Showing papers on "Bifurcation diagram published in 1985"
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01 Nov 1985
TL;DR: In this article, the restricted problem of three bodies is considered and the Hamiltonian Hopf bifurcation is shown to preserve normal forms for energy-momentum maps.
Abstract: Preliminaries.- Normal forms for Hamiltonian functions.- Fibration preserving normal forms for energy-momentum maps.- The Hamiltonian Hopf bifurcation.- Nonintegrable systems at resonance.- The restricted problem of three bodies.
292 citations
TL;DR: In this article, a periodic recurrence of a specific fine structure in the bifurcation set, which is closely connected with the nonlinear resonances of the system, is investigated.
198 citations
TL;DR: In this paper, a new class of instabilities for a plane-wave intracavity field in an optical ring resonator is identified, and a bifurcation diagram is given that organizes the important information, and global pictures are developed that describe the evolution of the attractor and its basin boundary.
Abstract: A new class of instabilities for a plane-wave intracavity field in an optical ring resonator is identified. Dynamical systems techniques are explained and applied to the map. A bifurcation diagram is given that organizes the important information, and global pictures are developed that describe the evolution of the attractor and its basin boundary. Anomalous behavior observed in earlier numerical studies is explained.
112 citations
TL;DR: In this article, the authors derived analytically the bifurcation diagram of the small amplitude ac forced Josephson junction and provided an enhanced picture of the dynamics of the ac forced case, as well as insightful explanation of the associated I-V characteristics.
Abstract: We study the dynamics of the Josephson junction circuit with both dc and ac current forcing, with emphasis on the ac case. Specifically, we derive analytically the bifurcation diagram of the small amplitude ac forced Josephson junction. We thus place on analytic grounds the qualitative, experimental, and simulation work of Belykh, Pedersen, and Soerensen; specially that which pertains to the regions of chaos. Combining previous results from the literature with our new results, we provide an enhanced picture of the dynamics of the ac forced case, as well as insightful explanation of the associated I-V characteristics. Explicit asymptotic formulae for the curves that separate the different regions in the bifurcation diagram are also given.
92 citations
TL;DR: In this paper, the families of simple periodic orbits in a three-dimensional system that represents the inner parts of a perturbed triaxial galaxy were studied and the regions where each family is stable, simply unstable, doubly unstable, or complex unstable.
Abstract: We study the families of simple periodic orbits in a three-dimensional system that represents the inner parts of a perturbed triaxial galaxy. The perturbations depend on two control parameters. We find the regions where each family is stable, simply unstable, doubly unstable, or complex unstable. the stable and simply unstable families produce other families by bifurcation. Several families reach a maximum (or minimum) perturbation and then are continued by other families. The bifurcations are direct or inverse. The transition from one type of bifurcation to the other is theoretically explained. Another important phenomenon is the splitting of one family into two, or the joining of two families into one. We do not have any complex instability in the limiting cases of two-dimensional motions (when one control parameter is zero).
88 citations
TL;DR: In this paper, the authors study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry and show that the linearized stability analysis leads to multiple eigenvalues for the most simple flows.
Abstract: We study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry. The difficulty is that the linearized stability analysis leads to multiple eigenvalues for the most simple flows. Only a consideration of the symmetry-group action on the critical eigenvectors allows us to derive and to solve the bifurcation equations. We recover through this analysis the different patterns which are observed in experiments as the Reynolds number is increased: Steady Taylor vortices and bifurcation of either wavy, or twisted vortices from the Taylor vortex flow in the case of co-rotating cylinders; spiral vortices in the case of (strongly) counterrotating cylinders, and ribbon-cells, which have not yet been observed in experiments. Then we show that, under natural assumptions on the loss of stability of these oscillatory flows, the next bifurcation leads to quasi-periodic flows without frequency locking, whose different patterns are studied.
