Nonlinear Perturbations Research Articles

Optical solitons, qualitative analysis, and chaotic behaviors to the highly dispersive nonlinear perturbation Schrödinger equation

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

An Efficient Computational Algorithm for a Class of Nonlinear Singular Perturbation Problems of Convection Diffusion Type with Cauchy Data

An Efficient Computational Algorithm for a Class of Nonlinear Singular Perturbation Problems of Convection Diffusion Type with Cauchy Data

Observing Array Designed for Improving the Short‐Term Prediction of Kuroshio Extension State Transition Processes

AbstractGiven the essential implications of Kuroshio Extension (KE) bimodality on oceanic dynamical environment and climate, the present study investigates the targeted observation schemes, based on the conditional nonlinear optimal perturbation (CNOP) method and a reduced‐gravity shallow‐water model, to improve the forecast skills of transition processes of KE bimodal states. To obtain a suitable observing array, the observation schemes, with different numbers of observation sites and observation distances between two sites, are designed. Furthermore, to demonstrate the superiority of the observing networks in predicting KE transition processes, two existing observation schemes and six random observation schemes are compared with the CNOP‐determined observing array. Based on this, a relatively optimal observing array with three sites and observation distance of 90 km is established, which is mainly located between 31°N and 33°N in the south of Japan. This targeted observing network is universal for two KE transition processes. The removal of initial errors on this array results in the mean prediction improvements of about 9.2% and 22.5% for KE transition processes from the low‐ to the high‐energy state and from the high‐ to the low‐energy state, respectively.

Evidence of 1+1D Photorefractive Stripe Solitons Deep in the Kerr Limit.

The Kerr nonlinearity allows for exact analytic soliton solutions in 1+1D. While nothing excludes that these solitons form in naturally occurring real-world 3D settings as solitary walls or stripes, their observation had previously been considered unfeasible because of the strong transverse instability intrinsic to the extended nonlinear perturbation. We report the observation of solitons that are fully compatible with the 1+1D Kerr paradigm limit hosted in a 2+1D system. The waves are stripe spatial solitons in bulk copper doped potassium-lithium-tantalate-niobate (KLTN) supported by unsaturated photorefractive screening nonlinearity. The parameters of the stripe solitons fit well, in the whole existence domain, with the 1+1D existence curve that we derive for the first time in closed form starting from the saturable model of propagation. Transverse instability, that accompanies the solitons embedded in the 3D system, is found to have a gain length much longer than the crystal. Findings establish our system as a versatile platform for investigating exact soliton solutions in bulk settings and in exploring the role of dimensionality at the transition from integrable to nonintegrable regimes of propagation.

The first kind of predictability problem of El Niño predictions in a multivariate coupled data‐driven model

AbstractEl Niño‐Southern Oscillation (ENSO) is the dominant atmosphere–ocean coupled mode of year‐to‐year variations in the tropical Pacific. It shows diverse spatiotemporal characteristics and casts major influences on seasonal predictions of global weather–climate extrema. Despite numerous dynamical and statistical models for ENSO prediction and predictability studies, they are commonly subjected to one‐to‐three issues among less skillful simulation of El Niño diversity, huge requirements of computational resources and a low robustness in statistics. Here, an efficient deep‐learning model involving nonlinear coupling of multiple variables is independently developed to study the predictability of two types of El Niño events related to initial uncertainty, which is the first kind of predictability problem. The model can skillfully simulate statistically robust features of observed El Niño diversity in terms of periodicity, amplitude, and seasonal phase‐locking. Using this model, we have revealed mathematically several new types of fastest‐growing initial errors in two types of El Niño predictions based on a novel concept of conditional nonlinear optimal perturbation (CNOP), especially including one that can strengthen central Pacific types of events, which is rarely investigated before. Moreover, CNOPs are superimposed into a numerical model, GFDL CM2p1, for comprehensive validation and growth mechanism mining, which demonstrates the consistent dynamical evolution of initial errors in both numerical and AI models. Our study represents the first attempt to explore the first kind of ENSO predictability problem from perspectives of nonlinear error‐evolving dynamics using a data‐driven model. This is of great importance as it offers us sufficient confidence to perform ENSO‐related (such as the Madden–Julian Oscillation, etc.) mechanisms and predictability studies in the future without strongly relying on dynamical numerical models.

