponding bifurcation problem.
Abstract: We consider a class of simple examples and investigate the connection between the solutions of the nonlinear wave equation and the solutions of the corresponding bifurcation problem.