Supercritical Pitchfork Bifurcation Research Articles

Dynamo transition in a five-mode helical model

We construct a five-mode helical dynamo model containing three velocity and two magnetic modes and solve it analytically. This model exhibits dynamo transition via supercritical pitchfork bifurcation. We show that the critical magnetic Reynolds number for dynamo transition (Rmc) asymptotes to constant values for very low and very high magnetic Prandtl numbers (Pm). Beyond dynamo transition, secondary bifurcations lead to periodic, quasi-periodic, and chaotic dynamo states as the forcing amplitude is increased and chaos appears through a quasi-periodic route.

Bifurcation Dynamics of a Particle-Encapsulating Droplet in Shear Flow.

To understand the behavior of composite fluid particles such as nucleated cells and double emulsions in flow, we study a finite-size particle encapsulated in a deforming droplet under shear flow as a model system. In addition to its concentric particle-droplet configuration, we numerically explore other eccentric and time-periodic equilibrium solutions, which emerge spontaneously via supercritical pitchfork and Hopf bifurcations. We present the loci of these solutions around the codimension-two point. We adopt a dynamic system approach to model and characterize the coupled behavior of the two bifurcations. By exploring the flow fields and hydrodynamic forces in detail, we identify the role of hydrodynamic particle-droplet interaction which gives rise to these bifurcations.

Model reduction and identification of nonlinear reactive sputter processes

A model for nonlinear reactive sputter processes is developed based on a system theoretic analysis. Hence, a physical model from the fields of thin film science and vacuum science is analysed and reduced in order to describe the input/output behavior of the process by a nonlinear Abel differential equation. Hereby the input/output behavior is related to a supercritical Pitchfork-bifurcation. The parameter identification of the reduced model is discussed and an experiment for model validation is presented. The experiment shows high conformity between the reduced model and the measured input/output behavior of the process. The reduced model can be used as basis for control design for process stabilization.

Combinatorial optimization using dynamical phase transitions in driven-dissipative systems.

The dynamics of driven-dissipative systems is shown to be well-fitted for achieving efficient combinatorial optimization. The proposed method can be applied to solve any combinatorial optimization problem that is equivalent to minimizing an Ising Hamiltonian. Moreover, the dynamics considered can be implemented using various physical systems as it is based on generic dynamics-the normal form of the supercritical pitchfork bifurcation. The computational principle of the proposed method relies on an hybrid analog-digital representation of the binary Ising spins by considering the gradient descent of a Lyapunov function that is the sum of an analog Ising Hamiltonian and archetypal single or double-well potentials. By gradually changing the shape of the latter potentials from a single to double well shape, it can be shown that the first nonzero steady states to become stable are associated with global minima of the Ising Hamiltonian, under the approximation that all analog spins have the same amplitude. In the more general case, the heterogeneity in amplitude between analog spins induces the stabilization of local minima, which reduces the quality of solutions to combinatorial optimization problems. However, we show that the heterogeneity in amplitude can be reduced by setting the parameters of the driving signal near a regime, called the dynamic phase transition, where the analog spins' DC components map more accurately the global minima of the Ising Hamiltonian which, in turn, increases the quality of solutions found. Last, we discuss the possibility of a physical implementation of the proposed method using networks of degenerate optical parametric oscillators.

Order Reconstruction for Nematics on Squares and Hexagons: A Landau--de Gennes Study

We construct an order reconstruction (OR-)type Landau--de Gennes critical point on a square domain of edge length $2\lambda$, motivated by the well order reconstruction solution numerically reported in [S. Kralj and A. Majumdar, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140276]. The OR critical point is distinguished by a uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small $\lambda$ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.

Symmetry-breaking oscillations in membrane optomechanics

We study the classical dynamics of a membrane inside a cavity in the situation where this optomechanical system possesses a reflection symmetry. Symmetry breaking occurs through supercritical and subcritical pitchfork bifurcations of the static fixed point solutions. Both bifurcations can be observed through variation of the laser-cavity detuning, which gives rise to a boomerang-like fixed point pattern with hysteresis. The symmetry-breaking fixed points evolve into self-sustained oscillations when the laser intensity is increased. In addition to the analysis of the accompanying Hopf bifurcations we describe these oscillations at finite amplitudes with an ansatz that fully accounts for the frequency shift relative to the natural membrane frequency. We complete our study by following the route to chaos for the membrane dynamics.

