Nonlinear Phase Dynamics Research Articles

Multi-field coupling in the scrape-off layer of tokamak plasma

Abstract We study a reduced electrostatic fluid model for the tokamak scrape-off layer (SOL), which incorporates temperature gradient and vorticity gradient as two free energy fields. Two scenarios of field coupling are addressed: (1) sheath condition; (2) vortex wave coupling. For the sheath condition induced field coupling, the poloidal E×B flow shear is coupled with the temperature gradient. Combining an eigenmode analysis and the nonlinear phase dynamics approach, our findings indicate that in the absence of a vorticity gradient, the overall effect of the sheath condition induced flow shear can either stabilize or destabilize the interchange mode, depending on the competition between the flow shear suppression and the temperature gradient driving. This is different from the case where the gradient drive and shear damping are decoupled. When the field coupling is mediated by wave interactions, by setting an idealized step-like temperature and vorticity profiles, a joint mode forms through resonant interaction between the interfacial waves driven by the temperature and vorticity gradients, respectively. Near the phase locking condition, the joint mode can be more unstable than pure temperature gradient driven mode.

The Lagrangian structure, the Euler equation, and second Newton’s law of ultrafast nonlinear optics

With a notable exception of space–time variable swap, pulse evolution equations of ultrafast optics are mathematically similar to the equations of quantum mechanics and beam dynamics. The Lagrangian structure of these equations is, however, much less intuitive. Here, we show that such a structure can be identified via a path-integral analysis of the pulse evolution equation, offering useful insights into the Lagrangian underpinning of ultrafast nonlinear optics. The Euler–Lagrange equation applied to the optical analog of the Lagrange function leads to a second-order differential equation that parallels in its structure second Newton’s law. Stationary-phase space–time paths found by solving such an equation reveal the hidden inner workings behind a vast class of field-waveform transformations in ultrafast optics, including the buildup of nonlinear phase, modulation instabilities, and soliton breather dynamics.

Phase and amplitude evolution in the network of triadic interactions of the Hasegawa–Wakatani system

The Hasegawa–Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices, is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking, where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However, when an additional triad, which is a reflection of the original one with respect to the y axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered, including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a “collective mean field,” individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation are discussed for the same parameters, and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.

Nonlinear motion regimes and phase dynamics of a free standing hybrid riser system subjected to ocean current and vessel motion

A novel structural dynamic model for free standing hybrid riser (FSHR) is established based on the absolute nodal coordinate formulation (ANCF) in the present study. The position and slope coordinates are used in global framework to avoid coordinate transformation, and an accurate geometric relationship is introduced to describe geometric nonlinearity. The current loads on the risers and buoyancy can are simulated by Morrison equation and wake oscillator, respectively. After case validations, the vortex-induced motion (VIM) regimes of the buoyancy can and the phase dynamics of the riser and jumper in a FSHR system, subjected to the ocean current and vessel motion, is numerically investigated. It is found that the VIM of the buoyancy can experiences multiple motion switches among the periodic, quasiperiodic, multiple periodic, transition I and transition II regimes under different vessel motions. The y-motion of the jumper and both x- and y-motions of the riser present the phase trapping phenomenon along the structures. The phase trapping and locking, phase drifting and slipping, and other states are observed for x-motion of the jumper, which form a shape of “fish head” with opening mouth towards lower amplitude or higher period of vessel motion in regime map.

Nonlinear phase dynamics of ideal kink mode in the presence of shear flow

We investigate nonlinear phase dynamics of an ideal kink mode, induced by E × B flow. Here the phase is the cross phase (θ c) between perturbed stream function of velocity () and magnetic field (), i.e. θ c = θ ϕ − θ ψ . A dimensionless parameter, analogous to the Richardson number, (γ kink: the normalized growth rate of the pure kink mode; : normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear. When R i > 1, θ c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R i ≤ 1, θ c enters a phase slipping or oscillating state, corresponding to a nonmodal solution. The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.

Nonlinear phase bores in drift wave-zonal flow dynamics

A minimal model of nonlinear phase dynamics in drift waves is shown to support phase bore solutions. Coupled nonlinear equations for amplitude, phase, and zonal flow are derived for the Hasegawa-Mima system and specialized to the case of spatiotemporally constant amplitude. In that limit, phase curvature (finite second derivative of the phase with respect to the radius) alone generates propagating shear flows. The phase field evolves nonlinearly by a competition between phase steepening and dispersion. The analytical solution of the model reveals that the phase bore solutions so obtained realize the concept of a phase slip in a concrete dynamical model of drift wave dynamics. The implications for phase turbulence are discussed.

Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation

Efficient and unconditionally stable high order time marching schemes are very important but not easy to construct for nonlinear phase dynamics. In this paper, we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for the Cahn-Hilliard equation with provable unconditional stability. In this scheme the nonlinear bulk force are treated explicitly with two second-order linear stabilization terms. The semi-discretized equation is a linear elliptic system with constant coefficients, thus robust and efficient solution procedures are guaranteed. Rigorous error analysis show that, when the time step-size is small enough, the scheme is second order accurate in time with aprefactor controlled by some lower degree polynomial of $1/\varepsilon$. Here $\varepsilon$ is the interface thickness parameter. Numerical results are presented to verify the accuracy and efficiency of the scheme.

On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank-Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.

Output consensus control of multi-agent systems with nonlinear non-minimum phase dynamics

ABSTRACTThis paper considers the output consensus problem in non-minimum phase nonlinear multi-agent systems. The main contribution of the paper is to guarantee achieving consensus in the presence of unstable zero dynamics. To achieve this goal, a consensus protocol consisting of two terms is proposed. The first term is a linear function of the states of each agent employed in order to overcome the non-minimum phase dynamics, and the second term is a function of the output of neighbouring agents which provides coupling among agents and guarantees output consensus in the network. The asymptotic stability of output consensus errors and the boundedness of the states of agents are also studied. A numerical example is presented to show the effectiveness of the proposed approach.

The influences of tissue anisotropy and source activity on power and phase stability of low-frequency EEG rhythms: a mathematical observation of the forward problem model

Spontaneous low-frequency brain rhythms could be associated with many neural functions, which were usually evaluated by physical parameters, such as power, frequency, and phase. However, how the parameters might be affected by the electrical features of brain tissue and the electrical activity of the rhythm source is not yet clearly known. To address this issue, the electrical significance of power and narrow-band phase stability (NBPS) was investigated by simulated rhythms. The rhythms were derived from the oscillatory field potentials (FPs) on a homogeneous sphere model of the solution to the electroencephalogram (EEG) forward problem. The sphere’s electrical feature was set as isotropic or anisotropic conductivity. The source was set as a quasi-static dipole current, whose activity was representative of a low-frequency sine oscillation with a nonlinear phase course, and the source location was changeable. After the instantaneous power and phase of simulated rhythms were estimated by the Hilbert transformation, the NBPS was calculated and the statistical properties of power and NBPS were analyzed. It was found that only nonlinear phase dynamics could lower the NBPS. However, power depended on many factors, such as conducting anisotropy, amplitude and positon of dipole current, and meshes on the sphere model. We hypothesized that NBPS would map the influence of nonlinear phase dynamics on the brain rhythms, but be independent of power. This research might highlight the nonlinear phase dynamics of intrinsic low-frequency oscillations in the brain. The results might be beneficial for the measurement and analysis of spontaneous EEG rhythms.

Simulation on relationship between power/phase stability of low frequency oscillatory potentials and activity of dipole current

