Dimensional Nonlinear Systems Research Articles

Sufficient variable selection of high dimensional nonparametric nonlinear systems based on Fourier spectrum of density-weighted derivative

Sufficient variable selection of high dimensional nonparametric nonlinear systems based on Fourier spectrum of density-weighted derivative

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Effects of spin torque within ferromagnetic infinite medium: The short-wave approximation and Painlevé analysis

In this paper, we investigate the effects of spin-transfer torques within the ferromagnetic infinite medium through the short-wave approximation method. As a result, we have derived the new (1+1) dimensional nonlinear evolution system, which describes the propagation of electromagnetic short waves within the ferromagnet in the presence of electric current density. Using the Painlevé analysis and Hirota’s bilinearization, we unearth the integrability properties of this new evolution system. In the wake of such an analysis, the typical class of excitations and its physical implications are presented. We remark that the current density acts on magnetization like an effective magnetic damping, which is important for the stabilization of magnetic information storage and data process elements.

Exact wave solutions of truncated M-fractional Boussinesq-Burgers system via an effective method

Abstract In this paper, we present distinct types of exact wave soliton solutions of an important fluid flow dynamic system called the truncated M-fractional (1+1)-dimensional nonlinear Boussinesq-Burgers system (BBS). This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, the proliferation of waves in shallow water, etc. We transform the nonlinear fractional system into a nonlinear ordinary differential equation by using a fractional transformation to obtain dark, bright, singular, dark-bright, dark-singular, bright-singular and periodic type solitons solutions by employing the modified extended tanh function method (METhFM). The use of fractional derivatives makes the solutions different from the existing solutions. The obtained results are useful in the optical fibers, fluid dynamics, ocean engineering and other related fields. To visualize the system’s behavior, some of the solutions are represented by two- and three-dimensional graphs which are obtained and verified with the help of Mathematica. The achieved results provide a better understanding of the behavior of the nonlinear fractional partial differential equations and the dynamics of BBS, which are not present in the literature and are helpful in future studies of the concerned system.

Mixed solutions and multiple solitary wave solutions for a (3 + 1)-dimensional nonlinear system

Mixed solutions and multiple solitary wave solutions for a (3 + 1)-dimensional nonlinear system

Event-triggered finite-time neural control for uncertain nonlinear systems with unknown disturbances and its application in SVC

In this article, an event-triggered finite-time neural control strategy is proposed for nonlinear power systems with unknown disturbances and static var compensator (SVC). We first transform the power system with SVC into a three-dimensional uncertain nonlinear system and then extend it to an [Formula: see text]-dimensional uncertain nonlinear system. The disturbance observer is established to estimate external disturbances and the unknown nonlinear terms are approximated by the radial basis function neural networks. Moreover, to avoid the complexity explosion problem in the traditional backstepping method, the command filtering technique is adopted, and the error caused by the command filters is compensated. The adaptive event-triggered finite-time controller ensures that all signals are bounded in finite time and excludes Zeno phenomena. In the end, the simulation for the two-area interconnected power system with SVC is presented to verify the availability and feasibility of the proposed approach.

Hydrological records can be used to reconstruct the resilience of watersheds to climatic extremes

Hydrologic resilience modeling is used in public watershed management to assess watershed ability to supply life-supporting ecoservices under extreme climatic and environmental conditions. Literature surveys criticize resilience models for failing to capture watershed dynamics and undergo adequate testing. Both shortcomings compromise their ability to provide management options reliably protecting water security under real-world conditions. We formulate an empirical protocol to establish real-world correspondence. The protocol applies empirical nonlinear dynamics to reconstruct hydrologic dynamics from watershed records, and analyze the response of reconstructed dynamics to extreme regional climatic conditions. We devise an AI-based early-warning system to forecast (out-of-sample) reconstructed hydrologic resilience dynamics. Application to the La Tejería (Spain) experimental watershed finds it to be a low dimensional nonlinear deterministic dynamic system responding to internal stressors by irregularly oscillating along a watershed attractor. Reconstructed and forecasted hydrologic resilience behavior faithfully captures monthly wet-cold/dry-warm weather patterns characterizing the Mediterranean region.

