Strong Nonlinear Regime Research Articles

Large basins of attraction for control-based continuation of unstable periodic states

Numerical continuation tools are nowadays standard to analyse nonlinear dynamical systems by numerical means. These powerful methods are unfortunately not available in real experiments without having access to an accurate mathematical model. Implementing such a concept in real world experiments using control and data processing to track unstable states and their bifurcations, requires robust control techniques with large basins and good global properties. Here we propose design principles for control techniques for periodic states which lead to large basins and which are robust, without the need to have access to a detailed mathematical model. Our analytic considerations for the control design will be based on weakly nonlinear analysis of periodically driven oscillator systems. We then demonstrate by numerical means that in strong nonlinear regimes successful control with large basins of attraction can be achieved when only plain time series data are available.

Electromagnetic TE- and TM-waves propagation in a plane waveguide covered with graphene characterized by nonlinear conductivity

Background. Guiding properties of waveguiding structures with graphene are of great importance for various applications and have been studied in many papers. In all such studies, graphene was characterized, as a rule, by linear surface conductivity. However, if the intensity of an electromagnetic wave is large enough, the interaction of graphene with the electromagnetic wave becomes nonlinear; in this case, it is more correct to describe graphene by nonlinear conductivity.
 Aim. This work is aimed at studying the influence of cubic nonlinearity of graphene, corresponding to the so-called self-action effects (not affecting the frequency of the incident wave), on the propagation of TE- and TM-polarized waves in the structure, which is a plain dielectric layer covered on one side by graphene.
 Methods. In this study, the guiding properties of the waveguide are studied using primarily an analytical approach. Thus, from Maxwell's equations, material equations and boundary conditions, a couple of dispersion equations for TE-and TM-polarized waves is derived and then its solvability is studied. In addition, some numerical experiments are carried out in the study.
 Results. The dispersion equations of the studied waveguiding structure for TE- and TM-polarized waves are derived in explicit form. Studying analytically obtained equations, conditions for waveguide parameters are found, providing the existence of a given number of waveguide modes. In addition, some numerical results are obtained in the paper, which give an idea of the influence of nonlinear effects on the electromagnetic waves propagating in the structure.
 Conclusion. The results obtained in this paper reveal two effects related to the cubic nonlinearity of graphene. Firstly, in a plain dielectric layer with graphene coating in the strong nonlinear regime TE-waves with longer wavelength and TM-waves with shorter wavelength propagate compared to electromagnetic waves that propagate in the same structure in the linear regime. Secondly, the strong cubic nonlinearity leads to a greater localization of the electromagnetic wave within the waveguiding structure.

Effective protocol for generating NOON states of resonator modes.

We propose a protocol for the generation of NOON states of resonator modes. The physical model is composed of two Kerr-nonlinear resonators and a four-level qudit. Using the off-resonant couplings between the resonators and the qudit, qudit-level-dependent frequency shifts on the two resonators are induced. The frequency shifts allow us to drive different resonators to the N-photon state when the qudit is in different intermediate levels. Consequently, the generation of NOON states with arbitrary photon number N can be completed in only three steps, i.e., driving the qudit to a superposition state of the two intermediate levels, driving one of the resonators to its N-photon state, and driving the qudit back to its ground level. Numerical simulations show that, in the regime of strong Kerr nonlinearity and coupling strengths, the protocol can produce the NOON state with high fidelity in the cases of different photon numbers. In addition, it is possible for the protocol to produce acceptable fidelity in the presence of systematic errors and decoherence factors. Therefore, the protocol may provide some useful perspectives for effective generation of photonic NOON states.