The physical parameters, e.g. power and phase, are usually employed in the neural analysis of brain rhythms, which are important in brain function and disease diagnosis. Though there has been extensive work, how both parameters are related to the electrical properties of brain tissue and the sources of brain rhythms has not been fully understood. To address the issue, a simulation is done based on the theory of dipole current. When referring to the solution to the forward problem in electroencephalograph, the brain is regarded as a homogenous sphere model, the electrical features of brain tissue are described by an isotropic electrical conductivity. The source of brain rhythms is simulated by the quasi-static dipole current whose activity is described as a sine oscillation at low frequency. The electrical field generated by the dipole current is considered to be quasi-static. By changing the amplitude and the phase time course of oscillatory dipole current, the distribution of potentials produced by the dipole current at a time-point could be calculated by applying the finite element method to the sphere model. Over a time period of sine oscillation, the oscillatory potentials regarded as the brain rhythms could be produced. Instantaneous power and phase of simulated rhythms are estimated by Hilbert transform, and then a method of phase stability in narrow-band is developed for a single oscillator. To highlight this method, three manners are employed to describe it, i.e., mean relative phase value termed phase preserved index, histogram on rose plane, and phase sorting with the help of EEGLAB. Finally the relationship between two physical parameters and the electrical features of brain tissue/the source activity of brain rhythms is investigated under the conditions of (an) isotropy of conductivity, linear or nonlinear phase dynamics and amplitude, eccentricity of dipole current, etc. The statistical methods of t-test and bootstrapping technology are performed respectively to show the significance of power and phase stability. It is obtained that the power of simulated rhythms decreases with the increase of electrical conductivity, and it is not only proportional to the square of the amplitude of dipole current, but also correlated with the anisotropy of conductivity and the locations of dipole current as well as meshes on the sphere model, however no relevance to other factors. On the contrary, the phase stability of simulated rhythms is correlated only with the non-linear time course of their own phase dynamics. The results imply that the power of brain rhythms is related to many factors such as brain tissue and amplitude of rhythm generator as well as placements of recording electrodes, but the phase stability is related only to the non-linear phase dynamics of brain rhythms. Thus, the electrical significance of the power is more complicated than that of the phase stability. This work might be helpful for understanding in depth the significance of both physical parameters from the perspective of electricity. The narrow-band phase stability of simulated rhythms could highlight the non-linear phase dynamics. It is hypothesized that the phase stability could not only map the synchrony in the neural activity as a custom means of phase coherence, but also reflect directly the non-linearity in phase dynamics, and the more divergent the phase dynamics, the lower the phase stability is, and vice verse. Therefore it is suggested that the phase stability of brain rhythms could be related closely to the non-linear factors to affect the phase dynamics of brain rhythms, e.g., the non-linear phase dynamics of rhythm generators. It is also suggested that both parameters of power and phase stability would offer more neural information.

An ultrafast reconfigurable nanophotonic switch using wavefront shaping of light in a nonlinear nanomaterial

AbstractWe demonstrate a new concept for reconfigurable nanophotonic devices exploiting ultrafast nonlinear control of shaped wavefronts in a multimode nanomaterial consisting of semiconductor nanowires. Femtosecond pulsed laser excitation of the nanowire mat is shown to provide an efficient nonlinear mechanism to control both destructive and constructive interference in a shaped wavefront. Modulations of up to 63% are induced by optical pumping, due to a combination of multimode dephasing and induced transient absorption. We show that part of the nonlinear phase dynamics can be inverted to provide a dynamical revival of the wavefront into an optimized spot with up to 18% increase of the peak to background ratio caused by pulsed laser excitation. The concepts of multimode nonlinear switching demonstrated here are generally extendable to other photonic and plasmonic systems and enable new avenues for ultrafast and reconfigurable nanophotonic devices.

Event-triggered feedback in noise-driven phase oscillators.

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.

Nonlocal Nonlinear Electro-Optic Phase Dynamics Demonstrating 10 Gb/s Chaos Communications

We report on successful 10 Gb/s transmission of a message hidden in a chaotic optical phase over more than 100 km of an installed fiber optic network. This represents the best performance to date for so-called optical chaos communication, a physical layer oriented optical data encryption technique. Such performances was achieved through the use of a recently developed electro-optic nonlinear delay phase dynamics, inspired from differential phase modulation techniques. The setup appears as a superior alternative to the most popular architectures, i.e., the ones involving laser rate equations subjected to delayed feedback. It is compatible with standard dispersion compensation techniques and optical amplification, as shown by two field experiments over installed fiber optic networks.

Effect of Fiber Dispersion on Broadband Chaos Communications Implemented by Electro-Optic Nonlinear Delay Phase Dynamics

We investigate theoretically and experimentally the detrimental effect of fiber dispersion on the synchronization of an optoelectronic phase chaos cryptosystem. We evaluate the root-mean square synchronization error and the cancellation spectra between the emitter and the receiver in order to characterize the quality of the optical fiber communication link. These two indicators explicitly show in temporal and spectral domain how fiber dispersion does negatively affect the phase chaos cancellation at the receiver stage. We demonstrate that the dispersion management techniques used in conventional optical fiber networks, such as dispersion-compensating modules/fibers or dispersion shifted fibers, are also efficient to strongly reduce the detrimental effects of fiber propagation in phase chaos communications. This compatibility therefore opens the way to a successful integration of more than 10-Gb/s phase chaos communications systems in existing networks, even when the fiber link spans over more than 100 km.

