Successive Bifurcations Research Articles

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.

Transition to chaos in a three-mode model of forced, dissipative flow in a rotating fluid

For a truncated triadic model of the barotropic vorticity equation of two-dimensional rotating fluid the mechanism of successive bifurcations to chaotic attractors is evidentiated. It is suggested that the transition to chaos takes place through a Feigenbaum sequence.

STEADY LAMINAR FLOW IN TWO SUCCESSIVE BIFURCATIONS

Abstract The steady-state vorticity transport equation was solved numerically for inspiratory flow in two-dimensional models of two successive bifurcations. Streamlines and velocity distributions were obtained for Reynolds numbers ranging from 1 to 800. For both symmetric and asymmetric airway models, the calculated axial velocity profiles in the parent channels of the second bifurcations are asymmetric.

Subharmonic emissions of a cavitating bubble

The nonlinear oscillations of a gas bubble pulsating in a liquid are critically dependent on the damping of these oscillations. We have used an accurate numerical formulation of the problem to investigate the subharmonic emissions from forced oscillations of the bubble. A systematic approach to chaotic behavior will be shown through successive bifurcation of the subharmonic emissions as the driving pressure is increased. [Work supported in part by the NSF and the ONR.]

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.

Quasiperiodicity in lasers with saturable absorbers

In this paper, we consider the mean-field equations for the laser with a saturable absorber (LSA) and concentrate on the low-intensity solutions. We show that the LSA equations may admit two successive bifurcations. The first bifurcation corresponds to the transition from the zero-intensity state to time-periodic intensities and is a Hopf bifurcation. The second bifurcation corresponds to the transition from these time-periodic intensities to quasiperiodic intensities which are characterized by two incommensurable frequencies. In order to describe these transitions, we investigate a particular limit of the parameters and propose a new perturbation method for solving the LSA equations. We give analytical conditions for the existence of both the primary and secondary bifurcations.

Period Doublings and Possible Chaos in Neural Models

A formal perturbation method is derived for the study of bifurcation near a degenerate zero eigenvalue. This results in a third order differential equation with a single quadratic nonlinearity. Numerical solutions of this show successive period doubling bifurcations and eventual “chaos.” The problem arises in the study of neural equations when an additional excitatory channel or cell is added to standard two-component models which oscillate. It also applies to a Van der Pol oscillator coupled to a simple RC circuit. Numerical simulations of a modified FitzHugh–Nagumo system agree qualitatively with the behavior of the bifurcation equations.

On the non-local nature of the behaviour of dissipative structures

The non-local nature of the behaviour of dissipative structure as the size of the space domain increases is studied. A class of systems is isolated, for which the continuum of inhomogeneous solutions, branching from the homogeneous solution, can be continued without limit as the size of the space domain increases; it is shown that, in general, this is not true. An example of a system is given, for which all possible types of successive bifurcations of dissipative structures are realized as the size of the space domain increases: stiff birth, stiff death, and secondary branching. All the predicted effects are modelled numerically.

Experimental evidence for chaotic behaviour of a parametrically forced pendulum

Experimental evidence is presented for chaotic type nonperiodic motions of a parametrically forced pendulum. A bifurcation diagram is measured directly, showing successive subharmonic bifurcations to ƒ/4, onset of a periodic motion and the appearance of periodic motions via intermittency. The experimentally determined threshold values of the amplitude of the driving force for the first period doublings and the onset of a periodic motion are found to be in good agreement with the theoretical predictions.

Successive Higher-Harmonic Bifurcations in Systems with Delayed Feedback

The scheme of bifurcations in systems described by a certain class of delay-differential equations is investigated in detail. It is shown that higher-harmonic oscillating states appear successively in the course of transition to developed chaos. First-order transitions between these states account for the frequency-locked anomaly recently observed by Hopf et al. in a hybrid optical bistable device.

Renormalization group analysis of bifurcations in area-preserving maps

Two-dimensional area-preserving maps can be represented by a generating function, the action. High orders of successive period-doubling bifurcations are studied by writing a renormalization group scheme for this action. Fixed points and eigenvalues of this scheme are found and interpreted.

On successive bifurcations of one-parameter families of maps

On successive bifurcations of one-parameter families of maps

Evidence for Universal Chaotic Behavior of a Driven Nonlinear Oscillator

A bifurcation diagram for a driven nonlinear semiconductor oscillator is measured directly, showing successive subharmonic bifurcations to $\frac{f}{32}$, onset of chaos, noise band merging, and extensive noise-free windows. The overall diagram closely resembles that computed for the logistic model. Measured values of universal numbers are reported, including effects of added noise.

