Hyperbolic Function Research Articles

Multiple scroll attractors and multistability in the collective dynamics of a four chain coupled hopfield inertial neuron network: analysis and circuit design investigations

This work deals with the chain bidirectional coupling of four inertial Hopfield neurons. Each of our cells taken alone is modeled by a second order differential equation having three resting points. Coupling these neurons helps increasing the number of fixed points that is related to the amount of memory assigned to the network. The system having a hyperbolic tangent as nonlinearity is investigated with the help of suitable nonlinear dynamical tools like bifurcation diagrams, Lyapunov exponent diagrams and phase portraits to resort the richness of the model. As the coupling adds the amount of equilibriums, it also helps in generating multiple scroll attractors. The system that was unable of oscillation presents firing patterns such as parallel branches, coexistence of up to sixteen attractors in the phase plane, extreme events and Hopf bifurcation to name a few. All these features are discovered when observing the coupling strengths, the dissipation coefficient and when programming initial states around sensitive equilibriums. The electronic version of the four-chain coupled inertial neurons system is provided and simulated on Pspice with the aim to confirm the results obtained in the numerical scheme.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

A class of analytic functions associated with sine hyperbolic functions

A class of analytic functions associated with sine hyperbolic functions

Fixed-time event-triggered sliding mode time-varying formation control for multiple surface vessels with fast convergence

ABSTRACT This paper proposes a new fixed-time sliding mode time-varying formation control strategy for multiple surface vessels (MSVs) to improve control performance and ensure navigation safety. First, a fixed-time extended state observer (FxESO) is designed to provide estimations of velocities and lumped disturbances. Then, the event-triggered fixed-time distributed time-varying formation control law is designed based on prescribed performance function (PPF) and fixed-time nonsingular terminal sliding mode (FxNTSM) manifold. Finally, system stability is analyzed based on Lyapunov theory. The salient features of the proposed scheme are as follows. First, a novel hyperbolic cosecant PPF is incorporated into the control scheme to enhance control performance. Then, the proposed scheme is combined with relative threshold event-triggered strategy to reduce unnecessary updates of actuators. Finally, a uniform exact differentiator is introduced to avoid the formation control law including the acceleration of adjacent vessel. Comparative results are given to demonstrate the superiority of the proposed scheme.

Resonant soliton molecules, asymmetric solitons and the other diverse wave solutions to the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation for liquid with gas bubbles

Under the present study, we focus on developing some exact solutions of the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation (KSE) for the liquid with gas bubbles. First, the resonant soliton molecules on the different planes are extracted via imposing the velocity resonance condition on the N-soliton solutions that constructed via the Hirota bilinear method. Besides, the asymmetric solitons are also derived by choosing the appropriate parameters. In the end, the diverse kinds of travelling wave solutions including the bright wave, dark wave, singular wave and periodic wave solutions that expressed by the trigonometric, hyperbolic and rational functions are constructed by employing three effective approaches, namely the variational method, Sub-equation method and Wang’s direct mapping method-II. The profiles of the attained solutions are depicted to present the dynamical performances. As far as we all know, the results of this research are all new and can enable us figure out the physical properties of the (3 + 1)-dimensional generalized KSE better.

Activation Functions for Neural Networks: Application and Performance-based Comparison

Past decade has seen explosive growth of Deep Learning (DL) algorithms based on Artificial Neural Networks (ANNs) and its applications in vast emerging domains to solve real world complex problems. The DL architecture uses Activation Functions (AFs), to perform the task of finding relationship between the input feature and the output. Essential building blocks of any ANN are AFs which bring the required non-linearity of the output in the Output layer of network. Layers of ANNs are combinations of linear and nonlinear AFs. Most extensively used AFs are Sigmoid, Hyperbolic Tangent (Tanh), Rectified Linear Unit (ReLU) etc to name a few. Choosing an AF for a particular AF depends on various factors such as Nature of Application, Design of ANN, Optimizers used in the network, Complexity of Data etc. This paper presents a survey on most widely used AFs along with the important consideration while selecting an AF on a specific problem domain. A broad guideline on selecting an AF based on the literature survey has been presented to help researchers in employing suitable AF in their problem domain.

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.

Exact solutions of the Landau–Ginzburg–Higgs equation utilizing the Jacobi elliptic functions

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations.

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

Abstract In this study, we use the Khater Method (KM) as an efficient analytical tool to solve (3+1)-dimensional fractional extended shallow water wave equations (FESWWEs) with conformable derivatives. The KM transforms fractional partial differential equations to ordinary differential equations (ODEs) via strategic variable transformation. Then, series-form solutions to these ODEs are proposed, which turn them into nonlinear algebraic systems. The solution to this set of algebraic equations yields shock travelling wave solutions expressed in hyperbolic, trigonometric, exponential, and rational functions. The study's findings are corroborated by 2D, 3D, and contour graphs that show the changing patterns of the detected shock travelling waves. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Exact solutions for travelling waves using Tanh method for two dimensional stochastic Allen–Cahn equation with multiplicative noise

AbstractThe research outlined here focuses on the hyperbolic tangent (tanh) method for deriving analytical solutions for travelling wave to the two-dimensional stochastic Allen–Cahn equation with multiplicative noise. The novelty of our work is to derive these exact solutions since the previous studies focused on the solutions of numerical nature. The tanh approach, which employs a finite tanh power series, is particularly adept at modeling travelling wave profiles. A key area of interest in this study is the effect of multiplicative noise on these travelling waves dynamics, especially how high levels of noise can lead to propagation failure of waves. The results demonstrate that for weak noise, the propagation of the travelling wave is basically unaffected, while the wave fails to propagate in the strong noise regime.