A Statistical Hypothesis Testing Strategy for Adaptively Blending Particle Filters and Ensemble Kalman Filters for Data Assimilation

Abstract Particle filters avoid parametric estimates for Bayesian posterior densities, which alleviates Gaussian assumptions in nonlinear regimes. These methods, however, are more sensitive to sampling errors than Gaussian-based techniques such as ensemble Kalman filters. A recent study by the authors introduced an iterative strategy for particle filters that match posterior moments—where iterations improve the filter’s ability to draw samples from non-Gaussian posterior densities. The iterations follow from a factorization of particle weights, providing a natural framework for combining particle filters with alternative filters to mitigate the impact of sampling errors. The current study introduces a novel approach to forming an adaptive hybrid data assimilation methodology, exploiting the theoretical strengths of nonparametric and parametric filters. At each data assimilation cycle, the iterative particle filter performs a sequence of updates while the prior sample distribution is non-Gaussian, then an ensemble Kalman filter provides the final adjustment when Gaussian distributions for marginal quantities are detected. The method employs the Shapiro–Wilk test to determine when to make the transition between filter algorithms, which has outstanding power for detecting departures from normality. Experiments using low-dimensional models demonstrate that the approach has a significant value, especially for nonhomogeneous observation networks and unknown model process errors. Moreover, hybrid factors are extended to consider marginals of more than one collocated variables using a test for multivariate normality. Findings from this study motivate the use of the proposed method for geophysical problems characterized by diverse observation networks and various dynamic instabilities, such as numerical weather prediction models. Significance Statement Data assimilation statistically processes observation errors and model forecast errors to provide optimal initial conditions for the forecast, playing a critical role in numerical weather forecasting. The ensemble Kalman filter, which has been widely adopted and developed in many operational centers, assumes Gaussianity of the prior distribution and solves a linear system of equations, leading to bias in strong nonlinear regimes. On the other hand, particle filters avoid many of those assumptions but are sensitive to sampling errors and are computationally expensive. We propose an adaptive hybrid strategy that combines their advantages and minimizes the disadvantages of the two methods. The hybrid particle filter–ensemble Kalman filter is achieved with the Shapiro–Wilk test to detect the Gaussianity of the ensemble members and determine the timing of the transition between these filter updates. Demonstrations in this study show that the proposed method is advantageous when observations are heterogeneous and when the model has an unknown bias. Furthermore, by extending the statistical hypothesis test to the test for multivariate normality, we consider marginals of more than one collocated variable. These results encourage further testing for real geophysical problems characterized by various dynamic instabilities, such as real numerical weather prediction models.

Measuring saturable nonlinearity in atomic vapor via direct spatial mapping.

We demonstrate a scheme to measure the saturable nonlinearity of atomic vapor by mapping its nonlinear response function onto a light beam profile. Our analysis shows that a part of a nonlinear optical solution solved in a model governing the nonlinear beam dynamics in atomic vapor can be used to perform this measurement, even in the presence of large absorption. A desired beam profile is achieved by an evolution of a well-known structured beam, namely the Airy beam. Using this simple yet effective method, we retrieve the saturable nonlinear response function of rubidium (Rb) atomic vapor in experiment, and employ it in light propagation simulation that reproduces well observed nonlinear dynamics, which nevertheless cannot be fitted in a strong nonlinear regime with an ideal Kerr approximation. Our method is applicable to a broad spectrum of materials featured with saturable nonlinearities.

Violation of the magnonic Wiedemann-Franz law in the strong nonlinear regime

The celebrated Wiedemann-Franz (WF) law which governs the relation between charge and heat transport traces back to the experimental discovery in 1853 by Wiedemann and Franz. Despite the fundamental difference of the quantum-statistical properties between fermions and bosons, the linear-in-$T$ behavior of the WF law at low temperatures has recently been found to be the universal property by the discovery of the WF law for magnon transport. However, the WF law is for the linear response, and whether or not the universal law is valid even in the nonlinear regime of Bose systems remains an open issue. Here we provide a solution to this fundamental challenge. We show that the ratio of the thermal to spin transport coefficient of magnons in topologically trivial insulating magnets exhibits a different behavior from the linear response and the universal law breaks down in the strong nonlinear regime. This finding is within experimental reach with current device and measurement technologies. Our discovery is the key ingredient in magnon-based spintronics, in the evaluation of the figure of merit for thermomagnetic conversion elements of spintronics devices.