Application of spiral enhanced whale optimization algorithm in solving optimization problems

The Whale Optimization Algorithm (WOA) is regarded as a classic metaheuristic algorithm, yet it suffers from limited population diversity, imbalance between exploitation and exploration, and low solution accuracy. In this paper, we propose the Spiral-Enhanced Whale Optimization Algorithm (SEWOA), which incorporates a nonlinear time-varying self-adaptive perturbation strategy and an Archimedean spiral structure into the original WOA. The Archimedean spiral structure enhances the diversity of the solution space, aiding the algorithm in escaping local optima. The nonlinear time-varying self-adaptive optimization dynamic perturbation strategy improves the algorithm’s local search capability and enhances solution accuracy. The effectiveness of the proposed algorithm is validated from multiple perspectives using CEC2014 test functions, CEC2017 test functions, and 23 benchmark test functions. The experimental results demonstrate that the enhanced Whale Optimization Algorithm significantly improves population diversity, balances global and local search, and enhances solution accuracy. Additionally, SEWOA exhibits excellent performance in solving three engineering design problems, showcasing its value and wide range of potential applications.

Finite Time Stability Analysis and Feedback Control for Takagi–Sugeno Fuzzy Time Delay Fractional-Order Systems

This study treats the problem of Finite Time Stability Analysis (FTSA) and Finite Time Feedback Control (FTFC), using a Linear Matrix Inequalities Approach (LMIA). It specifically focuses on Takagi–Sugeno fuzzy Time Delay Fractional-Order Systems (TDFOS) that include nonlinear perturbations and interval Time Varying Delays (ITVDs). We consider the case of the Caputo Tempered Fractional Derivative (CTFD), which generalizes the Caputo Fractional Derivative (CFD). Two main results are presented: a two-step procedure is provided, followed by a more relaxed single-step procedure. Two examples are presented to show the reduction in conservatism achieved by the proposed methods. The first is a numerical example, while the second involves the FTFC of an inverted pendulum, which exhibits both symmetrical dynamics for small angular displacements and asymmetrical dynamics for larger deviations.

Observer-Based Adaptive Fuzzy Control for Singular Systems with Nonlinear Perturbation and Actuator Saturation

Observer-Based Adaptive Fuzzy Control for Singular Systems with Nonlinear Perturbation and Actuator Saturation

Impacts of the Thermocline Feedback Uncertainty on El Niño Simulations in the Tropical Pacific

AbstractAs a key dynamic element of the Bjerknes feedback mechanism, the thermocline effect (TE) is critically important to El Niño modeling. In this study, the potential influence of TE‐related parametric uncertainties on El Niño is investigated using the conditional nonlinear optimal parametric perturbation (CNOP) method based on an intermediate coupled model (ICM). The optimal perturbation of the TE‐related parameter (OTEP), which substantially affects El Niño simulations, is estimated through the CNOP approach. Results reveal that the El Niño simulation is highly sensitive to the TE uncertainty in the eastern equatorial Pacific, with OTEP‐induced simulation errors demonstrating an El Niño‐like growth trend. On one hand, as indicated by the simulated El Niño intensity, the uncertainty in the TE in the eastern region can easily affect the strength of the Bjerknes feedback‐related thermocline effect and atmospheric circulation. On the other hand, the enhanced TE is highly favored to accelerate the growth of the SST error due to the air–sea interaction, thus severely affecting the El Niño simulations. Therefore, adequately representing the TE in the equatorial eastern Pacific is emphasized for effectively improving El Niño simulations.

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Ensemble forecasting of Indian Ocean Dipole events generated by conditional nonlinear optimal perturbation method

Ensemble forecasting of Indian Ocean Dipole events generated by conditional nonlinear optimal perturbation method

Electroactive differential growth and delayed instability in accelerated healing tissues

Guided by experiments contrasting electrically accelerated recovery with natural healing, this study formulates a model to investigate the importance of electroactive differential growth and morphological changes in tissue repair. It underscores the clinical potential of leveraging electroactive differential growth for improved healing outcomes. The study reveals that voltage stimulation significantly enhances the healing and growth of biological tissues, accelerating the regeneration process across various growth modalities and steering towards isotropic growth conditions that do not favor any specific growth pathways. Enhancing the electroelastic coupling parameters improves the efficacy of bioelectric devices, initiating contraction and fortification of biological tissues in alignment with the electric field. This process facilitates swift cell migration and proliferation, as well as oriented growth of tissue. In instances of strain stiffening at elevated strains, the extreme critical growth ratio aligns with the predictions of neo-Hookean models. Conversely, for tissues experiencing strain stiffening under moderate to very low strain conditions, the strain stiffening effect substantially delays the onset of electroelastic growth instability, ultimately producing a smooth, hyperelastic surface devoid of any unstable morphologies. Our investigation, grounded in nonlinear electroelastic field and perturbation theories, explores how electric fields influence differential growth and instability in biological tissues. We examine the interactions among dimensionless voltage, internal pressure, electroelastic coupling, radius ratio, and strain stiffening, revealing their effects on promoting growth and delaying instability. This framework offers insights into the mechanisms behind electroactive growth and its instabilities, contributing valuable knowledge to the tissue healing.

