Quadratic Nonlinearity Research Articles

Second‐Harmonic Generation via Nonlinear Raman–Nath Diffraction in an Optical Fibonacci Superlattice

AbstractNonlinear Raman–Nath diffraction is an effective method for frequency conversion in the presence of phase mismatch. In a structure featuring periodically modulated quadratic nonlinearity, the available frequency‐doubled patterns generated from nonlinear Raman–Nath diffraction constitute uniformly spaced points due to the one parametric decided reciprocal lattice vectors. Herein, nonlinear Raman–Nath second‐harmonic generation from the optical Fibonacci superlattice is theoretically investigated, and a propagation equation that describes the properties of frequency‐doubled waves is presented. Abundant diffraction peaks that are substantially more varied than the traditional Raman–Nath regime are observed. The dependence of the generated second‐harmonic signals on the structural parameters and fundamental laser performance is also comprehensively discussed.

Influence of cubic nonlinearity effect on quadratic solitons in boundary-constrained self-focusing oscillatory response function system

In this paper, we theoretically study the influence of cubic nonlinearity effect on quadratic solitons in the boundary-constrained self-focusing oscillatory response function system. Based on the Newton iteration approach, we numerically solve the nonlinear coupled-wave equations with both quadratic and cubic nonlinearity. Moreover, a family of quadratic solitons is obtained. By comparing the quadratic solitons with both quadratic and cubic nonlinearity with those with only quadratic nonlinearity, we find that the cubic nonlinearity changes the transverse distribution of the soliton profiles only slightly. However, because of the existence of the cubic nonlinearity, quadratic solitons can be found only in the strongly nonlocal case and general nonlocal case, and they cannot be found in the weakly nonlocal case, in which the quadratic solitons with only quadratic nonlinearity can be found. In addition, the existence of cubic nonlinearity reduces the number of extended half-periods of the quadratic solitons. Through the linear stability analysis of the obtained soliton solutions, it is found that the stability intervals of solitons are also shrunk due to the existence of the cubic nonlinearity. The results of the linear stability analysis are verified by the numerical simulations of soliton propagations through using the split-step Fourier method.

Polynomial Filtering Algorithms under Quadratic Nonlinearities in System and Measurement Equations: Comparison with Extended and Second-Order Kalman Filters1

The paper considers the filtering problem under quadratic nonlinearities both in system and measurement equations. A polynomial filter, which is a Kalman-type recursive algorithm, is proposed. The similarities between this algorithm and other Kalman-type algorithms such as extended and second-order Kalman filters are discussed. The procedure for estimating the performance and comparing the algorithms is presented. The advantages of proposed algorithm are illustrated using a navigation data processing example

Robustness Analysis and Controller Design Based on a Generalized Model of Nonisolated Multiphase DC–DC Converter

The nonisolated multiphase dc–dc converter (NMDC) has important research value and broad application prospect in fields such as smart grid and new energy vehicles because of its high power density and low output ripple. However, with the increase of the phase number, the parameter inconsistency among each phase will make the NMDC model quite complex. Moreover, parameter uncertainty and quadratic nonlinearity in the small-signal model can degrade control system performance, leading to a big challenge in controller design. In this paper, the generalized robust control model and the method of robust controller designing of NMDC concerning parameter uncertainty and quadratic nonlinearity is developed. Firstly, parameter uncertainty is described by convex polyhedra model. Secondly, the Lyapunov theorem is applied to solve the linear and quadratic nonlinearity of the control model. Finally, with the method of Linear Matrix Inequality (LMI) region pole configuration, the analytical solution of the control model is achieved. Based on the proposed model, the independent control of NMDC with reduced sensors, including current sharing, flexible power distribution and robust control of the NMDC are realized. Both simulation and experimental results verify the effectiveness of the controller. Compared with traditional controller, the controller proposed in this paper can be stable in more complicated working conditions, and has a faster adjustment time, suitable for multiple topologies as well.

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in a nonlinear Klein–Gordon model with quadratic and cubic nonlinearities. This equation describes the propagation of the envelope of slowly modulated wave packets approximated by a superposition of the fundamental, second, and zeroth harmonics. Although extended NLSEs are not generally Hamiltonian PDEs, the equation derived here is a Hamiltonian PDE that preserves the Hamiltonian structure of the original nonlinear Klein–Gordon equation. This could be achieved by expressing the fundamental harmonic and its first derivative in symplectic form, with the second and zeroth harmonics calculated from the variational principle. We demonstrate that the non-Hamiltonian form of the extended NLSE under discussion can be retrieved by a simple transformation of variables.