Time-delayed interactions on acoustically driven bubbly screens.

The influence of the compressibility effects is discussed, including the time delays on the dynamics of acoustically excited bubbly screens. In the linear regime, it is shown that the proposed model for the infinite bubbly screen recovers the results predicted by the effective medium theory (EMT) up to the second order without introducing any fitting parameter when the wavelength is large compared to the inter-bubble distance. However, the effect of boundaries on the finite bubbly screens is shown to lead to the appearance of multiple local resonances and characteristic periodic structures, which limit the applicability of the EMT. In addition, a local resonance phenomenon in the liquid spacings between the bubbles is observed for both the infinite and finite bubbly screens with crystal structures, and these effects vanish as the crystal structure is perturbed. In the nonlinear regime, the current model is treated with time-delay effects as a delay differential equation, which is directly solved numerically. The appearance of an optimal distance for the subharmonic emission for the crystal structures is shown, and the accuracy of the EMT in the strong nonlinear regime is discussed.

On the substantial anomalous absorption regime at the extraordinary wave decay into trapped and propagating upper hybrid waves

We analyze the strong nonlinear regime of low power-threshold parametric decay instability (PDI) of a pump extraordinary wave. The PDI leads to the excitation of two upper hybrid (UH) waves only one of which is trapped in the vicinity of the local maximum of a non-monotonic density profile. The pump depletion and the multi-step secondary decays of the localized daughter UH wave are treated as the mechanisms leading to the instability saturation. It is shown that in the case of the even-step secondary cascade decay of trapped primary UH wave the pump wave depletion is responsible for the transition of a nonlinear phenomenon into a stationary regime, and the strong anomalous absorption is possible.

Modeling linear and nonlinear fluid flow through sheared rough-walled joints taking into account boundary stiffness

Abstract This study modeled four hundred fluid flow cases through shear-induced aperture fields of a rough-walled joint with self-affine surface characteristics under different boundary stress and stiffness conditions by solving the Navier-Stokes equations. The results show that as the hydraulic gradient increases from 10−8 to 101, the fluid flow experiences changes characterized by a linear regime, a weak nonlinear regime and a strong nonlinear regime. The permeability is a constant in the linear regime and starts to decline when entering the weak nonlinear regime, and the declining rate gradually converges when reaching the strong nonlinear regime. Such transition is poorly indicated by the global Reynolds number alone; it is instead primarily governed by the magnitudes and distribution characters of local Reynolds numbers. The boundary stiffness has linear/nonlinear relations with the permeability, depending on the magnitude of the joint aperture, which is governed by surface roughness, joint compressive strength and mechanical boundary conditions. With increasing boundary stiffness from 0 to 2.0 GPa/m, the ratio of permeability in y-direction to that in x-direction increases from 2.49 to 26.49 by approximately a factor of 10, suggesting that anomalous flow may prevail in sheared zones with stiff surrounding confinements.

Dynamics of solitons for nonlinear quantum walks

We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703; Maeda et al 2018 Quantum Inf. Process. 17 215). It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703). In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.

Cross-correlation between photons and phonons in quadratically coupled optomechanical systems

We study photon, phonon statistics, and the cross-correlation between photons and phonons in a quadratically coupled optomechanical system. Photon blockade, phonon blockade, and strong anticorrelation between photons and phonons can be observed in the same parameter regime with the effective nonlinear coupling between the optical and the mechanical modes, enhanced by a strong optical driving field. Interestingly, an optimal value of the effective nonlinear coupling strength for the photon blockade is not within the strong nonlinear coupling regime. This abnormal phenomenon results from the destructive interference between different paths for two-photon excitation in the optical mode with a moderate effective nonlinear coupling strength. Furthermore, we show that phonon (photon) pairs or correlated photons and phonons can be generated in the strong nonlinear coupling regime with a proper detuning between the weak mechanical driving field and mechanical mode. Our results open up a way to generate anticorrelated and correlated photons and phonons, which may have important applications in quantum information processing.

