In this paper, we present and explore a nonlinear conformable M-fractional extension of the Boussinesq-Kaup system to examine the dynamics of shallow water waves. The proposed nonlinear conformable M-fractional system is considered an extension of the ferromagnetic spin chain equation, and it is able to describe the complicated dynamics of wave propagation in the presence of nonlinearity and memory effects. The suggested model is developed as a system of nonlinear partial differential equations with coupled conformable M-fractional derivatives providing an efficient way to consider nonlinearity, memory effects, and dispersal of waves. The direct algebraic method is an efficient but a simple method to deal with such complex nonlinear systems providing the exact traveling wave solutions. The inferred solutions of the analysis are then discussed by way of graphical representations with a view of explaining their physical meaning. In order to find out qualitative dynamics of the system, the analysis of phase portraits is conducted, which gives the opportunity to characterize equilibrium points in detail. The paper analyzes kink soliton dynamics in the conformable KPP equation, highlighting the impact of fractional derivatives on wave evolution. By employing bifurcation analysis, the research illustrates how sensitive wave patterns are to subtle parameter variations and initial conditions. These results mirror the complexity of actual shallow water waves. The integrated approach-combining fractional calculus with nonlinear and graphical analysis-provides a powerful tool for interpreting wave phenomena, confirming the methodology's utility in exploring diverse nonlinear dynamics.

This paper investigates the existence, uniqueness, and Ulam–Hyers stability of solutions for a class of impulsive multi-term fractional differential equations in nonreflexive Banach spaces. The considered system is formulated in the sense of Caputo fractional derivatives with respect to an increasing function and incorporates nonlinear boundary conditions together with impulsive effects. By transforming the given problem into an equivalent integral equation, sufficient conditions ensuring the existence and uniqueness of solutions are derived via Banach’s fixed point theorem. Furthermore, the Ulam–Hyers stability of the proposed model is established, guaranteeing that approximate solutions remain uniformly close to the exact solution under suitable conditions. The obtained results generalize and extend several existing contributions in the literature on fractional differential equations by including impulsive dynamics and the setting of nonreflexive Banach spaces. Finally, illustrative examples are presented to confirm the applicability and effectiveness of the theoretical results.

In this paper, the dynamical properties of a nonlinear Schrödinger-type oscillator motivated by nanobioscience are investigated, where natural fluctuations and external perturbations play a important role. The traveling wave reduction together with suitable parameter restrictions yields a lower order equation, which is further extended with periodic forcing and additive Gaussian noise to examine quasi-periodic, periodic, and chaotic oscillations. The analysis shows that small changes in initial conditions can produce dramatically different long term behaviors, ranging from nested invariant tori to fractalized strange attractors. Chaotic behavior is characterized by geometric transitions of the attractors caused by periodic external forcing and associated stretching folding mechanisms. Furthermore, the surrogate data analysis confirms that the observed transitions are genuine rather than noise induced artifacts, verifying the presence of deterministic chaotic dynamics.

This paper investigates the strict stability of multi-term non-impulsive pseudo-fractional differential equations defined with respect to different kernel functions. By employing Caputo-type fractional derivatives, we extend stability concepts from impulsive systems to the non-impulsive framework. Lyapunov-like functions are constructed to derive sufficient conditions that ensure strict stability. Moreover, comparison principles are established to relate the associated fractional operators to explicit stability bounds. An illustrative example is presented to illustrate the effectiveness of the theoretical results and to highlight the influence of the multi-term structure on the stability behavior of the system.

Multistable systems with non-smooth dynamics provide serious challenges to the engineering design. Using a suite of standard numerical tools, our analysis of a canonical non-smooth system: a resonant circuit with an asymmetric cubic restoring force fed by a discontinuous square wave voltage, offers two important contributions to the study of such systems. First, the discontinuous forcing itself causes the basins of attraction with riddled boundaries to be created, which is a fundamental sensitivity that cannot be predicted by local analysis. Second, our Lyapunov landscape analysis finds the coefficient of quadratic nonlinearity, l2, to be a significant critical stability switch, which can be directly used to control the predictability of the system. Lastly, we prove this control by achieving chaos synchronization and determining the critical coupling necessary to stabilize a problem stabilize synchronously as the required. This description of the global dynamics of the system is crucial for achieving robust performance in systems that incorporate both electronic and mechanical components.