Effects of quadratic coupling on optical response of a hybrid optomechanical cavity assisted by Kerr non-linear medium

We theoretically investigate the effect of non-linearity on the optical response of a hybrid optomechanical system. The optical cavity consists of a Kerr non-linear medium. We study how the presence of quadratic optomechanical coupling and Kerr nonlinearity in the conventional optomechanical system affects the system’s stability. It is shown that the presence of all the non-linearities in the system leads to seven real roots in the optical bistability. However, if any of the two non-linearities (quadratic optomechanical coupling or Kerr non-linearity) decreases, the roots in the optical bistability also decrease. Hence, quadratic optomechanical coupling and Kerr nonlinearity can be used as handles to control the optical bistability of the system. Therefore, we propose this system to be used as an all-optical switch in quantum information devices.

Asymptotic analysis of higher-order scattering transform of Gaussian processes

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein’s method to achieve the goal.

Second-order perturbation solution and analysis of nonlinear surface waves

The properties of ultrasonic nonlinear surface wave in the quasilinear region are investigated. In this work the governing equation of particle displacement potential is employed for surface wave in isotropic elastic solid with quadratic nonlinearity. Then, the quasilinear solution of the nonlinear surface wave is obtained by the perturbation method, and the absolute nonlinear parameter of the surface wave is derived. Subsequently, the main components of the second harmonic surface wave solution are discussed. A finite element model for the propagating nonlinear surface wave is developed, and simulation results of the nonlinear surface wave displacements agree well with the theoretical solutions, which indicates that the proposed theory is effective. Finally, the properties of wave propagation and the characteristic of the nonlinear parameter for the surface wave are analyzed based on the theoretical solutions. It is found that the second harmonic surface wave consists of cumulative and non-cumulative displacement terms. The cumulative displacement term is related to the self-interaction of the longitudinal wave component of the surface wave. However, its amplitude is larger than that of the pure longitudinal wave when the initial excitation conditions and propagation distances are the same. The nonlinear parameters for surface and longitudinal waves are related to each other, and an explicit relationship is found, which can be determined by the second-order elastic coefficients of the material. The propagation properties of nonlinear surface waves and the measurement method of absolute nonlinear parameters are also discussed, which will benefit the practical application of nonlinear surface waves.

Analysis of Nonlinear Characteristics for Forced Oscillation Affected by Quadratic Nonlinearity

Forced oscillation is challenging power system stability. Most previous research was based on linear models, so analysis of nonlinear characteristics of forced oscillation in power system is missing. Thus, this letter aims to theoretically reveal the effect of the quadratic nonlinearity on forced oscillation. A multi-scale technique is employed, by which the amplitude-frequency function of the steady response and the approximated first-order analytical expression of the dynamic response are derived for nonlinear forced oscillation. Interesting phenomena of frequency deviation and jumping caused by nonlinearity are found and explained. Above expressions and phenomena are verified by numerical simulation. Although revealing all characteristics of nonlinear forced oscillation in power system is difficult, the analysis results in this letter could offer insights.

Investigation of chaotic behavior and adaptive type-2 fuzzy controller approach for Permanent Magnet Synchronous Generator (PMSG) wind turbine system

<abstract> <p>This article begins with a dynamical analysis of the Permanent Magnet Synchronous Generator (PMSG) in a wind turbine system with quadratic nonlinearities. The dynamical behaviors of the PMSG are analyzed and examined using Poincare map, bifurcation model, and Lyapunov spectrum. Finally, an adaptive type-2 fuzzy controller is designed for different flow configurations of the PMSG. An analysis of the performance for the proposed approach is evaluated for effectiveness by simulating the PMSG. In addition, the proposed controller uses advantages of adaptive type-2 fuzzy controller in handling dynamic uncertainties to approximate unknown non-linear actions.</p> </abstract>

A New Hyperchaotic Two‐Scroll System: Bifurcation Study, Multistability, Circuit Simulation, and FPGA Realization

With the swift advancement of chaos theory, the modeling, chaotic oscillations, and engineering applications of chaotic and hyperchaotic systems are important topics in research. In this research paper, we elucidate our findings of a new four‐dimensional two‐scroll hyperchaotic system having only two quadratic nonlinearities and carry out a detailed bifurcation study of the proposed dynamical model. Also, an electronic circuit has been constructed for the new system using MultiSim (Version 14). The implementation of the new 4‐D hyperchaotic system in a field‐programmable gate array (FPGA) is performed herein by applying two numerical methods, viz. Forward Euler Method and Trapezoidal Method. The experimental results show a good match with the simulated hyperchaotic attractors. We also provide details of the hardware resources used for an FPGA Basys 3 Xilinx Artix‐7 XC7A35T‐ICPG236C.