Joint effects of thermal diffusion and diffusion thermo on MHD three dimensional nanofluid flow towards a stretching sheet

This communication reports the joint effects of Thermal Diffusion and Diffusion Thermo on viscous and incompressible three-dimensional nanofluid flow towards a stretching sheet in connection to the influence of a magnetic field. In this study, nanofluid model is employed for the effects of thermophoresis and Brownian motion. Following that, similarity variables are chosen to turn the dimensional nonlinear system into dimensionless expressions and the resultant transformed equations are solved numerically using Finite Element method. Special emphasis has been given to the parameters of physical interest. These findings are visually presented through graphical representations, providing a clear and insightful understanding involved in this flow scenario. In addition, the final results are examined in light of past research and it is determined that they meet the convergence standards to an exceedingly satisfactory degree. The study’s findings are beneficial for many technical and commercial endeavours.

New variable separation solutions and localized waves for (2+1)-dimensional nonlinear systems by a full variable separation approach

A full variable separation approach is firstly proposed for (2+1)-dimensional nonlinear systems by extending the well-established multilinear variable separation approach through the assumption that the expansion function is composed of full variable separated functions, namely, functions with respect to only one spacial or temporal argument. Taking the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, the Nizhnik-Novikov-Veselov equation and the dispersive long wave equation as examples, new full variable separation solutions are obtained with several arbitrary one dimensional functions. Especially, a common formula for some suitable physical quantities is discovered. By taking the arbitrary functions in different explicit expressions, the solutions can be used to describe plentiful novel nonlinear localized waves, which might be non-travelling waves as the spacial and temporal variables are fully separated into different functions. In particular, some new hybrid solitary waves, which can pulsate periodically, appear and/or decay with an adjustable lifetime, are discovered through the on-site interactions between a doubly periodic wave and a ring soliton, a four-humped dromion and a four-humped lump, and a doubly periodic wave and a cross type solitary wave. Nonlinear wave structures and their dynamical behaviours are discussed and graphically displayed in detail.

Investigation on the 1:2:3 internal resonance of high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces

The study of internal resonances is one of the main research areas in nonlinear dynamics. The energy transfer behaviors between multiple vibrational modes caused by internal resonances lead to complex nonlinear vibration behaviors. In this paper, the rotating blade is modeled as a functionally graded graphene platelet (FGGP) reinforced rotating pre-twisted plate. The nonlinear vibration of a high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces with 1:2:3 internal resonance is investigated. The axial excitations on the blade are primarily caused by axial aerodynamic force in the tip clearance and contact forces at the top of the blade. The subsonic airflow is treated as a transverse excitation and is calculated using the vortex lattice method. Rayleigh-Ritz method is used to obtain the mode shapes of the rotating composite pre-twisted blade. The three-degree-of-freedom ordinary differential equations of the rotating FGGP reinforced pre-twisted plate are derived by using Lagrange equation. Based on the multiscale method, the averaged equations of the rotating FGGP reinforced pre-twisted plate in the case of primary resonance and 1:2:3 internal resonance under axial and transverse excitations are acquired. We present a discussion on the amplitude-frequency response, force-amplitude response, bifurcations, and chaotic motions of a rotating FGGP reinforced pre-twisted cantilever plate under both axial and transverse excitations. A crescent-shaped independent resonance region is found in the amplitude-frequency response curves. The system exhibits complex nonlinear vibration behavior as axial or transverse excitation amplitude changes.