Investigating wave solutions and impact of nonlinearity: Comprehensive study of the KP-BBM model with bifurcation analysis.

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering.

The geometrically m$m$‐step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

AbstractIn this paper, we present some new results on the geometrically ‐step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bianabelian and strong bianabelian) geometrically ‐step solvable Grothendieck conjecture(s) for affine hyperbolic curves over fields finitely generated over the prime field. First of all, we show the conjecture over finite fields. Next, we show the geometrically ‐step solvable version of the Oda–Tamagawa good reduction criterion for hyperbolic curves. Finally, by using these two results, we show the conjecture over fields finitely generated over the prime field.

Exploration of soliton solution of coupled Drinfel’d–Sokolov–Wilson equation under conformable differential operator

The aim of this paper is to study the nonlinear couple Drinfel’d–Sokolov–Wilson (DSW) equation under conformable differential operator (CDO) that plays an important role in describing the shallow water in fluid mechanics, plasma physics, nonlinear optics and Bose–Einstein condensate (soliton formation). The Tanh-Coth and energy balance techniques are implemented in this work to explore some travelling wave solutions. These analytical methods yield various solutions for the considered DSW equation. The solutions obtained from both method are expressed in the form of hyperbolic and trigonometric function. Furthermore, the dynamic behaviour of the obtained solutions are illustrated, via 2D and 3D curves. Additionally, the effect of the conformable differential operator on dynamics of wave solutions is studied via 2D graphs.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson-Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behavior of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Command filter based input quantized adaptive tracking control for multi‐input and multi‐output non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints

SummaryThis paper addresses the problem of adaptive tracking control for multi‐input and multi‐output (MIMO) non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints. To tackle the interference of unmodeled dynamics, the dynamic signal generated by the auxiliary system is used. Hyperbolic tangent function is used as a nonlinear mapping tool to transform the constrained system into an unconstrained one. Hysteresis quantizer is introduced to mitigate the chattering phenomenon and quantization error in the quantization signal. The derivative of virtual signal can be approximated more efficiently by command filter. Furthermore, an error compensation mechanism is established to mitigate the error introduced by the command filter. Unknown nonlinear functions are approximated by radial basis function neural networks (RBFNNs). Stability analysis of the proposed controller is performed through the Lyapunov stability theory and the output tracking error can be constrained within a specified range. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method.

Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints

This paper investigates the adaptive neural network (NN) tracking control problem for stochastic nonlinear systems with multiple actuator constraints and full-state constraints. The issue of system full-state constraints is tackled by a generalized barrier Lyapunov function (GBLF), and the output constraints of the system are considered to be in the form of time-varying functions, which are more in line with the needs of real physical systems. The NN approximation technique is utilized to overcome the influence of the uncertainty term on controller design due to randomness. Based on the backstepping technique, a neural adaptive fixed-time tracking control strategy is designed. Under the designed control strategy, the tracking accuracy of the controlled system can reach the expectation in a fixed time. The multi-actuator constraints are converted into a generalized mathematical model to simplify the controller design process. Using the characteristics of the hyperbolic tangent function, a new function called practical virtual control signal is designed using the virtual control signal as the input. Due to the saturation constraint property of the hyperbolic tangent function, it is theoretically ensured that no state of the system exceeds the constraints through to the new form of the virtual controller. Using the adaptive controller constructed in this paper, the controlled system is semi-global fixed-time stabilized in probability (SGFSP). Finally, the effectiveness of the proposed control strategy is further verified by simulation examples.

A HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITY WITH THE KERNEL OF A HYPERBOLIC COSECANT FUNCTION

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible is discovered. Furthermore, two equivalent inequalities with the best constant factors were obtained.

Data-Driven Adaptive Controller Based on Hyperbolic Cost Function for Non-Affine Discrete-Time Systems with Variant Control Direction

As technology evolves, more complex non-affine systems are created. These complex systems are hard to model, whereas most controllers require information on systems to be designed. This information is hard to obtain for systems with varying control directions. Therefore, this study introduces a novel data-driven estimator and controller tailored for single-input single-output non-affine discrete-time systems. This approach focuses on cases when the control direction varies over time and the mathematical model of the system is completely unknown. The estimator and controller are constructed using a Multiple-input Fuzzy Rules Emulated Network framework. The weight vectors are updated through the gradient descent optimization method, which employs a unique cost function that multiplies the error by a hyperbolic tangent. The stability analyses demonstrate that both the estimator and controller converge to uniformly ultimately bounded (UUB) functions of Lyapunov. To validate the results, we show experimental tests of force control that were executed on the z-axis of a drive-controlled 3D scanning robot. This system has a varying control direction, and we also provide comparison results with a state-of-the-art controller. The results show a mean absolute percentage tracking error smaller than one percent on the steady state and the expected variation in the system’s control direction.