In this paper, a van der Pol-type oscillator with an additional external periodic force and a white Gaussian noise term is reported. Four possible scenarios with respect to the external periodic force and the white Gaussian noise term are discussed for three different oscillators, i.e. the classical van der Pol model and generalized models with supercritical and subcritical Hopf bifurcations, respectively. Then, a complete classification of phase portraits without a periodic force and a white Gaussian noise term is presented. Further, a complete dynamical analysis is performed for all three considered oscillators with additional external periodic force and without a white Gaussian noise term, respectively. In the meantime, a stiffness term is added and dynamical analysis is performed by utilizing different tools having the perturbed term with or without noise, respectively. For a deeper analysis, the Ott-Grebogi-Yorke (OGY) method is utilized with the noise term. The significance of stochastic effects in modeling nonlinear systems is highlighted, which show rich behavior such as quasi-periodicity, mild chaos, and fractal basin boundaries.

This article introduces certain algebraic properties of generalized (h˜1,h˜2)-pre-invex functions on Rℵ(0<ℵ≤1). A new fractal weighted integral identity is established and further employed to obtain several Ostrowski-type results in the fractal setting for functions whose first derivatives in the modulus belong to the generalized (h˜1,h˜2)-pre-invex functions’s class. An illustrative example is presented to validate the theoretical findings. Moreover, applications of the main results are derived in connection with generalized random variables and various special means, highlighting the effectiveness and potential scope of the proposed approach.

This paper introduces a systematic framework for classifying the parameter space of the Duffing oscillator into three distinct levels. The first level categorizes the parameter space based on statistical properties, identifying diverging, non-diverging, and fixed point attractors. The second level focuses on the nature of the eigenvalues. In this paper, nine distinct types and their corresponding parameter ranges are identified. Finally, the third level classifies chaotic and periodic behaviors using Lyapunov exponents. For a single initial state, 19965 cases are analyzed, revealing that most of the parameter space is diverging, with few regions of nondiverging cases, highlighting the importance of precise design decisions to remain in stable regions. With the variation in initial states, the category distribution also varies across the parameter space. To this end, the analysis of selected cases from classification results is conducted using tools: attractors, multistability, time series, Lyapunov exponents, binary mapping, return map, Poincar & eacute; map, and power spectrum. In addition, the representation of behavior across classified parameter space is obtained. Moreover, the design choices made to replicate the behavior of the Duffing oscillator in its electrical circuit counterpart are presented, ensuring alignment in their dynamic responses. Hardware experiments validated the chaotic and periodic behaviors under varying conditions. The comprehensive dataset including attractors, Lyapunov exponents, and categorized parameter space is created that can be used in the future and offers a valuable resource for engineers and researchers.

We introduce the generalized fractional-order Dirac delta distribution δGFODDF, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that δGFODDF reduces to the classical delta when η=1, while providing additional flexibility for 0<η<1. These results show that δGFODDF is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering.

This paper undertakes a theoretical analytic study of a (3+1)-dimensional NLHSM, thereby generalizing the (2+1)-dimensional NLHSM to three spatial dimensions with a view of capturing rich features in the NL electromagnetic wave systems. From the solutions gained by the IGREM method, a variety of dynamical behaviors, including bifurcation, multi-stable chaos, period, and quasi-period wave structures, are given. Both the cases of perturbation and no perturbation are considered to study the effect of interactions in the various wave phenomena. The complexity of stability and sensitivity of these systems is described through bifurcation analysis, which shows how the system proceeds from stable periodic oscillations to quasi-periodic and chaotic oscillations. The new ideas on the excitation and stabilization of electromagnetic waves in strongly nonlinear media can be useful in engineering, for example, for accurately modeling modern telecommunication systems and controlling waves in plasma physics. The analytic solutions offer a starting point to investigate wave behaviors in higher dimensions, which aids in the creation of new approaches to the management of waves in engineering. Users can gain physical wave behavior access to both simulations and their practical implementation through the generalized model for enhancing electromagnetic wave-based engineering systems.