Computational analysis of entropy generation in three-dimensional mixed convection flow with thermal dissipation and variable thermal conductivity

Entropy generation in steady three-dimensional thermal flow is investigated under the coupled influence of viscous dissipation and mixed convection. The stretching sheet with an exponential velocity profile induces the flow. In this examination, we presume that the fluid is incompressible and possesses a Newtonian behavior and temperature-dependent thermal conductivity. Using the appropriate classical similarity variables reduces the dimensional nonlinear dynamical system of governing equations to a dimensionless nonlinear dynamical set of ordinary differential equations. It’s important to note that the selection of an exponentially varying surface temperature is deliberate, as it enables precise similarity transformations to be established in situations involving both viscous dissipation and buoyancy forces. The simplified finite difference approach (SFDM) is used to solve the acquired nonlinear dynamical set of ordinary differential equations. The tridiagonal matrix approach is used to solve the resulting system of algebraic equations. In a particular situation, the present numerical simulation is numerically checked with the previously available results and observed a good agreement. The effects of physical parameters on velocity, temperature, surface stresses and heat tranfer rate, entropy generation, and the irreversibility parameter are investigated. The entropy generation rate exhibits a decreasing trend with respect to the mixed convection parameter in the vicinity of the stretching surface. Entropy reduction can be achieved by employing a working fluid with a low Prandtl number and by minimizing the fluid friction.

Exploring the new soliton solutions to the nonlinear M-fractional evolution equations in shallow water by three analytical techniques

This paper explores the new soliton solutions of the evolution equations named as truncated M-fractional (1+1)-dimensional non-linear Kaup-Boussinesq system by utilizing the expa function, modified simplest equation and Sardar sub-equation techniques. This system is used in the analysis of long waves in shallow water. The attained results involving trigonometric, hyperbolic and exponential functions. The effect of fractional order derivative is also discussed. Obtained results are very close to the approximate results due to the use of M-fractional derivative. Achieved results are verified by Mathematica tool. Few of the gained results are also explained through 2-D, 3-D and contour graphs. At the end, these techniques are straight forward, useful and effective to deal with non-linear FPDEs.

Annular rogue waves in whispering gallery mode optical resonators

Based on the variable-coefficient Lugiato-Lefever equation, nonlinear dynamics of rogue waves in whispering gallery mode optical resonators are studied. By means of the analytical solution in the absence of pump, dynamical behaviors of the rogue waves in the periodic system are discussed. Firstly, the first-order rogue wave under the condition of no pumping has obvious periodic nature, and the periodicity of width and amplitude of rogue wave can be consistent by controlling the wavefront curvature, and the energy of rogue wave remains stable under the small gain parameter, and the inverse diffraction effect makes the center position of rogue wave produce a small offset. In addition, the amplitudes of second-order rogue waves have more complicated periodicity and longer period lengths both with the good stability. At last, the dynamical behaviors of rogue wave after adding external excitation are investigated. Under the periodical external excitation with the lower intensity, rogue wave can achieve a longer stable transmission time, while the smaller wavefront curvature can effectively prolong the stable transmission time of rogue wave. Moreover, we investigate the influence of different forms of external excitation on rogue wave, and the periodical external excitation makes rogue waves more stable than the constant external excitation. These results extend the application of higher-order rogue waves in (2 + 1)-dimensional nonlinear systems, and have implications for the development of high-energy optical pulses.

An embedded Hamiltonian dynamic evolutionary neural network model for high-dimensional data recognition

Aiming at the limitation of Hamiltonian neural network in high-dimensional data processing with non-observable physical quantity system, an embedded Hamiltonian dynamic evolutionary neural network model (EHDN) is proposed in this paper. First, the dynamic physical quantities of high dimensional nonlinear systems are represented by constructing Hamiltonian functions. Then a symplectic integrator is designed to realize the time-varying evolution of network features based on Hamiltonian regular equation and variational numerical integration. Finally, the Hamiltonian dynamic evolutionary neural network is embedded in the convolutional neural network (CNN) to realize the Hamiltonian simulation evolution of convolutional features. Experimental results on two different datasets, CIFAR and Fashion-Mnist, show that compared with existing state-of-the-art (SOTA) methods, EHDN model can combine the advantages of Hamiltonian dynamics and convolutional network to improve the recognition accuracy and training stability in high-dimensional image classification tasks.