Nonlinear Phase Dynamics in a Driven Bosonic Josephson Junction

We study the collective dynamics of a driven two-mode Bose-Hubbard model in the Josephson interaction regime. The classical phase space is mixed, with chaotic and regular components, which determine the dynamical nature of the fringe visibility. For a weak off-resonant drive, where the chaotic component is small, the many-body dynamics corresponds to that of a Kapitza pendulum, with the relative phase φ between the condensates playing the role of the pendulum angle. Using a master equation approach we show that the modulation of the intersite potential barrier stabilizes the φ=π "inverted pendulum" coherent state, and protects the fringe visibility.

Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization

We demonstrate experimentally how nonlinear optical phase dynamics can be generated with an electro-optic delay oscillator. The presented architecture consists of a linear phase modulator, followed by a delay line, and a differential phase-shift keying demodulator (DPSK-d). The latter represents the nonlinear element of the oscillator effecting a nonlinear transformation. This nonlinearity is considered as nonlocal in time since it is ruled by an intrinsic differential delay, which is significantly greater than the typical phase variations. To study the effect of this specific nonlinearity, we characterize the dynamics in terms of the dependence of the relevant feedback gain parameter. Our results reveal the occurrence of regular GHz oscillations (approximately half of the DPSK-d free spectral range), as well as a pronounced broadband phase-chaotic dynamics. Beyond this, the observed dynamical phenomena offer potential for applications in the field of microwave photonics and, in particular, for the realization of novel chaos communication systems. High quality and broadband phase-chaos synchronization is also reported with an emitter-receiver pair of the setup.

Synchronization in a coupled two-layer quasigeostrophic model of baroclinic instability – Part 1: Master-slave configuration

Abstract. Synchronization is studied using a pair of diffusively-coupled, two-layer quasi-geostrophic systems each comprising a single baroclinic wave and a zonal flow. In particular, the coupling between the systems is in the well-known master-slave or one-way configuration. Nonlinear time series analysis, phase dynamics, and bifurcation diagrams are used to study the dynamics of the coupled system. Phase synchronization, imperfect synchronization (phase slips), or complete synchronization are found, depending upon the strength of coupling, when the systems are either in a periodic or a chaotic regime. The results of investigations when the dynamics of each system are in different regimes are also presented. These results also show evidence of phase synchronization and signs of chaos control.

I–V characteristics and optical properties in high-Tc superconductors with alternating junction parameters

We investigate the longitudinal phase dynamics in an intrinsic Josephson-junction array with alternating junction parameters on the basis of the capacitive coupling model. The coupled dynamical equations for the phase differences are derived. The linear dynamics in this system yields two collective phase modes propagating along the c -direction, which are identified with the two longitudinal Josephson-plasma modes observed in the T ∗ -structure high- T c cuprates. The non-linear phase dynamics in this system shows the existence of stable localized rotating modes, which leads to the multiple branch structure in the I – V characteristics. On the basis of the numerical results we predict the I – V characteristics of the T ∗ -structure high- T c cuprates.

Nonlinear oscillatory convection: A quantitative phase dynamics approach

The oscillatory instability of convective rolls, for both free-slip and rigid boundary conditions, is shown to be related to the translational and Galilean invariances of the Oberbeck-Boussinesq equations, which implies that the phase dynamics of the basic roll pattern is second order in time. In the free-slip case the invariance is exact and the instability comes in at zero wavenumber. It is argued that, for low Prandtl number fluids, the effect of the rigid boundaries is to weakly break the Galilean invariance, thereby shifting the instability's critical wavenumber to a finite value. We derive the equations governing the nonlinear phase dynamics and show that, when supercritical, the instability always saturates into travelling waves (as observed in experiments and numerical simulations). In the rigid case, the description is made quantitative by numerically computing both the linear and nonlinear coefficients of the phase equation. It is found that, depending of the values of Prandtl number P and basic roll pattern wavenumber α, the oscillatory instability can be supercritical or subcritical. In the case of mercury ( P = 0.025), our model predicts transition from supercritical onset to subcritical onset, for α slightly below the critical wavenumber for convection onset.