Higher-order Nonlinear Terms Research Articles

Hopf Bifurcation Analysis of a Nose Landing Gear System of Aircraft

The Hopf bifurcation problem of a nose landing gear system is investigated. Taking the nearly smooth landing gear system with high-order nonlinear terms as the research object, the nose landing gear system is simplified to a plane dynamic system considering nonlinear factors by using the center manifold simplification method. The first Lyapunov coefficient of the landing gear system is derived by using the classical bifurcation theory, and the Hopf bifurcation type of the system is determined according to its sign. The bifurcation diagram is obtained by solving the differential equations of motion numerically. It is revealed that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point. The influence of velocity and other factors on the shimmy oscillations of the landing gear system is analyzed. The dynamical behavior in the neighborhood of the Hopf bifurcation point is explored theoretically and numerically. The research results can provide a theoretical basis for the optimal design of aircraft landing gear systems.

A combination of XGBoost and neural network in LIBS spectrum processing for precise determination of critical elements in 620 iron ore samples of various origins

China is the worldwide largest importer of iron ores for supplying its iron and steel industries. The necessary inspections of iron ores not only concern total iron content but also involve minor elements, such as silicium, aluminum, magnesium, and calcium. Laser-induced breakdown spectroscopy (LIBS) promises in situ and on-site detection and analysis capabilities, which is required for efficient, reliable, and large-scale impartation inspections. Such opportunity represents at the same time, a challenge for the technique in terms of the quantitative analysis ability. A fundamental issue corresponds to the matrix effect in LIBS analysis due to a large diversity of iron ores in terms of mineralogical and chemical compositions. Demonstrated as effective to deal with the matrix effect in LIBS analyses of geological materials, the machine learning approach needs to be implemented within an optimized data processing pipeline, combining algorithms carefully chosen and tested. In this work, we developed a sequentially connected combination of an extreme gradient boosting (XGBoost) model and a back propagation neural network (BPNN) model, to effectively fit the functional relation between the concentrations and the spectra. The specificity of the approach consists in fitting the linear, low-order nonlinear, and high-order nonlinear terms in a sequential way to substantially improve the model prediction performances. The method has been developed and tested with a significant set of 620 collected iron ore samples of eight production countries and 35 different commercial brands, with a wide range of matrices in terms of mineralogical and chemical compositions. The samples underwent prior characterization using conventional laboratory analysis methods to establish the reference elemental concentrations. The results from this necessary step of laboratory investigation, with root mean square error for prediction (RMSEP) of 0.407 wt%, 0.372 wt%, 0.085 wt%, 0.131 wt%, and 0.114 wt%, respectively for total iron (TFe), SiO2, Al2O3, MgO, and CaO, show the potential for on-site inspections.

Multivariable adaptive decoupling control based on stochastic configuration networks with serial-parallel switching distribution

A multivariable adaptive decoupling control scheme is proposed based on stochastic configuration networks with serial-parallel switching distribution (SPSCN). Firstly, a linear controller is designed by combining a PID controller, feedback decoupling, and one-step optimal control. Secondly, a nonlinear controller is presented to deal with higher-order nonlinear terms and unknown external perturbations. SPSCN is used to improve the prediction accuracy of unmodeled dynamics. It combines uniform and normal search strategies in a serial-parallel fashion, aiming at improving the node quality and reducing the model complexity. The approximation performance of the SPSCN algorithm is demonstrated by performing approximation experiments with two functions and four benchmark datasets. Compared with the generalized minimum variance control (GMVC) algorithm in controlling the process of cement raw material decomposition, our proposed control scheme outperforms.

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

Quantitative derivation of a two-phase porous media system from the one-velocity Baer–Nunziato and Kapila systems

We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer–Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer–Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood–Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the overdamping phenomenon, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial O(ε) bounds.