Weakly nonlinear hyperbolic differential equation of the second order in Hilbert space

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn't inverse and can have nonclosed set of values. As an application we consider boundary-value problem for van der Pol equation in a separable Hilbert space.

An Innovative and Efficient Implementation of Matrix-Free Newton Krylov Method for Neutronics/Thermal-hydraulics Coupling Simulation

The core physical behavior of reactors is essentially the result of multi-physical fields coupling feedback. High-fidelity neutronics/thermal-hydraulics (N/TH) analysis can simulate and predict nuclear reactor core phenomena realistically, providing advanced and reliable technical means during the design and safety analysis of nuclear reactor. In this work, an efficient and robustness coupling method using power density as the coupling parameter, Matrix-Free Newton Krylov (MFNK) method, is successfully developed and innovatively implemented in HNET for high-fidelity N/TH coupling simulation. To enhance the efficiency and stability, the multi-level generalized equivalence theory-based CMFD (ML-gCMFD) iterative acceleration method and ML-gCMFD coupling acceleration method are proposed. In addition, the nonlinear preconditioning and hybrid perturbation size formula are implemented to further improve the convergence. Finally, to evaluate the numerical accuracy, convergence, efficiency and stability of MFNK method, a series of representative problems, including a three-dimensional (3D) single fuel pin problem, VERA Benchmark Problem 6, and VERA Benchmark Problem 7, are analyzed by comparing with the current N/TH coupling methods. Numerical results indicate that MFNK method can obtain strong stability, high convergence performance, and relatively high computational efficiency while ensuring high accuracy. It demonstrates that MFNK method has significant performance advantages and potential for high-fidelity N/TH coupling simulation.

An extension to ensemble forecast of conditional nonlinear optimal perturbation considering nonlinear interaction between initial and model parametric uncertainties

Initial and model uncertainties are the main sources of forecast errors, making the single and deterministic forecasts unreliable. To estimate these uncertainties, a growing consensus shifts towards ensemble forecasting, aiming to provide the probability density distribution of the atmosphere. However, current ensemble methods either focus on single-source uncertainties or employ a simple superposition of the two, neglecting the nonlinear interaction between them, and thus fail to reflect the real forecast uncertainty. Motivated by this, this study extends the CNOP approach, defined as the optimal growth considering nonlinear interaction between initial and model parameters, to the scenario of ensemble forecasts and proposes an orthogonal CNOPs method (O-CNOP-IPs). This method concerns the nonlinear effect of initial and model parametric uncertainties through a joint optimization strategy and enhances the estimation of this effect by providing diversity and independent CNOPs (via orthogonality). To evaluate the performance of O-CNOP-IPs, extensive experiments are conducted for North Atlantic Oscillation (NAO) ensemble forecasts in the realistically configured Community Earth System Model (CESM). Our findings reveal that the O-CNOP-IPs method outperforms existing methods in forecast skill and reliability, improving deterministic skill by 17.5 % and probabilistic skill by 52 %–63 %. Our dynamic analysis also unveils that this method undergoes rapid development in the early stage and effectively neutralizes errors in control forecasts, significantly enhancing the reliability of ensemble forecasts. It is expected that O-CNOP-IPs plays a significant role in accurately representing the forecast uncertainty of other high-impact weather and climate phenomena.