Simulink modeling and dynamic characteristics of discrete memristor chaotic system

<sec>In the last two years, the discrete memristor has been proposed, and it is in the early stages of research. Now, it is particularly important to use various simulation softwares to expand the applications of the discrete memristor model. Based on the difference operator, in this paper, a discrete memristor model with quadratic nonlinearity is constructed. The addition, subtraction, multiplication and division of the discrete memristor mathematical model are clarified, and the charge <i>q</i> is obtained by combining the discrete-time summation module, thereby realizing the Simulink simulation of the discrete memristor. The simulation results show that the designed memristor meets the three fingerprints of memristor, indicating that the designed discrete memristor belongs to generalized memristor.</sec><sec>Using memristors to construct chaotic systems is one of the current research hotspots, but most of the literature is about the introduction of continuous memristors into continuous chaotic systems. In this paper, the obtained discrete memristor is introduced into a three-dimensional chaotic map which is mentioned in a Sprott’s book titled as <i>Chaos and Time-Series Analysis</i>, and a new four-dimensional memristor chaotic map is designed. Meanwhile, the Simulink model of the chaotic map is established. It is found that attractors with different sizes and shapes can be observed by changing the parameters in the Simulink model, indicating that the changes of system parameters and memristor parameters can change the dynamic behavior of the system. The analyses of equilibria and equilibrium stability show that the four-dimensional chaotic map has infinite equilibrium points. The Lyapunov exponent spectra and bifurcation diagrams of the circuit imply that the map can transform between weak chaotic state, chaotic state, and hyperchaotic state. Meanwhile, the multistability and coexisting attractors are analyzed under different initial conditions. Moreover, by comparing the results of measuring the complexity, it is found that the chaotic map with discrete memristor has richer dynamical behaviors and higher complexity than the original map.</sec><sec>From the perspective of system modeling, in this paper the discrete memristor modeling and discrete memristor map designing are discussed based on the Matlab/Simulink. It further verifies the realizability and lays a foundation for the future applications of discrete memristor.</sec>

Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels

<abstract><p>In this work, we formulate a fractal fractional chaotic system with cubic and quadratic nonlinearities. A fractal fractional chaotic Lorenz type and financial systems are studied using the Caputo Fabrizo (CF) fractal fractional derivative. This study focuses on the characterization of the chaotic nature, and the effects of the fractal fractional-order derivative in the CF sense on the evolution and behavior of each proposed systems. The stability of the equilibrium points for the both systems are investigated using the Routh-Hurwitz criterion. The numerical scheme, which includes the discretization of the CF fractal-fractional derivative, is used to depict the phase portraits of the fractal fractional chaotic Lorenz system and the fractal fractional-order financial system. The simulation results presented in both cases include the two- and three-dimensional phase portraits to evaluate the applications of the proposed operators.</p></abstract>

ELECTROMAGNETIC RESPONSE OF LAYERED MAGNETO-ELECTRO-ELASTIC THIN RECTANGULAR PLATE UNDER MODERATELY LARGE DEFLECTION

The analytical solution of a layered thin magneto-electro-elastic rectangular plate is presented. The governing equation is based on classical laminate plate theory and von Karman's stress function; capturing the effects of moderately large deflection. Electromagnetic fields are determined in terms of mechanical unknowns by solving Maxwell's equations of electrostatics and magnetostatics. The condensation of electric and magnetic state into plate kinematics, coupled with stress function definition, derives the governing non-linear partial differential equation of motion. Solution for both simply supported and clamped transverse boundary condition is obtained using Galerkin method, which reduces the system into an ordinary differential equation of cubic and quadratic nonlinearity. Numerical results of laminated plate with constituent piezoelectric barium titanite and piezomagnetic cobalt ferrite is produced utilizing electromagnetic boundary and continuity conditions. The effect of transverse elastic boundary condition on through the thickness variation of electric and magnetic potential under linear and moderately large deflection is produced.