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

Virtual unmodeled dynamic and data-driven nonlinear robust predictive control

This study presents a novel approach for controlling an industrial process that exhibits uncertainty and significant nonlinear features. The proposed method utilizes a virtual unmodeled dynamic and data-driven nonlinear robust predictive control strategy. The representation of a controlled object involves a composite state space model that combines both linear and high-order nonlinear elements. Moreover, a robust model predictive controller is developed using the linear component. In addition, the notion of one-step optimal feedforward is used in combination with a compensating controller to handle the high-order nonlinear factor specifically. Subsequently, a compensation controller with incremental characteristics is developed for a modified version of the high-order nonlinear term. Furthermore, the stability conditions of the closed-loop system are derived, and an analysis is conducted on the stability and convergence of the proposed approach. The TTS20 three-capacity water tank was utilized in both simulations and practical scenarios. The study demonstrated that the suggested approach successfully reduces system output variations and enhances overall performance in response to unpredictable changes in the process’s dynamic features.

Third order nonlinear correlation of the electromagnetic vacuum at near-infrared frequencies

In recent years, electro-optic sampling, which is based on Pockel’s effect between an electromagnetic mode and a copropagating, phase-matched ultrashort probe, has been largely used for the investigation of broadband quantum states of light, especially in the mid-infrared and terahertz frequency range. The use of two mutually delayed femtosecond pulses at near-infrared frequencies allows the measurement of quantum electromagnetic radiation in different space-time points. Their correlation allows therefore direct access to the spectral content of a broadband quantum state at terahertz frequencies after Fourier transformation. In this work, we will prove experimentally and theoretically that when using strongly focused coherent ultrashort probes, the electro-optic sampling technique can be affected by the presence of a third-order nonlinear mixing of the probes’ electric field at near-infrared frequencies. Moreover, we will show that these third-order nonlinear phenomena can also influence correlation measurements of the quantum electromagnetic radiation. We will prove that the four-wave mixing of the coherent probes’ electric field with their own electromagnetic vacuum at near-infrared frequencies results in the generation of a higher-order nonlinear correlation term. The latter will be characterized experimentally, proving its local nature requiring the physical overlap of the two probes. The parameters regime where higher order nonlinear correlation results predominant with respect to electro-optic correlation of terahertz radiation is provided.

Evaluation of a high-order central-difference solver for highly compressible flows out of thermochemical equilibrium

A high-order shock-capturing central finite-difference scheme is evaluated for numerical simulations of hypersonic high-enthalpy flows out of thermochemical equilibrium. The scheme is an extension to thermochemical out-of-equilibrium flows of the technique presented in Sciacovelli et al. (2021) for high-speed flows in chemical nonequilibrium. It relies on a standard tenth-order accurate central-difference approximation of the inviscid fluxes, supplemented with a high-order accurate nonlinear artificial dissipation term of ninth-order accuracy in smooth flow regions. Close to flow discontinuities, a shock-capturing low-order term is activated based on a highly selective shock sensor. To enable robust and non oscillatory solutions in regions of strong discontinuities of the thermodynamic variables, including the vibrational temperature, a shock detector consisting of a combination of a pressure-based term and Ducros’ vorticity/dilatation sensor is applied to all conservation equations except that of vibrational energy. For the latter, a shock sensor based on the vibrational temperature itself is adopted instead, to account for the loose coupling between vibrational energy and pressure. The accuracy and robustness of the proposed approach is demonstrated for a variety of thermochemical non-equilibrium configurations, ranging from one-dimensional benchmarks to three-dimensional turbulent flows, for which the ILES capabilities of the selective high-order numerical dissipation are also showcased.

Modeling of Nonlinear Sea Wave Modulation in the Presence of Ice Coverage

A model accounting for the influence of ice coverage on the propagation of surface sea waves is suggested. The model includes higher-order linear and nonlinear terms in the equation of wave motion. The asymptotic solution is obtained to account for nonlinear modulated wave propagation and attenuation. Two kinds of attenuation are revealed. The influence of the higher-order nonlinear, dispersion, and dissipative terms on the shape and velocity of the modulated nonlinear wave is studied. Despite the presence of higher-order terms in the original equation, the modulated solitary wave solution contains free parameters, which is important for the possible generation of such waves.