Nonlinear characteristics of an autoparametric vibration system

The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.

Stability Analysis and Internal Heating Effect on Oscillatory Convection in a Viscoelastic Fluid Saturated Porous Medium Under Gravity Modulation

In this paper, we investigate the combined effect of internal heating and time periodic gravity modulation in a viscoelastic fluid saturated porous medium by reducing the problem into a complex non-autonomous Ginzgburg-Landau equation. Weak nonlinear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number is obtained in terms of the amplitude for oscillatory mode of convection. The influence of viscoelastic parameters on heat transfer has been discussed. Gravity modulation is found to have a destabilizing effect at low frequencies and a stabilizing effect at high frequencies. Finally, it is found that overstability advances the onset of convection, more with internal heating. The conditions for which the complex Ginzgburg-Landau equation undergoes Hopf bifurcation and the amplitude equation undergoes supercritical pitchfork bifurcation are studied.

Onset of double-diffusive Rayleigh-Bénard convection of a moderate Prandtl number binary mixture in cylindrical enclosures

A series of three-dimensional numerical simulations were conducted on double-diffusive Rayleigh-Bénard convection in cylindrical enclosures filled with an isopropanol/water mixture, with the aim of determining the critical threshold of Rayleigh-Bénard convection and the primary bifurcation flow patterns for different geometric dimensions. The Prandtl number of the fluid was 13.2, the aspect ratio of the cylindrical enclosure and the buoyancy ratio N varied from 0.5 to 4 and from −1 to 2, respectively. Results show that the critical Rayleigh number at the primary threshold for the onset of convection is influenced by both the aspect ratio and the buoyancy ratio. When 0<N<2, the solutal gradient has an accelerative effect to the flow destabilization and the critical Rayleigh number decreases with the increase of the buoyancy ratio. When −1<N<0, the fluid stability is enhanced, and the critical Rayleigh number increases with the increase of the absolute value of the buoyancy ratio. Furthermore, the buoyancy ratio has a direct effect on bifurcation sequences and flow pattern formations. When N=0, the system undergoes a supercritical pitchfork bifurcation and passes through various steady flow patterns. When 0<N<2, the flow bifurcates to a non-periodic oscillatory flow. However, when −1<N<0, the flow experiences a subcritical bifurcation and evolves into a periodic oscillatory flow pattern, consisting in an azimuthal rotation. Furthermore, the results have been compared with those obtained for a pure fluid to identify the effect of the buoyancy ratio.

Thermomagnetic instabilities in a vertical layer of ferrofluid: nonlinear analysis away from a critical point

A finite amplitude instability has been analysed to discover the exact mechanism leading to the appearance of stationary magnetoconvection patterns in a vertical layer of a non-conducting ferrofluid heated from the side and placed in an external magnetic field perpendicular to the walls. The physical results have been obtained using a version of a weakly nonlinear analysis that is based on the disturbance amplitude expansion. It enables a low-dimensional reduction of a full nonlinear problem in supercritical regimes away from a bifurcation point. The details of the reduction are given in comparison with traditional small-parameter expansions. It is also demonstrated that Squire’s transformation can be introduced for higher-order nonlinear terms thus reducing the full three-dimensional problem to its equivalent two-dimensional counterpart and enabling significant computational savings. The full three-dimensional instability patterns are subsequently recovered using the inverse transforms The analysed stationary thermomagnetic instability is shown to occur as a result of a supercritical pitchfork bifurcation.

