Onset Of Bifurcations Research Articles

Isochronous bifurcations of magnetic islands in tokamaks

On a rational magnetic surface, an isochronous bifurcation transforms one island chain into another chain with the same winding number. This transformation has been the subject of recent studies in tokamak plasmas. Namely, visco-resistive magnetohydrodynamic simulations of NSTX-U and DIII-D plasmas showed the onset of bifurcations with new magnetic isochronous islands for two competing helical perturbations on the same rational magnetic surface. To investigate these bifurcations, we use a cylindrical plasma model, with first-order correction for toroidicity, subject to externally applied magnetic perturbations, generated by a pair of resonant helical windings (RHWs) on the external wall and superposed to a helical current sheet (HCS) located on a rational plasma surface. We numerically integrate the magnetic field line equation and show that isochronous islands emerge when the perturbation created by the HCS increases. We present examples of such bifurcations on primary and secondary magnetic surfaces for different RHW configurations.

Bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem

We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number Q increases and compare it to the standard (non-magnetic) Rayleigh–Bénard problem. We compute steady states at a high Chandrasekhar number of Q=103 over a range of the Rayleigh number 0≤Ra≤105. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as Q increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at Q=103. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle–node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when Q increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.

NOVEL RESULTS ON BIFURCATIONS FOR A FRACTIONAL-ORDER NEURAL NETWORK WITH NEUTRAL DELAYS

In comparison with numerous available results with regard to bifurcations on account of fractional-order neural networks (FONNs) with the retarded delays, this paper is concerned with the stability and bifurcation of FONN with neutral delays. Delay-induced bifurcation results are firstly, procured and the bifurcation diagrams with respect to different delays are nicely plotted. Then it reveals that both time delay and fractional order have important effects in affecting the stability and bifurcation of the developed FONN. It further detects that FONN involving neutral delays can postpone the onset of bifurcations of the developed FONN compared with FONN with the retarded delays. It hints that neutral delays have a vital impact on the dynamics of FONNs in comparison with FONN with the retarded delays, which has outstanding performance in maintaining stability of FONNs. Eventually, numerical examples are addressed to check the availability of the developed results. The obtained criterion can be evaluated as an important contribution to the stability issue of the considered neural networks (NNs).

Hopf bifurcations in quaternary dynamical systems of rotating thermofluid mixtures, driven by spectrum characteristic coefficients

The paper is concerned with the onset of bifurcations in fluid mixtures. The instability of thermal conduction state in a rotating fluid layer heated and salted from below, is analyzed. The Taylor number threshold for the transition to Hopf bifurcations, in simple closed form, is obtained. The basic property of each coefficient of the spectrum equation to drive instability and type of bifurcation, is shown and applied.

A novel approach to finding mechanical properties of nanocrystal layers.

Flexible, bendable, stretchable devices represent the future of electronics for a wide range of real-world applications. Due to the fact that these technologies deviate significantly from traditional wafer technologies there is a need to understand and engineer material systems that allow large elastic deformations present in such devices, which requires knowledge about the mechanical properties of these material systems. Here we evaluate the mechanical properties of a bilayer polydimethylsiloxane (PDMS)/silicon nanocrystal system. By observing the formation of instabilities due to finite bending deformation and applying theoretical modeling, we estimated the neo-Hookean coefficient (analogous to shear modulus at low stress/strain) of the silicon nanocrystal film to be 345 ± 23 kPa. The method used here represents a novel approach to evaluating these properties and is widely applicable to many different combinations of systems of nanocrystals and elastomers.

Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems

Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many systems exhibit non-smooth dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems.

New approach of controlling cardiac alternans

The alternans of the cardiac action potential duration is a pathological rhythm. It is considered to be relating to the onset of ventricular fibrillation and sudden cardiac death. It is well known that, the predictive control is among the control methods that use the chaos to stabilize the unstable fixed point. Firstly, we show that alternans (or period-2 orbit) can be suppressed temporally by the predictive control of the periodic state of the system. Secondly, we determine an estimation of the size of a restricted attraction's basin of the unstable equilibrium point representing the unstable regular rhythm stabilized by the control. This result allows the application of predictive control after one beat of alternans. In particular, using predictive control of periodic dynamics, we can delay the onset of bifurcations and direct a trajectory to a desired target stationary state. Examples of the numerical results showing the stabilization of the unstable normal rhythm are given.

