Exact solutions and dynamical analysis of nonlinear electric transmission network via second-kind Jacobi elliptic function

Bifurcation, chaos, modulation instability, and solitons are important phenomena in nonlinear dynamical structures that help us understand complex physical processes. This work employs the Schrödinger equation with cubic nonlinearity (SECN), rising in superconductivity, quantum mechanics, optics, and plasma physics. We developed an ordinary differential arrangement using a traveling wave alteration and found the soliton outcome to the mentioned problem utilizing the unified solver technique. Then we obtain different categories of wave designs, including bright and dark solitons, with periodic, quasiperiodic, and chaotic behaviors. We examined the system’s planar dynamics to observe sensitivity, chaos, and bifurcation. The chaotic state involves various procedures such as multistability, recurrence diagrams, bifurcation shapes, fractal dimensions, return maps, Poincaré graphics, Lyapunov exponents, phase portraits, and strange attractors. These properties help us understand complicated behaviors like how waves become chaotic patterns. Our outcomes demonstrate a clear understanding of wave propagation and how energy is absorbed. These findings develop our knowledge of nonlinear problems and guide us in wave circulation in real-life situations. Therefore, our procedures are useful and operative offering a simple technique for studying nonlinear problems, which progresses our understanding of these equations’ sensitivity, waveform pattern, and propagation strategy.

This paper unearths solitary wave in a simplified electrical model practical of the nonlinear left-handed transmission electrical line having series capacitance and shunt inductance. The detailed study of the practical model digs out that by taken a good parameter of the lattice, the obtained equation from the evolution of the wave envelope can be reduced to a one-dimensional fractional nonlinear Schrödinger equation, which gives the pipe of the dark solitons propagation. By simulating transient circuits, the evolution of the soliton and modulational instability gain were demonstrated in the areas of perturbation frequency, pulsation, nonlinearity and signal strength with a continuous wave type input signal. It has also been shown that increased perturbation, nonlinearity and signal strength in simplified metamaterial models of short nonlinear transmission lines lead to the formation of Schrödinger dark solitons and increased instability gain in a continuous pulse structure. The establishment of Schrödinger solitons and the progressive distance of a point, i.e. the fractional parameter α = 1, could reduce the width of the signal bandwidth and the breaking of the symmetry of the instability gain in the simplified electrical model practical of the nonlinear left-handed transmission electrical line, so that the evolution of the system cannot be controlled due to the memory effect, which can find important practical applications in communication systems.

Chaotic-like behavior of modulated waves in a nonlinear discrete LC transmission line

Modulational instability induced by randomly varying coefficients for the nonlinear Schrödinger equation

M-shaped and other exotic solitons generated by cubic-quintic saturable nonlinearities in a nonlinear electrical transmission network with higher-order dispersion effects

Determination of largest Lyapunov exponents of vocal signal: application to unilateral laryngeal paralysis.

Modulation Instability is known as intrinsic property of a nonlinear medium like Kerr medium or photorefractive medium; through the such media, the system behavior is possible to transit form stationary regime to chaotic regime; this paper deals with Modulation Instability (MI) in a nonlinear medium and investigates the analogy of MI of optical nonlinear medium and the consequent chaotic regime based on extracting Lyapunov exponent through the power spectrum and equivalently intensity-time diagram of MI; the experimental observation truly confirms the results of MI as the route to the chaotic regime.

This research investigates the bifurcation theory of a generalized (3 + 1)-dimensional P-type nonlinear wave equation with an M-fractional derivative (M-fGP-NWE) and develops its soliton solutions. The model is initially transformed into an ordinary differential equation form using a wave variable. By employing a Galilean transformation, a dynamical system of equations is obtained. The phase portrait and Hamiltonian function are examined under different parametric conditions, facilitating the identification of homoclinic and heteroclinic orbits. These orbits illustrate solitary, bell-shaped, periodic wave solutions for certain parameter values. The modified simple equation (MSE) method is employed to derive soliton solutions for the M-fractional generalized (3 + 1)-dimensional P-type nonlinear wave equation. The resultant solutions are articulated in hyperbolic, trigonometric, and exponential forms under parametric circumstances. Complex wave phenomena are further exemplified through detailed 3D, 2D, and density graphs for certain parameter values. Additionally, we also analyse the modulation Instability of the proposed model. The computational results and visual depictions validate the efficiency and dependability of the MSE method, highlighting its efficacy as a flexible solution for complex fractional differential equations.

This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor–Corrector Adams–Bashforth–Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge–Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin–Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag–Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents’ diagrams.

The ‘Cabaret’ finite differences method allows the performance of fast and conclusive modelling of the microcavity temporal dynamics with group velocity dispersion, cross- and self- phase modulation included. The proposed scheme and model allow us to study the dynamics of a cavity with two counter-propagating pulses, second order dispersion, modulation instability, Rayleigh scattering and other effects along the cavity roundtrips number up to 106…7. The possibility of Lyapunov exponents mapping for the Raman and SBS fibre lasers and cavities is discussed.

Railwayfreight cars operating in heavy-load and complex outdoor environments are frequently subject to adverse conditions such as haze, temperature fluctuations, and transmission interference, which significantly degrade the quality of the acquired images and introduce substantial noise. Furthermore, the structural complexity of freight cars, coupled with the small size, diversity, and complex structure of defect areas, poses serious challenges for image denoising. Specifically, it becomes extremely difficult to remove noise while simultaneously preserving fine-grained textures and edge details. These challenges distinguish railway freight car image denoising from conventional image restoration tasks, necessitating the design of specialized algorithms that can achieve both effective noise suppression and precise structural detail preservation. To address the challenges of incomplete denoising and poor preservation of details and edge information in railway freight car images, this paper proposes a novel image denoising algorithm named the Nonlinear Activation-Free Network based on Multi-Scale Edge Enhancement and Fusion (NAF-MEEF). The algorithm constructs a Multi-scale Edge Enhancement Initialization Layer to strengthen edge information at multiple scales. Additionally, it employs a Nonlinear Activation-Free feature extractor that effectively captures local and global image information. Leveraging the network's multi-branch parallelism, a Multi-scale Rotation Fusion Attention Mechanism is developed to perform weight analysis on information across various scales and dimensions. To ensure consistency in image details and structure, this paper introduces a fusion loss function. The experimental results show that compared with recent advanced methods, the proposed algorithm has better noise suppression and edge preservation performance. The proposed method achieves significant denoising performance on railway freight car images affected by Gaussian, composite, and simulated real-world noise, with PSNR gains of 1.20 dB, 1.45 dB, and 0.69 dB, and SSIM improvements of 2.23%, 2.72%, and 1.08%, respectively. On public benchmarks, it attains average PSNRs of 30.34 dB (Set12) and 28.94 dB (BSD68), outperforming several state-of-the-art methods. In addition, this method also performs well in railway image dehazing tasks and demonstrates good generalization ability in denoising tests of remote sensing ship images, further proving its robustness and practical application value in diverse image restoration tasks.

Two-neuron-based non-autonomous memristive Hopfield neural network: Numerical analyses and hardware experiments

Lyapunov exponents spectrum estimation of fractional order nonlinear systems using Cloned Dynamics

A dynamical model is proposed and the corresponding characteristic aligns are derived for voltage-mode bi-frequency controlled switching converter operating in discontinuous conduction mode. According to the dynamical model, the border-collision bifurcation and multi-period behaviors, such as period-2, period-3, period-4, and so on, are studied using bifurcation diagrams as the circuit parameters are varied. It is found that the converter behaves along the bifurcation route of period-1, multi-period, and period-1, and the change of period state is induced by border-collision bifurcation. Based on the characteristic equation, the converter stability is investigated by the Lyapunov exponent. It is shown that Lyapunov exponent is always smaller than zero with the variation of circuit parameters and the converter operates in stable period state all the time. Also, it is validated that period-3 behavior of voltage-mode bi-frequency controlled switching converter does not predicate its inevitable chaos. Time-domain waveforms, phase portraits, and frequency spectra of voltage-mode bi-frequency controlled switching converter are analyzed by circuit simulation, which validates the feasibility of dynamical model and the correctness of theoretical analysis. Simulation results are verified by experiments in this paper.

In the last 20 years, the chaotic system has been widely studied. A new six-dimension hyper chaotic system is introduced in this paper. We used a new chaotic system based on a six-dimension for the purpose of increasing chaos in the system, where the new system has twelve positive parameters, complicated chaotic dynamics behaviors and gives an analysis of the new systems. The basic characteristics and dynamic behavior of this system are investigated with a presence of chaotic attractor, dissipativity, symmetry, equilibrium points, Lyapunov Exponents, Kaplan-Yorke dimension, waveform analysis and sensitivity toward initial conditions. The results of the analysis exhibit that the new system contains three unstable equilibrium points and the six Lyapunov exponents. Maxim non-negative Lyapunov Exponent (MLE) is obtained as 1.04609 and Kaplan-Yorke are obtained as 4.00695, and the new system characteristics with, unstable, high complexity, and unpredictability, The new system dynamics is simulated utilizing MATHEMATICA program. the phase portraits and the qualitative properties of the new hyper chaotic system have been described at the detail.