Secure Communication Scheme based on A New Hyperchaotic System

This study introduces a new continuous time differential system, which contains ten terms with three quadratic nonlinearities. The new system can demonstrate hyperchaotic, chaotic, quasi-periodic, and periodic behaviors for its different parameter values. All theoretical and numerical analysis are investigated to confirm the complex hyperchaotic behavior of our proposed model using many tools that include Kaplan-Yorke dimension, equilibrium points stability, bifurcation diagrams, and Lyapunov exponents. By means of Multisim software, the authors also designed an electronic circuit to confirm our proposed systems’ physical feasibility. MATLAB and Multisim simulation results excellently agree with each other, which validate the feasibility of our new ten terms hyperchaotic system and make it very desirable to use in different domains especially in chaotic-based communication. Furthermore, by employing the drive response synchronisation, we developed a secure communication strategy for the proposed system. Findings from the proposed scheme show that the proposed approach was successful in completing the encryption and decryption procedure.

Near-room-temperature reversible switching of quadratic optical nonlinearities in a one-dimensional perovskite-like hybrid

The switching of quadratic nonlinear optical (NLO) effects between two or more NLO states of solid-state materials represents an intriguing new branch in the field of photoelectrics and optics. While structural phase transitions have shown potential in this field, near-room-temperature reversible NLO switches have rarely been reported. To exploit new NLO switching materials within the structurally flexible class of hybrid perovskites, here, we synthesize a one-dimensional perovskite-like hybrid, (MP)PbBr3 (where MP+ is a 1-methylpyrrolidinium cation), through a facile solution method, which exhibits strong second harmonic generation (SHG) activities with an intensity of ~1.6 times as large as potassium dihydrogen phosphate. Intriguingly, (MP)PbBr3 enables the near-room temperature reversible switching of SHG properties, showing a large NLO switching contrast of up to ~40 between its SHG-active and SHG-inactive phases, beyond most of its liquid counterparts. Further microscopic structural analyses reveal that the dynamic ordering of the organic MP+ cation and inorganic chain-like skeleton triggers its centrosymmetric (P63/mmc) to acentric (P212121) phase transition at 316 K upon cooling, resulting in a crucial contribution to its NLO switching properties. This work illustrates the potential of this material as a candidate for solid-state NLO switches and will promote the development of NLO materials within the family of low-dimensional hybrid perovskites.

Quantum squeezing in coupled waveguide networks with quadratic and qubic nonlinearity

Quantum squeezing in coupled waveguide networks with quadratic and qubic nonlinearity

A SEMI-ANALYTIC APPROACH FOR SOLVING FISHER’S REACTION-DIFFUSION EQUATION BY METHOD OF LINES USING REPRODUCING KERNEL HILBERT SPACE METHOD

Many nonlinear systems are described with the nonlinear Fisher’s reaction-diffusion equation. The purpose of this work is to propose the method of lines to find out the solution of the Fisher’s reaction-diffusion equation in one dimension with quadratic and cubic nonlinearity using reproducing kernel Hilbert space method. In this method, the partial derivatives of the space variable are discretized to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Four test examples are given to demonstrate the technique’s efficacy and applicability. The results are compared with the exact and existing numerical solutions by calculating the error norms L2 and L∞ at various time levels. It has been discovered that the recommended approach is not only simple to use, but also produces superior outcomes

Global wellposedness for 2D quasilinear wave without Lorentz

We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

This paper exploits the modified simple equation and dynamical system schemes to integrate the Klein–Gordon (KG) model amid quadratic nonlinearity arising in nonlinear optics, quantum theories, and solid state physics. By implementing the modified simple equation (MSE) technique, we develop some disguise adaptation of analytical solutions in terms of hyperbolic, exponential, and trigonometric functions with some special parameters. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of all phase portraits, we originate bounded and unbounded solitary, periodic, and periodic rogue‐type wave solutions of the KG model. These two schemes extract widespread classes of solitary, periodic, and periodic rogue‐type wave solutions for the KG model jointly due to restrictions on parameters. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its nature. We illustrate some dynamical features of the acquired solutions via the 3D, 2D, and contour graphics.