Approximate fully analytical Fourier series solution to the forced and damped Helmholtz–Duffing oscillator

• A new method to obtain approximate fully analytical steady state solutions of NLODEs is presented. • The method is applied to the forced and damped Helmholtz–Duffing oscillator. • We obtain improved approximate fully analytical Fourier series solutions valid for strong nonlinearities . • The solutions describe very well the principal and superharmonic resonances. • The method allows obtaining explicit expressions for the coefficients of the Fourier series solution. We obtain improved approximate fully analytical steady state solutions for the ordinary differential equation of the forced and damped anharmonic oscillator with quadratic and cubic nonlinearities. The solutions, are obtained in the form of a truncated Fourier series using the harmonic balance method (HBM). In order to obtain explicit analytical expressions for the Fourier coefficients we introduce the hypothesis of weak interdependence between the equations in the system of nonlinear algebraic equations produced by the HBM. Comparison with the numerical solution shows that the analytical results for the first harmonics which we investigated are valid in comparatively strong nonlinear regimes, where previously obtained expressions are completely inaccurate. Our expressions are correct over a wide range of the physical parameters, such as excitation frequency , nonlinearity and forcing. The proposed method of solution is expected to be suitable for other nonlinear ordinary differential equations.

Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III

Nonlinear rheological properties of chiral crystal cholesteryl oleyl carbonate (COC) in blue phase III (BPIII) were investigated under different shear deformations: large amplitude oscillatory shear, step shear deformation, and continuous shear flow. Rheology of the liquid crystal is significantly affected by structural rearrangement of defects under shear flow. One of the examples on the defect-mediated rheology is the blue phase rheology. Blue phase is characterized by three dimensional network structure of the disclination lines. It has been numerically studied that the rheological behavior of the blue phase is dominated by destruction and creation of the disclination networks. In this study, we find that the nonlinear viscoelasticity of BPIII is characterized by the fracture of the disclination networks. Depending on the degree of the fracture, the nonlinear viscoelasticity is divided into two regimes; the weak nonlinear regime where the disclination network locally fractures but still shows elastic response, and the strong nonlinear regime where the shear deformation breaks up the networks, which results in a loss of the elasticity. Continuous shear deformation reveals that a series of the fracture process delays with shear rate. The shear rate dependence suggests that force balance between the elastic force acting on the disclination lines and the viscous force determines the fracture behavior.

Zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian with a gain cavity

Under the weak driving limitation, we find that the strong photon blockade can be triggered when all the eigenvalues of the non-Hermitian Hamiltonian are equal to zero, so we name this the zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian. The possibility of realizing single- and two-photon blockades is investigated in the cavity QED system. A negative effective decay rate of the cavity is required, so the gain medium is needed to compensate the intrinsic loss rate of the cavity. The photon antibunching can be realized in both strong and weak nonlinear regimes, which is different from the conventional photon blockade and unconventional photon blockade. Our scheme has paved an avenue towards the study of photon blockades.

Temperature and frequency dependent mean free paths of renormalized phonons in nonlinear lattices

Unraveling general properties of renormalized phonons are of fundamental relevance to the heat transport in the regime of strong nonlinearity. In this work, we directly study the temperature and frequency dependent mean free path (MFP) of renormalized phonons with the newly developed numerical tuning fork method. The typical 1D nonlinear lattices such as Fermi–Pasta–Ulam β lattice and lattice are investigated in detail. Interestingly, it is found that the MFPs are inversely proportional to the frequencies of renormalized phonons rather than the square of phonon frequencies predicted by existing phonon scattering theory.