Computational tractable guaranteed numerical method to study the stability of [formula omitted]-dimensional time-independent nonlinear systems with bounded perturbation

The stability analysis of nonlinear continuous systems often requires manual calculation, which can become time-consuming when dealing with complex systems. Some works use positive invariant sets to discuss stability. These sets can be numerically approximated using Interval analysis but the computational complexity is exponential. In this paper, we propose a computational tractable numerical but guaranteed method based on Interval analysis to verify the robust positive invariance of ellipsoids to automatize the study of n-dimensional nonlinear systems’ stability. This method relies on a fast enclosure of a state integration by an Euler method. Interval analysis guarantees the results of the developed algorithms. Several examples show the effectiveness of the proposed approaches on n-dimensional non-asymptotical continuous systems subject to bounded perturbation.

Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system

In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution u instead of its potential uy. This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.

Evolving Tangent Hyperbolic memristor based 6D chaotic model with fractional order derivative: Analysis and applications

The nonlinear dynamics of fractional order systems and memristor based circuits have attracted an escalating attention in recent years. This paper reports on a novel six dimensional nonlinear dynamical system with a tangent hyperbolic memristor circuit and amalgamated image encryption. The dynamical system is analyzed using standard tools, including phase portraits, equilibrium points, Eigenvalues, Routh–Hurwitz stability criterion, Lyapunov exponents and bifurcation diagrams. The analysis suggests that the developed system is chaotic in nature and has exciting new 2D and 3D trajectories that have not been reported before. The chaotic system is numerically solved for different values of the fractional order q, and the results of equilibrium points and Lyapunov exponents are mentioned in tables. Fractional order circuits are designed for q=1 and q=0.99 utilizing the approximation fractional order technique of the transfer function. The developed circuit is generalized in a way that the circuit can work for both the integer and non-integer fractional order values by just introducing the fractional block to make it work for q=0.99, and removing it with a conventional integrator capacitor to work for q=1. The circuit simulation and numerical results were compared, and signifying that they were both in good agreement with each other. Lastly, the random number generated from the chaotic system is utilized to scramble the image via the scheme of Amalgamated Image Encryption. The scrambled image was tested using different image security test algorithms to support the idea that the chaotic system and image can together form an advantageous key.

Finite Dimensional Koopman Form of Polynomial Nonlinear Systems*

The Koopman framework is a popular approach to transform a finite dimensional nonlinear system into an infinite dimensional, but linear model through a lifting process using so-called observable functions. While there is an extensive theory on infinite dimensional representations in the operator sense, there are few constructive results on how to select the observables to realize them. When it comes to the possibility of finite Koopman representations, which are highly important from a practical point of view, there is no constructive theory. Hence, in practice, often a data-based method and ad-hoc choice of the observable functions is used. When truncating to a finite number of basis, there is also no clear indication of the introduced approximation error. In this paper, we propose a systematic method to compute the finite dimensional Koopman embedding of a specific class of polynomial nonlinear systems in continuous-time, such that the embedding can fully represent the dynamics of the nonlinear system without any approximation.

Rogue wave excitations of the (2 + 1)-dimensional nonlinear Zakharov system

Rogue wave excitations of the (2 + 1)-dimensional nonlinear Zakharov system

Novel soliton solutions, local excitations and fractal structures of nonlinear breaking soliton equation

The model of ([Formula: see text])-dimensional breaking soliton equation is proposed, and the novel soliton solutions are derived by using the variable separation and extended mapping approach of Riccati equation and then, the propagation dynamics of the solutions which include multi-dromions, ring breather, fusion solitons and other multi-solitons are also constructed. Furthermore, the rich local excitations and fractal structures are investigated. The simulation results verify the proposed scheme. It can get more abundant local excitation structures than the ([Formula: see text])-dimensional nonlinear systems and it has potential values for certain applications in optical field.