On the Transition to Turbulence in Magneto-Hydrodynamic Models of Confined Plasmas

Starting from a model of an interchange-unstable plasma in a shearless magnetic field, the successive bifurcations and the transition to turbulence of the system are studied in various approximations. Numerical calculations using the finite element method and a spectral method with mode truncation yield converging results in certain domains of parameters. In the last part of the paper it is shown that a tearing-unstable plasma in a sheared magnetic field has a formally similar description and therefore should also be investigated in terms of bifurcation theory.

Analyse numérique du comportement d'une solution presque périodique d'une équation différentielle périodique par une méthode des sections

After having reminded the reader of the cross section method used by Jean ( Int. J. Non-Linear Mech. 15, 367–376, 1980) to localise almost periodic solutions to a T-periodic ordinary differential equation, we give a numerical approximation of the said method. Numerical results are then presented together with evidence of successive bifurcations of the initial torus of solutions into double, quadruple, etc.

Nonlinear Dynamical Models of Plasma Turbulence

An exact pole-type truncation expansion technique has been used to study nonlinear dynamical systems which can be considered models for plasma turbulence. One of the models describes the mode coupling saturation of a linearly unstable, one-dimensional homogeneous plasma. Another describes the situation where waves with k < 0 are completely damped. The solutions exhibit a rich variety of complex behaviors including successive bifurcations to increasingly complicated periodic motion and the onset of chaotic behavior, as well quasi-periodic-like behavior with highly complex topology.

Diffusion-induced morphogenesis in the development of Dictyostelium

A reaction-diffusion model for the regulation of cAMP in Dictyostelium discoideum is analyzed. As a parameter declines with starvation, the model sequentially yields pulse relaying, spiral waves, target patterns, streaming and sorting, directed locomotion, and tissue buckling, closely matching the observed morphogenetic sequence. These morphologies appear through successive bifurcations of a single reaction-diffusion system and do not require the expression of new genetic information.

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of 1 δ = 1 8.721097200 …, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = −4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.

Period-doubling bifurcations in a simple model

Periodic solutions in a simple model, whose solution shows successive period-doubling bifurcations leading to chaotic motion, are calculated by using the harmonic balance method. The result is in good agreement with that of computer simulation.

Do successive bifurcations in Hamiltonian systems have the same universal ratio?

DO SUCCESSIVE BIFURCATIONS IN HAMILTO1NIA1N SYSTEMS ETC. 4 ~ consider the fami ly of per iodic orbits (3) q~ = 2~q~, for different values of h. Fo r small values of h this fami ly is stable. A t a par t icu lar va lue h = h 1 (table I) a new fami ly of simple periodic orbits bifurcates f rom i t ; the fami ly (3) becomes unstable for h larger t h a n hx, while the new fami ly is stable. As h increases beyond a TABLE I. -Bifurcations o] the straight-line solutions o] Hamiltonian (2). 1 0.111 361 163 61 9.382 16 0.115 289 767 88 9.542 57 2 0.123 546 301 252 9.25635 O. 123 982 430 124 9.266 78 3 0.124 842 951 250 9.226 57 0.124 890 191 665 9.227 05 4 0.124 982 978 639 9.221 48 0.124 988 099 306 9.220 69 5 0.124 998 154 162 9.220 75 0.124 998 709 459 9.218 40 6 0.124 999 799 816 996 9.220 47 0.124 999 860 038 637 9.220 84 7 0.124 999 978 289 277 9.221 97 0.124 999 984 821 187 9.218 49 8 0.124 999 997 645 762 0.124 999 998 353 438 va lue h = hi (> hi) the fami ly (3) becomes again stable and a new uns table fami ly appears for h > hi. The fami ly (3) becomes again unstable at h = h~ (> hi) etc. There is an inf ini ty of successive t ransi t ions f rom stabi l i ty to ins tabi l i ty at hi, h2, ha ... and f rom ins tabi l i ty to s tabi l i ty at hi, h~, h~, ... such tha t (4) h i c~ is the escape energy h~ = 0.125. I n the fol lowing we define the va lue h ~ h k (5) a~ = h ~ h~+l and a similar va lue 0rl~ The values of hk, hl, 6k, (5~ are g iven in table I. The error of the numbers g iven is at most about one in the last digit . W e notice tha t the b i furcat ion rat ios O k, Of t end to the number 9.22. The accuracy is no t so good as in de te rmina t ions of s imilar rat ios g iven by o ther authors. However , i t is obvious t h a t the difference f rom the (~ universal ~> n u m b e r for conservat ive systems = 8.7210972 is significant, a l though bo th numbers are of the same order. (6) R . C . CHUROttlLL, G. PEOELZI a n d D. L . ROD : i n Stochastic Bshaviour in Classical and Quantum Hamiltonian Sysiems, e d i t e d b y G. CASATI a n d J . FORD (Ber l in , 1979) , p . 76.