Orbital stability of periodic wave solution for Eckhaus-Kundu equation

In this paper, we mainly study the orbital stability of periodic traveling wave solution for the Eckhaus-Kundu equation with quintic nonlinearity, which is not a standard Hamilton system. Considering the studied equation is not a standard Hamilton system, the method presented by M. Grillakis and others for proving orbital stability cannot be applied directly, and this equation has two higher order nonlinear terms. So, by constructing three conserved quantities, using detailed spectral analysis and appropriate techniques, we overcome the complexity of the studied equation developed in calculation and proof, then, a conclusion on the orbital stability of the dn periodic wave solution for the Eckhaus-Kundu equation is obtained. As an extension of the proof for the above results, we also prove the orbital stability of the solitary wave for the studied Eckhaus-Kundu equation.

Soliton molecules for combined mKdV-type bilinear equation

Starting from the multi-soliton solutions obtained by the Hirota bilinear method, the soliton molecule structures for the combined mKdV-type bilinear equation are investigated using the velocity resonance mechanism. The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper. With particular selections of the involved arbitrary parameters, especially the wave numbers, it is confirmed that, besides the usual multi-bright soliton molecules, the multi-dark soliton molecules and the mixed multi-bright-dark soliton molecules can also be obtained. In addition, we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions. The results demonstrate that when N ≥ 4, the combined mKdV-type bilinear equation no longer admits soliton molecules comprising more than four solitons.

A generalized nonlinear Schrödinger involving the weak nonlocality: Its Jacobi elliptic function solutions and modulational instability

The present paper investigates the features of specific waves in a weakly nonlocal nonlinear medium. More precisely, a generalized nonlinear Schrödinger equation (gNLSE) involving a high-order dispersion term, a high-order nonlinear term, and the effect of a term that accounts for the weak nonlocality is considered. By applying the Jacobi elliptic function (JEF) method, a wide range of Jacobi elliptic function solutions to the governing model is first derived. The characteristics of modulation instability (MI) and modulated wave (MW) topological and non-topological solitons are then analyzed by giving a series of two- and three-dimensional graphs. A major accomplishment of the study is the discovery of stable and unstable zones, as well as the control of the amplitude of topological and non-topological solitons.

A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons

In this paper, we deal with the Cauchy problem for a generalized two-component Camassa-Holm system with waltzing peakons and complex higher-order nonlinear terms. By the classical Friedrichs regularization method and the transport equation theory, the local well-posedness of solutions for the generalized coupled Camassa-Holm system in nonhomogeneous Besov spaces and the critical Besov space B^{5/2}_{2,1}times B^{5/2}_{2,1} was obtained. Besides the propagation behaviors of compactly supported solutions, the global existence and precise blow-up mechanism for the strong solutions of this system are determined. In addition to wave breaking, the another one of the most essential property of this equation is the existence of waltzing peakons and multi-peaked solitray was also obtained.

Multiple mixed state variable incremental integration for reconstructing extreme multistability in a novel memristive hyperchaotic jerk system with multiple cubic nonlinearity

Memristor-based chaotic systems with infinite equilibria are interesting because they generate extreme multistability. Their initial state-dependent dynamics can be explained in a reduced-dimension model by converting the incremental integration of the state variables into system parameters. However, this approach cannot solve memristive systems in the presence of nonlinear terms other than the memristor term. In addition, the converted state variables may suffer from a degree of divergence. To allow simpler mechanistic analysis and physical implementation of extreme multistability phenomena, this paper uses a multiple mixed state variable incremental integration (MMSVII) method, which successfully reconstructs a four-dimensional hyperchaotic jerk system with multiple cubic nonlinearities except for the memristor term in a three-dimensional model using a clever linear state variable mapping that eliminates the divergence of the state variables. Finally, the simulation circuit of the reduced-dimension system is constructed using Multisim simulation software and the simulation results are consistent with the MATLAB numerical simulation results. The results show that the method of MMSVII proposed in this paper is useful for analyzing extreme multistable systems with multiple higher-order nonlinear terms.