Rotating pendulum with smooth and discontinuous dynamics

This paper focuses on a novel rotating pendulum system which is derived from the coupling rotational model of simple pendulum and mass-spring oscillator. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting smooth and discontinuous characteristics due to the geometry configuration. The equilibrium stability and bifurcations of the unperturbed system are explored showing the subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and saddle-node bifurcation. Furthermore, the complex transitions of multiple well dynamics and various kinds of singular orbits are demonstrated by using nonlinear dynamical technique in both smooth and discontinuous case. For the perturbed system, the primary resonance of small angle oscillation is investigated analytically by introducing the complete Jacobian elliptic integrals. Finally, complicated resonant structures of period, quasi-period and stochastic phenomena are presented for the system with unique harmonic perturbation. Bifurcation diagram, Poincaré section and basin analysis are carried out to demonstrate the co-existing different types of periodic motion and the transitions of chaotic motion between smooth and discontinuous case.

On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type

This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.

Stability and boundary equilibrium bifurcations of modified Chua’s circuit with smooth degree of 3

Chua’s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua’s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the supercritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous (PWSC) systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. Therefore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.

Collision and annihilation of relative equilibrium points around asteroids with a changing parameter

In this work, we investigate the bifurcations of relative equilibria in the gravitational potential of asteroids. A theorem concerning a conserved quantity, which is about the eigenvalues and number of relative equilibria, is presented and proved. The conserved quantity can restrict the number of non-degenerate equilibria in the gravitational potential of an asteroid. It is concluded that the number of non-degenerate equilibria in the gravitational field of an asteroid varies in pairs and is an odd number. In addition, the conserved quantity can also restrict the kinds of bifurcations of relative equilibria in the gravitational potential of an asteroid when the parameter varies. Furthermore, studies have shown that there exist transcritical bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations, saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative equilibria in the gravitational potential of asteroids. It is found that for the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the number of relative equilibria changes from 7 to 5 to 3 to 1, and the bifurcations for the relative equilibria are saddle-node bifurcations and saddle-saddle bifurcations.

Dynamics of Rocking Semicircular, Parabolic, and Semi-Elliptical Disks: Equilibria, Stability, and Natural Frequencies

This paper performs a theoretical and experimental investigation of the natural frequency and stability of rocking semicircular, parabolic, and semi-elliptical disks. Horace Lamb's method for deriving the natural frequency of an arbitrary rocking disk is applied to three shapes with semicircular, parabolic, and semi-elliptical cross sections, respectively. For the case of the semicircular disk, the system's equation of motion is derived to verify Lamb's method. Additionally, the rocking semicircular disk is found to always have one stable equilibrium position. For the cases of the parabolic and semi-elliptical disks, this investigation reveals a supercritical pitchfork bifurcation for changes in a single geometric parameter which indicates that the systems can exhibit bistable behavior. Comparisons between experimental validation and theory show good agreement.

Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams

This paper investigates the three-dimensional motion characteristics of perfect and imperfect Timoshenko microbeams under mechanical and thermal forces; the mechanical properties of the microbeam are considered temperature-dependent. The centerline of the microbeam is considered to be extensible and the equations of motion for the longitudinal, transverse, and rotational motions are derived by means of the extended Hamilton’s principle and the modified couple stress theory. These three coupled nonlinear partial differential equations are discretized by means of Galerkin’s technique, yielding a set of second-order nonlinear ordinary differential equations. These equations are solved by means of the pseudo-arclength continuation technique and via an eigenvalue analysis, for the nonlinear and linear analyses, respectively. The geometrically perfect microbeam remains stable at its original static equilibrium position up to the temperature when it loses stability by divergence via a supercritical pitchfork bifurcation; the post-buckling state is obtained and resonant response over it is analysed. For the initially imperfect microbeam, as the temperature is increased, the initial curvature amplitude increases and no instabilities occur; the resonant response of the system over the new deflected configuration is examined numerically. The effect of different parameters on the nonlinear behaviour of the system is studied.

Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity.

