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Owing to valuable significance of bioconvective transport phenomenon in interaction of nanoparticles, different applications are suggested in field of bio-technology, bio-fuels, fertilizers and soil sciences. It is well emphasized fact that thermal outcomes of nanofluids can be boosted under the consideration of various thermal sources. The aim of current research is to test the induction of induced magnetic force in bioconvective transport of non-Newtonian nanofluid. The rheological impact of non-Newtonian materials is observed by using Casson fluid with suspension of microorganisms. The chemical reaction effected are interpreted. The thermal conductivity of material is assumed to be fluctuated with temperature fluctuation. The flow pattern is endorsed by stretching surface following the stagnation point flow. Under the defined flow assumptions, the problem is formulated. A computational software with shooting technique is used to present the simulations. A comprehensive analysis for problem is presented. It is claimed that the interpretation of induced magnetic force exclusively enhanced the thermal phenomenon.

In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, R0 was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data.

This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen is observed. The NABC method illustrates chaotic systems. We used the LNIM method to find analytical solutions for fractional Chen systems. The method stands out for its user-friendliness and numerical stability. The proposed methods are effective and yield analytical solutions and detection of chaotic behavior. Simultaneously, this results in a more precise understanding of the system. As a result, we may apply the approach to different systems and achieve more accurate findings. Furthermore, it has been demonstrated to be effective in accurately identifying instances through the exhibition of attractor chaos. Future applications in science and engineering can utilize these two methods to find numerical simulations and solutions to a variety of models.

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders α. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different α show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems.

Soliton theory research has a substantial impact on the application of nonlinear sciences in different fields. This has led to a significant increase in the focus of researchers on the study of solitary waves in recent years. This study explores the diverse dynamic behaviors exhibited by soliton solutions within the framework of the soliton neuron model. In neuroscience, this model is regarded as an important tool for comprehending the initiation and propagation of action potentials along axons through the application of a thermodynamic theory of nerve pulse transmission. The model proposed herein suggests that signals that propagate through the cell membrane can be represented as solitons, or solitary sound pulses. In order to analyze these soliton solutions, the nonlinear differential equation is transformed into the corresponding ordinary differential equation using a wave transformation. The wave profiles of the soliton neuron model are derived by using the Kumar-Malik method, multivariate generalized exponential rational integral function method, and Riccati modified extended simple equation method. These methods are implemented to extract a diverse array of soliton solutions, such as mixed, dark, bright-dark, singular, bright, complex, and combined solitons. This examination is focused on specific nonlinear phenomena of the proposed model. Numerous graphs are incorporated to clarify the behavior of solutions across a diverse range of parameter values. By validating the effectiveness of current methodologies and elucidating the nonlinear dynamic characteristics of a system, this research makes a substantial contribution to the domains of nonlinear science and higher-dimensional nonlinear wave fields. The insights presented in this paper can be implemented to address analytical challenges in various nonlinear systems in biological, technological, and physical systems to facilitate the comparison of computational and experimental data.

In this study, we investigate the -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal processing, geophysics, fluid dynamics, and plasma physics. Given its broad utility, a rigorous mathematical analysis of the JM equation is essential. The primary objective of this work is to derive exact soliton solutions using the Modified Sub-Equation (MSE) and Modified Auxiliary Equation (MAE) techniques. These solutions are computed using Maple 18, and encompass a variety of wave structures, including bright solitons, kink solitons, periodic waves, and singular solitons. The potential applications of these solutions span diverse domains, such as nonlinear dynamics, fiber optics, ocean engineering, software engineering, electrical engineering, and other areas of physical science. Through numerical simulations, we visualize the physical characteristics of the obtained soliton solutions using three distinct graphical formats: 3D surface plots, 2D contour plots, and line plots, based on the selection of specific parameter values. Our results demonstrate that the MSE and MAE techniques are not only efficient but also straightforward in extracting soliton solutions for the JM equation, outperforming other existing methods. Furthermore, the solutions presented in this study are novel, representing contributions that have not been previously reported in the literature.

In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and cold plasma physics. We describe the fundamental properties of fractional calculus and its application to the STF-mBBM equation. Utilizing beta derivatives, we enhance our understanding of the intricate wave dynamics involved. Through the modified -expansion method (M -EM), we derive periodic, and kink singular soliton solutions and represent them graphically. We present the influence of the fractional parameter on traveling wave with 2 D , 3 D , surface and contour plots, providing a thorough understanding of the physical phenomena associated with the fractional model. In addition, we utilize the Hamiltonian property to analyze the chaotic dynamics of the solutions we’ve acquired. We perform two types of analysis using the Galilean transformation: a local sensitivity examination is conducted to see how the model responds to changes in individual input factors, and a global sensitivity examination is conducted to comprehend the correlation between the variability in the results and the variability in each input variable throughout its whole range of significance. This comprehensive approach allows us to determine traveling wave solutions effectively, offering new insights into the non-linear dynamical behavior of the system. The findings from this study are unique and significant for further exploration of the equation, offering valuable insights for future researchers.

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.

In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science.

In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems.