SOME PROPERTIES OF HYPERBOLIC TRIGONOMETRIC FUNCTIONS

We propose a three-neuron heterogeneous cyclic Hopfield neural network (het-CHNN) utilizing three different activation functions: the hyperbolic tangent, sine, and cosine functions. The network’s globally uniformly ultimate boundedness is proved theoretically, and its chaotic dynamics are explored through numerical simulations and analog experiments. The numerical results demonstrate that the het-CHNN displays chaotic dynamics and multi-scroll chaotic attractors. Subsequently, the het-CHNN is implemented in an analog circuit, and hardware experiments are performed to verify the previous numerical results. Notably, the het-CHNN successfully resolves the issue of the absence of chaos in a three-neuron CHNN and currently appears to be the simplest three-neuron Hopfield neural network (HNN) that can generate chaos.

In the article, we establish several Lazarevic and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means.

The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by using time series, bifurcation diagrams, Lyapunov exponents' plots, Poincaré sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors of the oscillator.

Forex indicators are one way of examining market data. One of the best methods is using approximations of quadratic curves. For example, the Parabolic SAR indicator is an indicator that approximates the trend by a parabola. Of course, not all indicators necessarily map the direction of the trading trend with a curve. But discovering a clear curved path for the trading trend is definitely the dream of many great analysts. The problem that we are dealing with here is actually trying to generalize the idea of approximating the direction of the trading process by using hyperbolic functions. Suppose there are two fixed points in the movement of the financial market and the path length between these two points is also a fixed number, using numerical approximation methods, we try to calculate the maximum and minimum amount of financial market movement. The main idea that is in our minds is the approximation of financial market trends by using hyperbolic functions. We approximate the path between these two points with a hyperbolic sine function. Finally, we try to find the maximum or minimum points of this hyperbolic sine function numerically.

In this paper, a circuit technique to extend the measuring range of a linear variable differential transformer (LVDT) is proposed. The transfer characteristic of the LVDT contains the odd function form of the cubic polynomial. Therefore, the measuring range of a commercial LVDT is linear in a narrow range compared to its physical dimensions. The wide measuring range of the LVDT requires a large structure of the LVDT, which increases the scale and the cost of the measurement system. The measuring range of the LVDT can be linearly extended to the maximum of the stroke range using the proposed technique. The realization of the proposed technique is based on the use of the hyperbolic sine (sinh) function of the electronic circuit building block, named the class AB bipolar amplifier. The class AB bipolar amplifier can be obtained by the current feedback operational amplifier (CFOA). The circuit of the proposed technique requires two CFOAs and an operational transconductance amplifier (OTA) as the active devices and all devices used in the proposed technique to synthesize the sinh function are commercially available. The proposed technique exhibits an ability to compensate for the nonlinear characteristic of the LVDT without digital components. The proposed technique is attractive in terms of its simple circuit configuration, small size, and low cost. The linear range extension of the LVDT used in this paper is significantly increased with a maximum error of about 18.3 μm of 6.2 mm at the full stroke range or the full-scale percentage error of about 0.295%. The results indicate that the proposed technique provides excellent performance to extend the measuring range of the LVDT without modifying the LVDT structure.

In the present study, an analytical solution for the static analysis of orthotropic plates under combined thermal and mechanical loading has been developed using the hyperbolic shear deformation theory. The hyperbolic sine and cosine function is used in the displacement field in terms of thickness coordinate to represent the effect of transverse shear deformation. The most important feature of the theory is that transverse shear stresses can be accurately calculated by the direct constitutive equation approach. The present theory contains the same number of dependent variables as in the first-order shear deformation theory. The governing equilibrium equations and boundary conditions of the theory are obtained using the principle of a virtual work. Simply supported orthotropic plates subjected to a sinusoidal and uniformly distributed thermal and mechanical load is considered for the detail numerical study. Navier solution technique is employed for the simply supported plates. The program has been developed in FORTRAN. Results are obtained for displacements and stresses. The present solutions are compared with those obtained by other researchers. The results are in good agreement with the solution presented by other shear deformation theories.

: This report documents several continuous functions that can approximate a discrete step function. The functions considered include the cosine, cubic polynomial, bezier polynomial, and hyperbolic tangent. Based on the error condition used, the Hyperbolic Tangent (Tanh) and Cosine functions provide the best approximations depending on the degree of smoothness desired.

ABSTRACTBy employing the Hirota bilinear (HB) method and starting from the solution with arbitrary functions, we independently analyze the relationship between abundant wave behavior and the superposition formulas of solutions for a (3+1)‐dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. Concurrently, we correlate findings with the behaviors of oceanic waves, yielding some valuable insights. By taking the arbitrary functions as exponential functions, polynomial functions, hyperbolic cosine functions, trigonometric functions, and mixed function, numerous solutions to the equation are obtained as follows: the rogue wave solution, the multikink wave solution, the periodic wave solution, and interaction solutions. Through asymptotic analysis, the superposition of exponential functions leads to the increase in the kink wave numbers. For the rogue wave as the rational function solution, reformulating parameter relations using vectors facilitates the precise determination of its emergence time. Additionally, the impacts of parameters and the order of the rational function on the rogue‐wave shape are independently analyzed. Finally, we examine amplitude variations in rogue and periodic waves under the gradual superposition of kink waves. All types of wave profiles are visually displayed in the form of graphics by selecting appropriate parameters.

Application of the least-squares inversion method: Fourier series versus waveform inversion

The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

1 - Variables and Functions

A novel autonomous 5-D hyperjerk RC circuit with hyperbolic sine function is proposed in this paper. Compared to some existing 5-D systems like the 5-D Sprott B system, the 5-D Lorentz, and the Lorentz-like systems, the new system is the simplest 5-D system with complex dynamics reported to date. Its simplicity mainly relies on its nonlinear part which is synthetized using only two semiconductor diodes. The system displays only one equilibrium point and can exhibit both periodic and chaotic dynamical behavior. The complex dynamics of the system is investigated by means of bifurcation analysis. In particular, the striking phenomenon of multistability is revealed showing up to seven coexisting attractors in phase space depending solely on the system's initial state. To the best of author's knowledge, this rich dynamics has not yet been revealed in any 5-D dynamical system in general or particularly in any hyperjerk system. Pspice circuit simulations are performed to verify theoretical/numerical analysis.

Pseudorandom number generators (PRNGs) are fundamental components in cryptographic algorithms. The new concept of multi-PRNG introduced in this article consists of a unique generator capable of producing multiple streams of pseudorandom numbers. Multiscroll chaotic systems are known for generating multiple scrolls within a single attractor. With the aforementioned this article introduces a novel field-programmable gate array (FPGA) implementation of a multi-PRNG based on a multiscroll chaotic memristive Hopfield neural network (MHNN). The main contribution of this work is the generation of multiple spatially dependent PRNG streams from a chaotic multiscroll system by dividing the phase space of the attractor into sub-phase spaces. Each scroll in the multiscroll attractor functions as an independent PRNG. This innovative approach to generating multiple PRNGs from multiscroll chaotic systems is unprecedented in the existing literature. The 5-D MHNN chaotic model used in this work employs hyperbolic tangent and sine functions, which were implemented through a hardware-efficient CORDIC approach. Besides, The FPGA implementation to produce the chaotic time series leverages the Euler method with 32-bit fixed-point arithmetic, selected for its simplicity and low resource utilization. Finally, The randomness of the binary sequences produced by the multi-PRNG is rigorously validated using the NIST SP 800-22a and TestU01 suites, confirming their potential for cryptographic applications.

CHAPTER XII - INVERSE CIRCULAR FUNCTIONS, HYPERBOLIC FUNCTIONS AND INVERSE HYPERBOLIC FUNCTIONS

This work explores the hyperbolic cosine and hyperbolic secant functions within the framework of the maximum entropy principle, deriving these probability distribution functions from first principles. The resulting maximum entropy solutions are applied to various physical systems, including the repulsive oscillator and solitary wave solutions of the advection equation, using the method of moments. Additionally, a different moment analysis using experimental and theoretical inputs is employed to address non-linear systems described by the non-linear Schrödinger equation, non-linear diffusion equation, and Korteweg–de Vries equation, demonstrating the versatility of this approach. These findings demonstrate the broad applicability of maximum entropy methods in solving different differential equations, with potential implications for future research in non-linear dynamics and transport physics.