Primary Hopf Bifurcation Research Articles

A new scaling for the flow instability past symmetric bluff bodies

A scaling law to predict the onset of the primary Hopf instability in the steady flow past two-dimensional symmetric bluff bodies is proposed. It uses a measure of the spatial extent of the separation bubble as the length scale and the largest reverse-flow speed within it as the velocity scale. The ensuing Reynolds number, evaluated at the onset of the primary Hopf bifurcation, collapses quite nicely for bodies of different shape and aspect ratio even when a small angle of attack perturbs the symmetry; its relative variation is one order of magnitude smaller than that of the usual Reynolds number defined with the free-stream velocity and the cross-stream body size. With the new scaling, it can be roughly assessed whether the steady flow past a two-dimensional bluff body is absolutely and globally unstable to two-dimensional perturbations without a computationally expensive stability analysis: only the inspection of the base flow is required. More importantly, the scaling provides an insight into the flow mechanism that produces the instability.

Nonlinear dynamics of continuously tunable optoelectronic oscillators based on stimulated Brillouin amplification.

We present a theoretical analysis for tunable optoelectronic oscillators (OEOs) based on stimulated Brillouin scattering (SBS). A pump laser is used to generate a Brillouin gain which selectively amplifies a phase-modulated and contra-propagating laser signal. The radiofrequency beatnote generated after photodetection is filtered, amplified and fed back to the phase modulator to close the optoelectronic loop. Tunability is readily achieved by the adjustable detuning of the pump and signal lasers. OEOs based on stimulated Brillouin scattering have been successfully demonstrated at the experimental level, and they feature competitive phase noise performances along with continuous tunability for the output radiofrequency signal, up to the millimeter-wave band. However, the nonlinear dynamics of SBS-based OEOs remains largely unexplored at this date. In this article, we propose a model that describes the temporal dynamics of the microwave envelope, thereby allowing us to track the dynamics of the amplitude and phase of the radiofrequency signal. The corresponding nonlinear and time-delayed differential equation is then analyzed to reveal the underlying bifurcation behavior that emerges as the feedback gain is increased. It is shown that after the primary Hopf bifurcation that triggers the microwave oscillations, the system undergoes a secondary Neimark-Sacker bifurcation before fully developed chaos emerges for the highest gain values. We also propose a model for the chipscale version of this SBS-based OEO where the delay line is replaced by a highly nonlinear waveguide. The numerical simulations are found to be in excellent agreement with the analytical study.

Resonances between fundamental frequencies for lasers with large delayed feedbacks.

High-order frequency locking phenomena were recently observed using semiconductor lasers subject to large delayed feedbacks. Specifically, the relaxation oscillation (RO) frequency and a harmonic of the feedback-loop round-trip frequency coincided with the ratios 1:5 to 1:11. By analyzing the rate equations for the dynamical degrees of freedom in a laser subject to a delayed optoelectronic feedback, we show that the onset of a two-frequency train of pulses occurs through two successive bifurcations. While the first bifurcation is a primary Hopf bifurcation to the ROs, a secondary Hopf bifurcation leads to a two-frequency regime where a low frequency, proportional to the inverse of the delay, is resonant with the RO frequency. We derive an amplitude equation, valid near the first Hopf bifurcation point, and numerically observe the frequency locking. We mathematically explain this phenomenon by formulating a closed system of ordinary differential equations from our amplitude equation. Our findings motivate experiments with particular attention to the first two bifurcations. We observe experimentally (1) the frequency locking phenomenon as we pass the secondary bifurcation point and (2) the nearly constant slow period as the two-frequency oscillations grow in amplitude. Our results analytically confirm previous observations of frequency locking phenomena for lasers subject to a delayed optical feedback.

A normal form method for the determination of oscillations characteristics near the primary Hopf bifurcation in bandpass optoelectronic oscillators: Theory and experiment.

We propose a framework for the analysis of the integro-differential delay Ikeda equations ruling the dynamics of bandpass optoelectronic oscillators (OEOs). Our framework is based on the normal form reduction of OEOs and helps in the determination of the amplitude and the frequency of the primary Hopf limit-cycles as a function of the time delay and other parameters. The study is carried for both the negative and the positive slopes of the sinusoidal transfer function, and our analytical results are confirmed by the numerical and experimental data.

Centre-manifold reduction of bifurcating flows

AbstractIn this paper we describe a general and systematic approach to the centre-manifold reduction and normal form computation of flows undergoing complicated bifurcations. The proposed algorithm is based on the theoretical work of Coullet & Spiegel (SIAM J. Appl. Maths, vol. 43(4), 1983, pp. 776–821) and can be used to approximate centre manifolds of arbitrary dimension for large-scale dynamical systems depending on a scalar parameter. Compared with the classical multiple-scale technique frequently employed in hydrodynamic stability, the proposed method can be coded in a rather general way without any need to resort to the introduction and tuning of additional time scales. The method is applied to the dynamical system described by the incompressible Navier–Stokes equations showing that high-order, weakly nonlinear models of bifurcating flows can be derived automatically, even for multiple codimension bifurcations. We first validate the method on the primary Hopf bifurcation of the flow past a circular cylinder and after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylinders.