Effects of thermo hydrodynamic (THD) floating ring bearing model on rotordynamic bifurcation

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.

A Hamiltonian approach to model and analyse networks of nonlinear oscillators: Applications to gyroscopes and energy harvesters

Over the past twelve years, ideas and methods from nonlinear dynamics system theory, in particular, group theoretical methods in bifurcation theory, have been used to study, design, and fabricate novel engineering technologies. For instance, the existence and stability of heteroclinic cycles in coupled bistable systems has been exploited to develop and deploy highly sensitive, low-power, magnetic and electric field sensors. Also, patterns of behaviour in networks of oscillators with certain symmetry groups have been extensively studied and the results have been applied to conceptualize a multifrequency up /down converter, a channelizer to lock into incoming signals, and a microwave signal generator at the nanoscale. In this manuscript, a review of the most recent work on modelling and analysis of two seemingly different systems, an array of gyroscopes and an array of energy harvesters, is presented. Empirical values of operational parameters suggest that damping and external forcing occur at a lower scale compared to other parameters, so that the individual units can be treated as Hamiltonian systems. Casting the governing equations in Hamiltonian form leads to a common approach to study both arrays. More importantly, the approach yields analytical expressions for the onset of bifurcations to synchronized oscillations. The expressions are valid for arrays of any size and the ensuing synchronized oscillations are critical to enhance performance.

Bifurcations in dissipative fermionic dynamics

The Boltzmann-Langevin One-Body model (BLOB), is a novel one-body transport approach, based on the solution of the Boltzmann-Langevin equation in three dimensions; it is used to handle large-amplitude phase-space fluctuations and has a broad applicability for dissipative fermionic dynamics. We study the occurrence of bifurcations in the dynamical trajectories describing heavy-ion collisions at Fermi energies.The model, applied to dilute systems formed in such collisions, reveals to be closer to the observation than previous attempts to include a Langevin term in Boltzmann theories. The onset of bifurcations and bimodal behaviour in dynamical trajectories, determines the fragment-formation mechanism. In particular, in the proximity of a threshold, fluctuations between two energetically favourable mechanisms stand out, so that when evolving from the same entrance channel, a variety of exit channels is accessible.This description gives quantitative indications about two threshold situations which characterise heavy-ion collisions at Fermi energies. First, the fusion-to-multifragmentation threshold in central collisions, where the system either reverts to a compact shape, or splits into several pieces of similar sizes. Second, the transition from binary mechanisms to neck fragmentation (in general, ternary channels), in peripheral collisions.

Static and Dynamic Aspects of Voltage-Power relationships in Electric Power Systems

This paper investigates various aspects of voltage-power relationships in power systems. Conditions for feasible generating points are analytically derived for a simple power system with a detailed generator model. The set of Hopf Bifurcation points for the sample system are visualized on the feasible operating surface. The effect of damping on the onset of bifurcations in the system is briefly investigated. The role of different generator models in isolating bifurcations and subsidiary phenomena in a multi-machine system are also presented. The effect of nonlinear indices of voltage dependent load models on system bifurcations is dealt with as also the effect of AVR gains.

Detecting the onset of bifurcations and their precursors from noisy data

We study the problem of the detection of noise-induced precursors of periodic motion instabilities in stochastic dynamical systems. In particular, we concentrate on the period-doubling bifurcation. We have developed a statistical method to detect the onset of bifurcations and their precursors based on the previously established topological recurrence technique.

Transition to Chaos for Ballooning Modes Stabilized by Finite Larmor Radius Effects

The nonlinear dynamics of interacting ballooning modes, stabilized by finite Larmor radius effects is analyzed in terms of a set of equations, which exhibit stochastic properties. These are explicitly shown to depend on the balance between shear and driving pressure force. The onset of bifurcations and chaotic behaviour are directly identified with certain values of parameters characterizing the physical system.