Stabilization of Highly Nonlinear Stochastic Time-Varying Coupled Systems via Aperiodically Intermittent Control

Highly non-linear stochastic time-varying coupled systems are first considered in which the coupling structure is time-variant. It is worth pointing out that aperiodically intermittent control (AIC) is considered to achieve the stability of highly non-linear stochastic time-varying coupled systems. Since the existing research methods for processing AIC are not suitable for stochastic highly non-linear systems, a new Halanay-type differential inequality with higher-order nonlinear terms that extends the existing Halanay-type differential inequalities is established. Then, with the help of the Lyapunov method, some techniques of inequalities and the graph theory, two stabilization criteria are presented to guarantee the exponential stabilization for highly non-linear stochastic time-varying coupled systems. Finally, as an application of our results, the modified time-varying coupled Van der Pol-Duffing oscillators are studied via AIC with numerical simulations provided.

Stability analysis of hybrid high-order nonlinear multiple time-delayed coupled systems via aperiodically intermittent control

New results on the stability of hybrid high-order nonlinear multiple time-delayed coupled systems (HHNCSs) are presented by aperiodically intermittent control (AIC). The model considered in this paper includes Markovian switching and multiple time delays, which make the high-order nonlinear coupled systems more accurately simulate the actual models. In addition, Halanay-type differential inequalities are powerful tools when investigating the stability of time-delayed systems with AIC. However, existing Halanay-type differential inequalities are not applicable for HHNCSs, since high-order nonlinear terms exist. Therefore, a novel Halanay-type differential inequality is established which not only generalizes the classic Halanay inequality but also develops the applications of AIC under the condition of high-order nonlinearity. On the foundation of this innovative Halanay-type differential inequality, sufficient conditions are obtained by employing the graph theory and the Lyapunov method. Finally, the obtained theoretical results can be applied to modified coupled Van Pol–Duffing oscillators and some simulation results are given to demonstrate the feasibility and validity of our results.

Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x^{2}y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: dot{x}=a(y - x), dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y, dot{z}= -cz + y^{2}, which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S_{x} = {(x, x, frac{x^{2}}{c})|xin mathbb {R}, cne 0} are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.

Application of Multiscale Method in Nonlinear Vibration Problem of Multi-joint Manipulator

In this paper, the nonlinear two-joint manipulator system is taken as the research object. Firstly, the dynamic equation of the system is established by using the Lagrange method. In the case of ignoring some high-order nonlinear terms in the equations, the multi-scale method in the nonlinear vibration approximate analytical method is used to solve the nonlinear multi-joint manipulator system model, and the approximate analytical solution of the system is obtained. The theoretical analysis results are verified by numerical calculation. The results show that the analytical solution is in good agreement with the numerical solution, which verifies the effectiveness and accuracy of the multi-scale method.

Jacobian-elliptic-function and rogue-periodic-wave solutions of a high-order nonlinear Schrödinger equation in an inhomogeneous optical fiber

Optical fiber communication plays an important role in the modern communication. Under investigation is a high-order nonlinear Schrödinger equation with the additional high-order dispersion and nonlinear terms in an inhomogeneous optical fiber. We give the Jacobian-elliptic-function solutions of that equation. Based on the nonlinearization of the existing Lax pair, we obtain the values of the spectral parameters corresponding to that Lax pair. We construct the linearly-independent and non-periodic solutions of that Lax pair. We take the Jacobian-elliptic-function solutions as the seed solutions, and substitute the linearly-independent and non-periodic solutions into the Darboux transformation to obtain the rogue-periodic-wave solutions. For the one-fold rogue-dn-periodic wave, when the parameter γ of the equation increases, we find that the period and maximum amplitude of the one-fold rogue-dn-periodic wave remain unchanged, while the minimum amplitude decreases. When the parameter η of the equation increases, we find that the maximum amplitude of the one-fold rogue-dn-periodic wave remains unchanged, while the period becomes the smaller. For the one-fold rogue-cn-periodic wave, the same conclusions can be observed. For the two-fold rogue-cn-periodic wave, when the parameter γ of the equation increases, we find that the period and maximum amplitude of the two-fold rogue-cn-periodic wave remain unchanged, while the minimum amplitude rises. When the parameter η of the equation increases, we find that the maximum amplitude of the two-fold rogue-cn-periodic wave remains unchanged, while the period becomes the smaller.