68 citations
TL;DR: In this article, it was proved that the stochastic normal form of a general system undergoing a Hopf bifurcation contains new types of terms which did not appear in the deterministic normal form.
67 citations
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TL;DR: In this paper, the degenerate Hopf bifurcation was studied for systems of ODE using singularity theory methods, and the power of these methods is illustrated by Case Study 2, where they presented the analysis by Labouriau [1983] of degenerate hopf-furcation in the clamped Hodgkin-Huxley equations.
Abstract: The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs. In this chapter, we study Hopf bifurcation for systems of ODE using singularity theory methods. The principal advantage of these methods is that they adapt well to degenerate Hopf bifurcations; i.e., cases where one or more of the hypotheses of the traditional theory fail. The power of these methods is illustrated by Case Study 2, where we present the analysis by Labouriau [1983] of degenerate Hopf bifurcation in the clamped Hodgkin-Huxley equations.
66 citations
TL;DR: The unstable-unstable pair bifurcation is an example of the crisis route to chaos as discussed by the authors, in which two unstable fixed points or periodic orbits of the same period coalesce and disappear as a system paremeter is raised.
Abstract: The unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points or periodic orbits of the same period coalesce and disappear as a system paremeter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reached during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, (T), satisfies
61 citations
TL;DR: In this article, a new degree for S 1 -invariant gradient maps where the classical degree gives little information was constructed and applied to obtain a global bifurcation theorem which applies to cases where classical results give limited information.
Abstract: We construct a new degree for S 1 -invariant gradient maps where the classical degree gives little information. The main technical result needed is a new result on generic homotopies. We apply this degree to obtain a global bifurcation theorem which applies to cases where classical results give limited information. We apply our results to obtain a bifurcation theorem for periodic solutions of Hamiltonian systems and for a problem in elasticity. We also obtain new results on bifurcation for elliptic equations on domains with an S 1 symmetry.
TL;DR: In this article, the bifurcation of capillary-gravity waves is analyzed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two and the existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurbation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.
Abstract: The bifurcation and secondary bifurcation of capillary-gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two. The existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurcation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.
TL;DR: The bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.
TL;DR: In this paper, Liapunov-Schmidt et al. presented an alternative method for the analysis of Hopf bifurcations in a class of FDE with unbounded delay.
TL;DR: It is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ρ) bifurcates from the unique (normalized) critical point (1, 0).
Abstract: The existence of a stable positive equilibrium state for the density ρ of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ρ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.
TL;DR: Chow and Hale as discussed by the authors used a third order Picard-Fuchs equation to show that a certain two parameter family of vector fields for parameter values in a certain cone has a unique limit cycle, which is born from a Hopf bifurcation and dies in a saddle connection.
TL;DR: In this article, conditions are derived which ensure persistence of Hopf bifurcation under singular perturbations of the vector field, and the results justify use of reduced-order models in the study of nonlinear oscillations via Hopfbifurcation via nonlinear systems.
Abstract: Hopf bifurcation is the simplest way in which periodic solutions can emerge from an equilibrium point of an autonomous one-parameter family of ordinary differential equations The phenomenon occurs if the linearized system has a pair of complex-conjugate simple eigenvalues which cross the imaginary axis transversely for some value of the parameter In this paper, conditions are derived which ensure persistence of the Hopf bifurcation under singular perturbations of the vector field The results justify use of reduced-order models in the study of nonlinear oscillations via Hopf bifurcation Both single parameter and multiparameter singular perturbation problems are considered In the single parameter case, we show how Fenichel's center manifold theorem for singularly perturbed systems can be used to prove regular degeneration of the bifurcated periodic solutions and to study their stability In the multiparameter case, we obtain a novel asymptotic formula for the eigenvalues of the perturbed system This formula is valid regardless of the relative magnitudes of the small parameters, and the results on multiparameter singularly perturbed Hopf bifurcation apply to two time scales as well as multiple time-scale systems
TL;DR: In this article, the generic properties of the nonlinear interaction of three drift waves with finite k∥ are investigated, and different types of stationary or quasistationary states are characterized by the bifurcation diagram in γ1, κ parameter space, where γ 1 measures the mode excitation and κ the parallel wavenumber.