In this research, we investigate the phenomenon of multistability and complex dynamic behaviors in plasma waves by utilizing advanced mathematical techniques. We examine how fractional-order derivatives influence plasma wave stability by applying the fractional diffusion–reaction model, the framework of nonlinear dynamical systems, and the (G′G2) method. The principal direction of our work is associated with different forms of oscillations in the plasma wave: non-linear periodic, solitons, and kink waves. This leads to the study of small amplitude pulses and solitary waves, which are significant in plasma activities. Using bifurcation analysis, we discuss how these waves appear and develop under different conditions, as well as determine which conditions generate the chaotic behavior or highly complex patterns of waves. We study the details of transitions between waves and their chaotic behavior to characterize the laws that govern their plasma environment. Moreover, we have used non-linear modeling and numerical simulations to understand in detail the complex patterns and the factors of stability underlying the phenomena of plasma waves. In addition, our study also investigates the correspondence between non-linearity, multi-stability, and the birth of complex structures such as solitons and kink waves. The solutions of the dynamical system produced by the proposed nonlinear model generate different patterns of response based on system parameter variation. These patterns include oscillations and decay behaviors. Research results about system stability and solution convergence under various parameter settings provide an extended performance evaluation of the proposed method through a better understanding of system dynamics. They increase our understanding of chaotic behavior in plasma systems and pave the way for applications in plasma physics and energy systems, as well as advanced technologies.

The objective of this work is to study the dynamics and multiple stability of electromagnetic waves interacting with the medium affected by the Higgs field, both in disturbed and stable conditions. The numerical analysis uses an enhanced approach known as the improved Generalized Riccati Equation Mapping Method (GREMM), which is suitable for such systems' analysis because of strong, complex behaviors. In this way, one investigates the changes in the wave configurations and stability preferences for different circumstances of perturbation and the impact of the Higgs field. The following set of relations presents stability relations, where depending on the parameter values, the system can be either uni- or bistable, or multistable, displaying more than one stable state. From the phase portraits, it can be seen that the proposed system shows periodic behavior, which tends to the multistability. Particularly, we propagate disturbances that alter the behavior of electromagnetic waves and their coupling with the substratum Higgs field; this gives insight into the nature of wave-particle correlations in that type of physics system. The research emphasizes the importance of shallow nonlinear phenomena and the notion of the Higgs field regarding the nature of electromagnetic waves and their possible usage in various areas, from physics to the domain of advanced engineering. Thus, this study enriches the understanding of nonlinear waves and provides a solid ground for analyzing similar waves in other physical problems.

This paper introduces a new mathematical analysis for approximating and predicting nonlinear dynamical systems through fractional calculus coupled with diffusion-reaction equations. The idea is to discuss specific aspects of system behaviors, memory effects, and historical dependencies for fractional-order systems. We apply the Generalized Riccati Equation Mapping Method (GREMM) for imaging the nonlinear system and developing more insights into the complex systems’ behaviors as well as memory effects, especially in the case of fractional-order systems. This methodology, thus, allows us to get a better understanding. More than that, the current approaches introduced to understand the behavior of sophisticated systems do so far better compared to the traditional methodologies to study and make a prognosis on the behavior of solitary waves in these systems. The extended GREMM is also important for the analysis as it allows for the investigation of complex wave processes, such as solitary waves and solitons. We begin by describing kink waves and pulse waves, as well as linear and nonlinear periodic waves, and how they behave. We also discuss bifurcation analysis in which a relatively small change in the value of the parameters leads to large changes in the behavior, up to chaos by introducing fractional derivatives, and the method assists in the advancement of the analysis, the prediction, and the control of many processes in science and engineering disciplines, such as biology, chemistry, and physics. It provides better approximations. As compared to other methods, this method can provide better predictions of nonlinear phenomena for controlling power interactions and any other system demonstrating nonlinear behavior.

In this paper, we establish several general local fractional integral inequalities via generalized (h1,h2) preinvex mapping on fractal sets. By considering different parameter values, we develop particular applications of our result, such as midpoint-type inequality, generalized trapezoidal-type inequality, and generalized Simpson-type inequality. We present applications of the derived inequalities in numerical quadrature formulas, providing error estimates.

Stability analysis is recognized as a critical aspect of the study of dynamical systems, which encompasses the identification of the system attractors, the assessment of their stability or instability, and the determination of the basins of attraction of these attractors. A notable characteristic of this novel system is its multiple attractors, which are attributed to varying initial conditions. A prominent feature of the system under investigation is its ability to exhibit multiple attractors, which arise from varying initial conditions. The dynamical properties of the proposed system have been thoroughly examined through the evaluation of Lyapunov exponents, bifurcation diagrams, and multi-stability. Various chaos detection tools, including the Poincaré map, power spectrum, and time series analysis, have been employed to identify chaotic behavior within the system. To enhance understanding and interpretation, comprehensive numerical and graphical illustrations have been provided. In addition, a novel overlapping technique has been employed to compute the common area between two data sets for the first time, offering a detailed qualitative examination of the complex dynamics of the system. The results obtained are novel and informative, contributing valuable information for future research and serving as a resource for ongoing studies in the field.