Effect of solid body degrees of freedom on the path instabilities of freely falling or rising flat cylinders

In 2007 Fernandes et al. reported a surprising delay of the onset of instabilities for freely rising cylinders of aspect ratio (χ=diameter/thickness of the cylinder) 10 and more. At Reynolds numbers at which instabilities develop in the wake of the same body kept fixed, the path was still found to be vertical and the wake axisymmetric. In this paper, we explain this delay by investigating numerically the transition scenario of the solid–fluid system represented by the freely moving body interacting with the ambient fluid by hydrodynamic forces. We show that the free body degrees of freedom can have a stabilizing effect on the onset of the primary bifurcation. This effect explains, however, only partly the experimental observation. We show that the primary bifurcation is followed by a sequence of weakly oscillating, virtually unobservable, bifurcating states before the observable path oscillations set in. For aspect ratio smaller than 8, the free body degrees of freedom destabilize the system in agreement with expectations. The primary bifurcation is, however, a Hopf bifurcation instead of regular one in the wake of a fixed body. In our study we focus to the intermediate interval of aspect ratio between 8 and 10. We show that, for χ>8, the primary Hopf bifurcation is replaced by a new one with much lower frequency and leading to weakly oscillating periodic oscillations, later (for χ>9) the Hopf bifurcation is replaced by a regular one disappearing again for very thin cylinders (χ>10).

NONLINEAR DYNAMIC BEHAVIOR OF A ROTOR ACTIVE MAGNETIC BEARING

The nonlinear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The vibration of the rotor is modeled by a coupled second order nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Their approximate solutions are sought applying the method of multiple scales in the case of primary resonance. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency response curves. Choosing the Hopf bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. The three types of primary Hopf bifurcations are found for the first time in the rotor-AMB system. It is shown that the limit cycles undergo cyclic fold, period doubling bifurcations, and intermittent chaotic attractor, whereas the chaotic attractors undergo explosive bifurcation and boundary crises. In the regime of multiple coexisting solutions, multiple stable equilibriums, periodic solutions and chaotic solutions are the most interesting phenomena observed.

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.

Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We study a model for chatter in twist drills derived by Stone and Askari [Dynam. Sys., 17, 1 (2002), 65–85], in which a linear vibration mode interacts with nonlinear cutting forces. This results in a delay differential equation describing an oscillator that is nonlinear in damping and with cross-terms in the damping and the delay. Linear stability analysis of the model with significant nonlinear terms is performed, and an analysis of the nonlinear stability of the primary Hopf bifurcation is observed. The latter is done via the construction, using symbolic algebra, of a two-dimensional centre manifold in the infinite dimensional space employing an algorithm developed by Campbell and Belair [SIAM J. Appl. Math., 54, 5 (1994), 1402–1424; and Can. Appl. Math. Quart., 3, 2 (1995), 137–154]. Our analysis shows that the stability of the Hopf bifurcation depends on the type of vibration in question and on the cutting speed. These results are confirmed numerically and further bifurcations in the high-speed limit are also explored numerically, with tantalizing results that could be the basis of much future work.

Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis

In the present work, we study the onset of double-diffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless double-diffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case Le⩾1 and for ε Le2<1 the first primary bifurcation is always a Hopf bifurcation for any A and φ except for φ=+90°. For ε Le2>1 only stationary primary bifurcations exist. In the case Le<1, zones where stationary and oscillatory primary bifurcations exist are separated by a curve depending on A and φ. The last part of this work consists of a series of numerical simulations. The onset of stationary and oscillatory convection is obtained numerically at the critical Rayleigh number predicted by linear analysis. We also verified the frequency of oscillations for several sets of dimensionless parameters. The numerical simulations show multiple subcritical solutions.

Experimental determination of the stability diagram of a lamellar eutectic growth front

We present an experimental study of the growth patterns of directionally solidified thin samples of the lamellar eutectic alloy CBr${}_{4}$-C${}_{2}$Cl${}_{6}$ as a function of the pattern wavelength $\ensuremath{\lambda}$, the solidification velocity $V$, and the alloy concentration $C$, within the so-called planar coupled zone of the parameter space. Capillary-anisotropy effects and three-dimensional (3D) effects are minimized by an appropriate choice of the eutectic grain size, the eutectic grain orientation, and the sample thickness. We first verify the old proposition made by Jackson and Hunt [Trans. AIME 236, 1129 (1996)] that the basic patterns (i.e., the stationary, spatially periodic, reflection-symmetric, 2D patterns) of our system are stable over a finite range of $\ensuremath{\lambda}$ at given $V$ and $C$, the lower bound of which is determined by a local, lamella-termination instability. We show that the upper bound of the basic state stability range is marked by a primary Hopf bifurcation toward an oscillatory state. The nature of the oscillatory state, and the threshold value for the bifurcation, depend on $C$. Other, secondary bifurcations occur at higher $\ensuremath{\lambda}$. In total, we identify six different types of low-symmetry extended growth patterns: the already-known steady symmetry-broken, or ``tilted'' state [K. Kassner and C. Misbah, Phys. Rev. A 44, 6533 (1991); G. Faivre and J. Mergy, ibid., 45, 7320 (1992)], and five new types of oscillatory and/or tilted states. We determine the stability domains of the various states in the plane $(C,\ensuremath{\lambda}{V}^{1/2})$, and characterize the various primary and secondary bifurcations of our system. Our experimental results are in good quantitative agreement with the stability diagram numerically calculated by Karma and Sarkissian [Metall. Mater. Trans. 27A, 635 (1996)] in the frame of a 2D model without capillary anisotropy.

Sequence of bifurcations in an 'antigen-antibody' reaction.

The aperiodically singularly disturbed system of ordinary differential equations describing the process of the formation of the antibody complex initiated by antigen binding was analysed. The numerical and analytical examinations showed that there are solutions which correspond to primary Hopf bifurcation and subsequent finite number of Feigenbaum's period-doubling bifurcations. The last leads to chaotization of the system.

Spatially heterogeneous discrete waves in predator-prey communities over a patchy environment

A model motivated by the earlier work of J. D. Murray and D. Stirzaker is proposed to describe the dynamics of predator-prey communities over a patchy environment. The model is a system of delay differential equations. It is shown that if only one time lag is incorporated into the model, then a branch of spatially homogeneous periodic solutions occur as a primary Hopf bifurcation. However, if two time lags are used to measure different delayed factors in the process of growth, decay, and predator consumption of the prey population, then stable spatially heterogeneous periodic solutions (discrete waves or phase-locked oscillations) may exist. The utilized method is based on center manifold, normal form, and equivariant Hopf bifurcation theory.

Nonlinear stability of the geomagnetic field

By subjecting a model axisymmetric magnetic field to a nonaxisymmetric perturbation, or instability, linear stability analyses can provide an indirect measure of the toroidal field strength in the Earth's core. It is thought that such field strengths have an upper bound of the order of 50 Gauss. We have constructed a model of the Earth's core to investigate how magnetic field instabilities evolve nonlinearly and find that at reasonably low viscosities, the instability evolves to a finite amplitude and rotates rigidly, as predicted from the theory of bifurcations in rotating systems. As the field strength is increased the solution adopts a completely different spatial configuration and has a different characteristic frequency to the branch found at the lower field strength. The primary Hopf bifurcation is thought to be of a subcritical nature which accords with a previous weakly nonlinear analysis. We conjecture that such an instability could provide the geomagnetic field with a mechanism for evolution to a different state at field strengths lower than the strength at the point of marginal stability.

Periodic structure beyond a hope bifurcation

AbstractA Hopf bifurcation where a stable fixed pointer bifurcates into a stable periodic orbit as a parameter passes through a critical value, occurs frequently in nonlinear problems. Here, the Maynard Smith nonlinear map in population dynamics involving a parameter k is analysed with the help of numerical experiments in order to determine the periodic structure of the map beyond the primary Hopf bifurcation of period 6 which occurs at k = 1. The interesting result obtained is that in the parameter range from k = 1 to the value for blow‐up of all initial conditions, there are successive windows of periods 7, 8 and 9, the last containing a secondary Hopf bifurcation of period 4.

Patterns at Primary Hopf Bifurcations of a Plexus of Identical Oscillators

Mathematically, an oscillator is a system of ordinary differential equations with a periodic limit cycle, usually arising from a Hopf bifurcation for a stationary solution. A plexus is a collection of such oscillators, coupled to each other via one-way or two-way communication channels. Plexuses model a wide number of phenomena in the mathematically oriented sciences. The primary Hopf bifurcation theory of such a plexus is developed. There are features of the bifurcating solutions which depend only on the combinatorics of the plexus and not on the details of the individual oscillators. Such features are called the pattern of the bifurcation. The relation of patterns to the modes of small oscillations of mechanical systems is discussed. More detailed information is developed for weak coupling. The theory is used to analyze two particular types of plexuses; viz, rings and lines of oscillators. It is then shown that the structure of a plexus permits numerical bifurcation computations to be made more efficient, compared to straightforward bifurcation computations. A numerical algorithm is developed to this end. The numerical methods are extended to incorporate higher-order information so as to determine whether the bifurcating branch is sub- or supercritical. Finally some effects of detuning are considered.