In this study, we analyze the solution characteristics and dynamics of a variable-order fractional (V-OF) Arneodo system using the Liouville–Caputo fractional operator with variable order. The V-OF operator is used to describe the time-dependent memory effect in the system, which leads to more complex and diverse dynamics compared to integer-order systems. In this work, numerical simulations are performed to observe the effect of the order functions on the dynamic behaviors of the system. In addition, the phase portraits, time series graphs, and three-dimensional diagrams are used to analyze the dynamic behaviors and different types of oscillations present in the system. Furthermore, the bifurcations, chaotic behaviors, and stability of the system with variable orders are studied, and it is found that the system has more complex dynamics compared to the integer-order case. In this case, the Lyapunov exponents indicate that the system under investigation is sensitive to the initial conditions, and the memory effect can control the chaotic oscillation depending on the order of the functions.

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term.

The hydromangetic stability of bounded fluid jet under the influence of the electromagnetic (with toroidal varying field) force has been developed. A general dispersion relation valid for all modes of perturbation is derived. The geometric factor q which is the radii ratio of the tenuous-fluid regions plays an important role for stabilizing the model. The axial and transvers magnetic fields interior and exterior the fluid jet are stabilizing. The magnetic fields decrease the streaming destabilizing domains and at the same time give a sort or rigidity to the fluid molecules. For any value of the applying magnetic field strength, the instability character of the streaming model could be suppressed and dispersed. These results are confirmed numerically upon using computer programs.

In this paper, we investigate the oscillatory properties of fourth-order neutral delay differential equation solutions in the canonical situation. To our knowledge, this equation has received minimal research. We prove new improved features and relationships for the solution and the accompanying function. Based on these relationships, oscillation theorems were developed that guarantee oscillation of all solutions to the considered equation. The comparison principle used in our results is one of the most significant methods for investigating the oscillatory behavior of delay differential equations. The findings of our study extend and develop a number of previous findings in the literature.

This study covers the oscillatory characteristics of solutions of delay differential equations. In order to test the oscillation of all solutions, we establish three distinct criteria and take into account the noncanonical situation. The purpose of establishing various criteria is to achieve a number of benefits in the oscillation test, namely applying the criterion to Euler's equation yields the sharp condition, the influence of delay function of the criterion, and applying to some cases that have not been studied previously. In previous works that treated the studied equation in its general form, only classical criteria were obtained for the canonical case. Finally, we demonstrate that our findings are a generalization and extension of earlier findings in the literature through discussion and examples.