Entropy production-based nonlinear optimal perturbation for subsonic flows around an airfoil

The extraction and time evolution of optimal perturbation (OP) offers abundant physical insights in fluid dynamics. Nonlinear OP (NLOP) analysis provides an approach for obtaining the trajectory to induce the maximum changes in the flow field. In an extension into unsteady flow field, we tracked the changes of trajectory by an application of initial perturbation field in the compressible Navier–Stokes equation, and we focused on the entropy production (EP) to characterize the trajectory. We proposed entropy production-based NLOP (EP-NLOP) analysis for compressible flows and investigated the effect of evaluation function on the extracted Ops using the subsonic flow around an airfoil. Compared with the conventional disturbance energy (DE-) based NLOP (DE-NLOP) analysis, we demonstrated that the OPs with different spatial wavelength and concentration regions were successfully extracted due to the different spatial sensitivity of evaluation function. In the EP-NLOP analysis, the spatial distribution of OP extracted the larger energy dissipation upstream of the separation points for the short evaluation time. For the long evaluation time, EP-NLOP analysis extracted the transient-time evolution of interacting separation vortices, attributing the multiple wavelengths of OPs. These differences in the OPs offer promising insights into fluid dynamics.

Dynamics of the Davydov’s soliton in external oscillating magnetic field

The influence of an external oscillating in time magnetic field on the dynamics of the Davydov’s soliton is studied. It is shown that soliton parameters depend on the amplitude and frequency of the magnetic field, as well as on the field orientation with respect to the molecular chain axis. Electron dynamics in the parallel magnetic field is a composition of the “free” soliton coherent propagation along the molecular chain and oscillatory movement in the transverse direction. The latter is described by the functions of the harmonic oscillator with the cyclotron frequency depending on the field frequency. In a perpendicular field the soliton dynamics is a superposition of (i) the free electron plane wave in the plane perpendicular to the molecular chain, and (ii) propagation along the chain which is described by the modified Nonlinear Schrödinger equation with an extra term depending on the field. This equation is solved using the nonlinear perturbation theory. It is shown that the soliton velocity and phase are given by expressions which include the terms oscillating in time with the frequency of the main harmonic, given by the external field frequency, and higher multiple harmonics. Such complex effects of external time-depending magnetic fields on the dynamics of solitons modify the charge transport in low-dimensional molecular systems, which in its turn, can affect the functioning of the devices based on such compounds. These results suggest also the physical mechanism of therapeutic effects of oscillating magnetic fields, based on the field influence on the dynamics of solitons which provide charge transport through biological macromolecules in the redox processes.

Nonlinear transient disturbance growth analysis for the compressible external flow and application to a flow around circular cylinder

The property of nonlinear optimal perturbation (NLOP) analysis for the flow around the circular cylinder was investigated with a comparison with the linear optimal perturbation analysis (LOP). Our analysis revealed that NLOP is highly dependent on the initial disturbance norm and highlighted the limitation of linear prediction in triggering instability in the flow field. This NLOP shows an asymmetric and deformed vortex structure on the cylinder wall, with the angle of the maximum wall vorticity shifting in relation to the initial disturbance norm. We also observed that the circulation and disturbance vorticity at the shear layer in boundary layer efficiently contributed to larger disturbance growth of the separation vortex behind the cylinder compared with the LOP.

Finite time adaptive recoil controller design for deepwater drilling riser systems

In this paper, we propose a finite-time model reference adaptive controller to attenuate the recoil response of deepwater drilling riser systems following an emergency disconnection. Firstly, a coupled model that characterizes the recoil movements of the riser, wave-excited heave motion, and the frictional resistance due to drilling fluid discharge is developed. Secondly, an adaptive control law is presented, with a portion of the adaptive gain determined through linear matrix inequalities. Thirdly, the stability criteria and updating rules that ensure finite settling time for the recoil control system is established. Then, the effectiveness and robustness of the controller against nonlinear systematic perturbations are verified. It is worth pointing out that: (1) The proposed adaptive controller significantly attenuates the recoil responses of the riser in finite time. (2) Compared to other controllers, the finite-time adaptive controller provides faster convergence for both error states and adaptive gains. (3) Perturbation cases involving nonlinear and unmodeled dynamics are examined, demonstrating the controller’s effectiveness and robustness, which ensures the safety of the drilling riser system.

Exponential stability of second order time delay semilinear systems with bilinear control

ABSTRACT This paper explores the stabilisation of delayed bilinear hyperbolic systems, examining scenarios with delays in bilinear control and state. Our contributions are twofold: First, we establish existence results under specific conditions for these systems. Second, we design bilinear controls ensuring exponential stability under Lipschitz nonlinear perturbations. For control delay, stability is demonstrated using Lyapunov's abstract function. In contrast, for state delay, we show an equivalence between the stabilisation of the delayed system and the observability of the corresponding undamped system. Our theoretical findings are exemplified through practical models, emphasising their applicability in real-world scenarios.