The geometric content of chaos in nonlinear systems with multiple stabilities of high order is a challenge to computation. We introduce a single algorithmic framework to overcome this difficulty in the present study, where a parametrically forced oscillator with cubic–quintic nonlinearities is considered as an example. The framework starts with the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which is a self-learned algorithm that extracts an interpretable and correct model by simply analyzing time-series data. The resulting parsimonious model is well-validated, and besides being highly predictive, it also offers a solid base on which one can conduct further investigations. Based on this tested paradigm, we propose a unified diagnostic pathway that includes bifurcation analysis, computation of the Lyapunov exponent, power spectral analysis, and recurrence mapping to formally describe the dynamical features of the system. The main characteristic of the framework is an effective algorithm of computational basin analysis, which is able to display attractor basins and expose the fine scale riddled structures and fractal structures that are the indicators of extreme sensitivity to initial conditions. The primary contribution of this work is a comprehensive dynamical analysis of the DM-CQDO, revealing the intricate structure of its stability landscape and multi-stability. This integrated workflow identifies the period-doubling cascade as the primary route to chaos and quantifies the stabilizing effects of key system parameters. This study demonstrates a systematic methodology for applying a combination of data-driven discovery and classical analysis to investigate the complex dynamics of parametrically forced, high-order nonlinear systems.

Stochastic differential equations are practical tools for modeling systems in which stochastic effects prevail, distinguishing it from deterministic models. Qualitative and quantitative analyses of a specific observed model are possible with the help of a thorough discrimination framework of such systems. The effectiveness of the method is supported by exact results derived from the model using necessary constraint conditions. This study looks at how model parameters influence solution behavior with two and three dimensions. In addition, numerical studies are conducted to validate the theoretical findings and determine the stability of the system under different circumstances. Therefore, when the model is reformulated as a dynamical system, we get the Hamiltonian and topological characteristics, bifurcation theory, Lyapunov coefficients, quasiperiodic, and chaos. The analysis of the sustained chaotic behavior by outer forms of control offers greater insight into the dynamics of the proposed model.The findings further indicate possible uses of this model in areas such as climatology where stochastic disturbances play a major role in system behavior. Hence, the current study shares enough methodological improvement for analytical problems in engineering, physics, and mathematics, especially the non-linearities solved with stochastic models.

In this paper a detailed investigation of the non-linear Schrödinger equation is presented. A comprehensive set of tools, such as chaotic attractors, Lyapunov exponents, multistability, Poincaré maps, and phase portraits are employed for the analysis of the non-perturbed and perturbed dynamical systems. In addition, the system power frequency response is analyzed, and the sensitivity to time delays is examined with return maps. To this end, the study introduces a novel classification for multi-stability and Lyapunov exponents to analyze and categorize the complex behavior of the system. The developed algorithm explores the wide parameters and initial state space, providing a complete classification of system behavior. The chaotic, quasi-periodic and periodic behaviors of the perturbed dynamical system are observed. Furthermore, the paper compares the two soliton solutions by analyzing their outcomes across a defined range of parameters to gain valuable insights into the state under which the solutions exhibit identical behavior. The classification algorithms are provided in an organized manner, and complete results are made available at the GitHub repository, ensuring that future researchers can expand upon these results. The proposed classification algorithms enable the engineers to select parameters informed by classification results. Moreover, by analyzing the results of the soliton overlap, engineers can leverage soliton co-existence behaviors to improve the resilience and performance of the system in various operating conditions. The proposed methodology has applications in optical fibers and other engineering fields.

Recently, there has been growing interest among researchers in extending and generalizing the concept of convexity to incorporate fractional theory and related inequalities. One notable extension which involves studying convexity on fractal sets is the (h - m)-convexity, which generalizes the h-convexity and m-convexity concepts. In this paper, we present a new notion of generalized (h - m)-preinvex functions, which is a generalization of convexity on fractal sets, along with various characteristics for the newly presented functions. Utilizing the new concept, we derive new Hermite-Hadamard-type inequalities and establish a new identity for generalized (h - m)-preinvex functions with parameters involving local fractional integrals. In addition, we derive some general local fractional integral inequalities for generalized (h - m)-preinvex functions. We utilize our findings in practical applications by selecting particular values for the variables to construct various generalizations of midpoint, trapezoidal, and Simpson type inequalities. The findings of this work provide essential extensions and generalizations of earlier studies conducted in the area.

The stochastic dynamical rho 4 equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical rho 4 equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems' dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems' dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations.