In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation by using first integral method; we found new types of hyperbolic solutions and trigonometric solutions by symbolic computation.

This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical simulations to see how different fractional-order functions alter the qualitative behavior of the system. We demonstrate this via phase portraits and time-series responses. The research analyzes bifurcation development, chaotic oscillations, and stability transition and demonstrates dynamic patterns impossible to describe with integer-order models. Lyapunov exponent analysis also demonstrates system sensitivity to initial conditions and small disturbances. The outcomes confirm that the variable-order procedure provides a faithful representation of nonlinear and intricate processes of engineering and physical sciences, pointing out the dominant role of memory effects on the transitions among periodic, quasi-periodic, and chaotic regimes.

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 $\alpha$. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different $\alpha$ 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.

In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving electric circuits, population growth, vibrational beams, and heat conduction. The Mohanad transform is defined and introduced in this study, along with its fundamental qualities,including linearity and convolution. It is also discussed in connection with other integral transforms and how it is used in derivatives. Additionally, we use the Mohanad transform to solve a few systems of ordinary differential equations (ODEs) and review its properties in this paper. Determining the concentration of a chemical reactant (material) in a series is a physical chemistry problem that we use in the application part. We achieve this by developing a model based on ordinary differential equations (ODEs) and then solving them using the Mohanad transform. This research proves that, with little computational effort, we can get the exact solutions of ordinary differential equations (ODEs) via the Mohanad transform. We used graphs and tables to show our answer.

We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order $\alpha \left(t\right)$ as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a windowed estimate of the largest Lyapunov exponent, drives both static and dynamic order modulation laws. The presence and uniqueness of solutions are demonstrated through two distinct methodologies: a piecewise constant-order decomposition with an explicit convergence rate and a direct contraction-mapping argument on the variable-order Volterra operator. Local stability is analyzed via Matignon’s spectral criterion under a quasi-static (frozen-time) approximation. The modulation laws are designed to steer $\alpha \left(t\right)$ below the critical order ${\alpha }_c\approx 0.8632$, at which the nontrivial equilibria $E_{1,2}=(\pm \sqrt{5.5},0,0)$ become locally asymptotically stable. A second-order predictor–corrector scheme attains its expected convergence rate. A controlled ablation study over 200 Monte Carlo runs demonstrates that the proposed laws reduce the terminal tracking error by 81% relative to the best fixed-order baseline, while requiring approximately eight orders of magnitude less control effort than classical active control. Hybrid synchronization (complete in $(u,v)$ and anti-synchronization in w) is successfully achieved in the variable-order setting.