Abstract: The generic properties of the nonlinear interaction of three drift waves with finite k∥ are investigated. The different types of stationary or quasistationary states are characterized by the bifurcation diagram in γ1, κ parameter space, where γ1 measures the mode excitation and κ the parallel wavenumber. The transition to turbulence corresponds exactly to the Ruelle–Takens picture: steady state→periodic solution→doubly periodic solution→turbulence, in contrast to the period‐doubling route usually observed in low‐dimensional dynamic systems. The transition to k∥=0, the model of Horton and Terry [Phys. Fluids 25, 491 (1982)], occurs at very small values of k∥.
TL;DR: An algorithm for the computation of a Hopf bifurcation point based on a direct method, i.e. an augmented time independent system is solved and the bandstructure of the Jacobian matrix is exploited.
TL;DR: With a singular perturbation problem occurring in chemical reaction processes, substantial changes of the bifurcation diagram due to discretization are demonstrated and it is shown that a discrete system can possess any number of solutions.
Abstract: With a singular perturbation problem occurring in chemical reaction processes, substantial changes of the bifurcation diagram due to discretization are demonstrated. It is shown that a discrete system can possess any number of solutions, whereas the underlying continuous problem has exactly one solution. In addition to that, there is no way to favor one of the various discrete solutions as the one approximating the continuous solution.
TL;DR: In this paper, the authors study the five mode equations used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance and show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue.
Abstract: We study the five mode equations which are used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance. We show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue. A center manifold reduction is performed to fix the three-dimensional submanifold in parameter space where this degeneracy occurs. The associated Takens-normal form is given. By unfolding the normal form we obtain all structurally stable phase portraits near this bifurcation point and display them in the form of bifurcation diagrams with the laser pumping rate as a distinguished bifurcation parameter. These diagrams allow a unifying analytical and geometrical description of many different numerical solutions of the equations describing a laser with absorber. In particular, they yield the connection of the small amplitude periodic solutions with passiveQ-switching and suggest new bifurcation processes, which one can expect to occur for physical parameters near the critical submanifold. The existence of a codimension four bifurcation is indicated.
TL;DR: In this paper, a bifurcation sequence from a periodic to a quasiperiodic regime, leading ultimately to a steady state, is reported in an experimental study of the Belousov-Zhabotinsky reaction.
TL;DR: In this article, a systematic approach is presented for predicting all possible phase-plane diagrams of a system of two ordinary differential equations with widely separated time scales, and of the sequence of phase plane diagrams obtained by varying a parameter, i.e. bifurcation diagrams.
TL;DR: In this paper, a mechanical system with a forced nonlinear torsion pendulum was investigated and the state diagram was given as a function of both the external driving frequency and damping parameter.
TL;DR: In this paper, the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise and a universal time scale for all such processes is established.
TL;DR: The sensitivity of the bifurcation diagrams to imperfections is analyzed in this article, where a complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurbations is presented.
TL;DR: In this article, a reaction-diffusion equation related to some mathematical models of gasless combustion of solid fuel is studied, and a suitable version of the Hopf bifurcation theorem is developed and the existence of time periodic solutions for values of the parameter near some critical value.
TL;DR: This work identifies the conditions leading to input multiplicity and presents an efficient technique for dividing the parameter space into regions with different number of input states as well as with different types of bifurcation diagrams.
Abstract: Input multiplicity is a situation in which different inputs can give the same output. In this work we identify the conditions leading to input multiplicity and present an efficient technique for dividing the parameter space into regions with different number of input states as well as with different types of bifurcation diagrams. It is shown that bounds on the values of the manipulated inputs have an important impact on the number of feasible solutions. The technique is illustrated with examples.