Collapse of the surface dusty plasma waves under the plasma–beam instability

The nonlinear dynamics of the dusty plasma–dusty beam instability is investigated in the dusty plasma waveguides bounded by dielectrics. The dusty plasma includes the positive ions as the light component and the negative dust as the heavy component. A beam of dust particles moves along the waveguide. The set of hydrodynamic equations for the dust and beam particles, namely, the continuity equations and the equations for the momentum jointly with the Poisson one are used. The Boltzmann distribution is used for the ions. The electric and hydrodynamic boundary conditions are applied at the interfaces. The simulations have demonstrated that the dusty sound waves of small amplitudes are the subject to amplification with a high increment due to the convective instability, even when the concentration of the beam particles is ≤0.1 of the uniform dust concentration. The amplification very rapidly transits to the regime of strong surface nonlinearity, and near the interfaces the variations of the dust concentration reach extremely high values, where the collapse of the beam dust component occurs.

Intermittency and emergence of coherent structures in wave turbulence of a vibrating plate.

We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long-time numerical simulations makes this system extremely valuable for wave turbulence studies. The purely 2D character of dynamics of the elastic plate makes it much simpler to handle compared to much more complex 3D physical systems that are typical of geo- and astrophysical issues (ocean surface or internal waves, magnetized plasmas or strongly rotating and/or stratified flows). When the forcing is small the observed wave turbulence is consistent with the predictions of the weak turbulent theory. Here we focus on the case of stronger forcing for which coherent structures can be observed. These structures look similar to the folds and D-cones that are commonly observed for strongly deformed static thin elastic sheets (crumpled paper) except that they evolve dynamically in our forced system. We describe their evolution and show that their emergence is associated with statistical intermittency (lack of self similarity) of strongly nonlinear wave turbulence. This behavior is reminiscent of intermittency in Navier-Stokes turbulence. Experimental data show hints of the weak to strong turbulence transition. However, due to technical limitations and dissipation, the strong nonlinear regime remains out of reach of experiments and therefore has been explored numerically.

Nondegenerate Parametric Resonance in a Tunable Superconducting Cavity

We develop a theory for non-degenerate parametric resonance in a tunable superconducting cavity. We focus on nonlinear effects that are caused by nonlinear Josephson elements connected to the cavity. We analyze parametric amplification in a strong nonlinear regime at the parametric instability threshold, and calculate maximum gain values. Above the threshold, in the parametric oscillator regime the linear cavity response diverges at the oscillator frequency at all pump strengths. We show that this divergence is related to the continuous degeneracy of the free oscillator state with respect to the phase. Applying on-resonance input lifts the degeneracy and removes the divergence. We also investigate the quantum noise squeezing. It is shown that in the strong amplification regime the noise undergoes four-mode squeezing, and that in this regime the output signal to noise ratio can significantly exceed the input value. We also analyze the intermode frequency conversion and identify parameters at which full conversion is achieved.

Nematicon-enhanced spontaneous symmetry breaking

ABSTRACTWe investigate topological and optical spontaneous symmetry breaking in nematic liquid crystals subject to the Fréedericksz transition. Specular nematicon states couple to mirrored distributions of the director due to symmetry breaking, with transverse velocities controlled by beam power in the strong nonlinear regime. Hysteresis in transverse velocity versus incidence angle is observed in the soliton regime.

The mise en scéne of memristive networks: effective memory, dynamics and learning

We discuss the properties of the dynamics of purely memristive circuits using a recently derived consistent equation for the internal memory variables of the involved memristors. In particular, we show that the number of independent memory states in a memristive circuit is constrained by the circuit conservation laws, and that the dynamics preserves these symmetry by means of a projection on the physical subspace. Moreover, we discuss other symmetries of the dynamics under various transformations of the involved variables, and study the weak and strong non-linear regimes of the dynamics. In the strong regime, we derive a conservation law for the internal memory variable. We also provide a condition on the reality of the eigenvalues of Lyapunov matrices. The Lyapunov matrix describes the dynamics close to a fixed point, for which show that the eigenvalues can be imaginary only for mixtures of passive and active components. Our last result concerns the weak non-linear regime, showing that the internal memory dynamics can be interpreted as a constrained gradient descent, and provide the functional being minimized. This latter result provides another direct connection between memristors and learning.