Spatiotemporal waves of synchronized activity are known to arise in oscillatory neural networks with lateral inhibitory coupling. How such patterns respond to dynamic changes in coupling strength is largely unexplored. The present study uses analysis and simulation to investigate the evolution of wave patterns when the strength of lateral inhibition is varied dynamically. Neural synchronization was modeled by a spatial ring of Kuramoto oscillators with Mexican hat lateral coupling. Broad bands of coexisting stable wave solutions were observed at all levels of inhibition. The stability of these waves was formally analyzed in both the infinite ring and the finite ring. The broad range of multi-stability predicted hysteresis in transitions between neighboring wave solutions when inhibition is slowly varied. Numerical simulation confirmed the predicted transitions when inhibition was ramped down from a high initial value. However, non-wave solutions emerged from the uniform solution when inhibition was ramped upward from zero. These solutions correspond to spatially periodic deviations of phase that we call ripple states. Numerical continuation showed that stable ripple states emerge from synchrony via a supercritical pitchfork bifurcation. The normal form of this bifurcation was derived analytically, and its predictions compared against the numerical results. Ripple states were also found to bifurcate from wave solutions, but these were locally unstable. Simulation also confirmed the existence of hysteresis and ripple states in two spatial dimensions. Our findings show that spatial synchronization patterns can remain structurally stable despite substantial changes in network connectivity.

On the Secondary Instability of Forced and Unforced Laminar Separation Bubbles

Previous studies demonstrate that the primary instability of laminar separation bubbles (LSB) on a flat-plate in the absence of external forcing is a three-dimensional centrifugal one. This work develops a weakly non-linear expansion of the associated symmetry-breaking bifurcation, showing that it corresponds to a supercritical pitchfork bifurcation. The secondary instability of the fully 3D bifurcated LSB is then investigated by means of the temporal instability of 3D global modes, computed either as solutions of a 3D eigenvalue problem, or based on a WKB approximation and the existence of local regions of absolute instability of the cross-stream planes. Both methodologies recover an amplified global oscillator, originated by the spanwise velocity gradients, that can explain the origin of the unsteadiness observed in numerical simulations of unforced LSBs with peak reversed flows below 15%. Finally, the manner in which the three-dimensionalization of the LSB due to the primary instability affects its amplifier behavior is investigated by means of 3D Parabolized Stability Equations computations. The 3D LSBs are found to distort the incoming plane waves into a more amplified system of oblique waves.

Breaking an epigenetic chromatin switch: curious features of hysteresis in Saccharomyces cerevisiae telomeric silencing.

In addition to gene network switches, local epigenetic modifications to DNA and histones play an important role in all-or-none cellular decision-making. Here, we study the dynamical design of a well-characterized epigenetic chromatin switch: the yeast SIR system, in order to understand the origin of the stability of epigenetic states. We study hysteresis in this system by perturbing it with a histone deacetylase inhibitor. We find that SIR silencing has many characteristics of a non-linear bistable system, as observed in conventional genetic switches, which are based on activities of a few promoters affecting each other through the abundance of their gene products. Quite remarkably, our experiments in yeast telomeric silencing show a very distinctive pattern when it comes to the transition from bistability to monostability. In particular, the loss of the stable silenced state, upon increasing the inhibitor concentration, does not seem to show the expected saddle node behavior, instead looking like a supercritical pitchfork bifurcation. In other words, the ‘off’ state merges with the ‘on’ state at a threshold concentration leading to a single state, as opposed to the two states remaining distinct up to the threshold and exhibiting a discontinuous jump from the ‘off’ to the ‘on’ state. We argue that this is an inevitable consequence of silenced and active regions coexisting with dynamic domain boundaries. The experimental observations in our study therefore have broad implications for the understanding of chromatin silencing in yeast and beyond.

The motion of a 2D pendulum in a channel subjected to an incoming flow

AbstractThe flow around a tethered cylinder subjected to an incoming flow transverse to its main axis and confined in a channel is investigated by means of global stability analysis of the full coupled body–fluid problem. When the cylinder is strongly confined (ratio of cylinder diameter to cell height,$D/H=0.66$) we retrieve the confinement-induced instability (CIV) discovered by Seminet al. (J. Fluid Mech., vol. 690, 2011, pp. 345–365), which sets in at a Reynolds number below the vortex-induced vibration threshold. For a moderately confined case ($D/H=0.3$), a new steady static instability is discovered, referred to as confinement-induced divergence (CID). This instability saturates into an asymmetric steady solution through a supercritical pitchfork bifurcation. In addition, the CIV and CID instabilities are studied via a reduced model obtained by considering a quasi-static response of the fluid, allowing for tracing back the physical mechanisms responsible for the instabilities.