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.

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.

Fractional-order chaotic systems have received increasing attention over the past few years due to their ability to effectively model memory and complexity in nonlinear dynamics. Nonetheless, most of the research conducted so far has been on constant-order formulations, which still have some limitations in terms of adaptability and reality. Thus, to evade these limitations, we present a recently designed four-dimensional hyperchaotic Chen system with variable-order fractional (VOF) derivatives in the Liouville–Caputo sense. In comparison with constant-order systems, the new system possesses excellent performance in numerous aspects. Firstly, with the use of variable-order derivatives, the system becomes more adaptive and flexible, allowing the chaotic dynamics of the system to evolve with changing fractional orders. Secondly, large-scale numerical simulations are conducted, where phase portrait orbits and time series for differences in VOF directly illustrate the effect of the order function on the system’s behavior. Thirdly, qualitative analysis is performed with the help of phase portraits, time series, and Lyapunov exponents to confirm the system’s hyperchaotic behavior and sensitivity to initial and control parameters. Finally, the model developed demonstrates a wide range of dynamic behaviors, which confirms the sufficient efficiency of VOF calculus for modeling complicated nonlinear processes. Numerous analyses indicate that this research not only shows meaningful findings but also provides thoughtful methodologies that might result in subsequent research on fractional-order chaotic systems.

This work applies two efficient techniques to investigate fractional-order systems: the Residual Power Series Method (RPSM) and the Caputo fractional derivative (CFD). Both are utilized, especially to evaluate chaotic behavior and investigate the complex dynamics associated with a four-dimensional fractional-order chaotic system. Reliable and efficient numerical simula-tions with the complexity of chaotic behavior are obtained using the CFD approach. Furthermore, derived analytical solutions of the fractional-order 4D system using the RPSM are obtained. This approach is highly preferred as it can handle several beginning conditions, is computationally efficient, and is numerically stable, hence producing rather precise results. Combining RPSM’sanalytical capacity with CFD’s strong numerical accuracy offers a comprehensive understanding of the system dynamics. Although RPSM is best for stability and simplicity of use, the CFD technique is especially helpful because of its great accuracy in predicting chaotic behavior. The suggested methods precisely define their dynamics, provide exact solutions, and clearly identify chaotic attractors. For instance, representative parameter values used in our analysis are (ν = 0.95, 0.99, 1). These results show how well CFD- and RPSM-based methods represent and solve challenging problems in engineering and scientific research.

A Banach space A, an open subset V of A, and an open subset U of A' are considered. Our definition introduces novel categories of topological algebras of holomorphic functions on A. We demonstrate the equality of the two sets of holomorphic functions (Hwv(V)) and (Hw*vk(U)) under specific assumptions. We demonstrated that norm-dense Pgi(A) is found in Pw(A) and norm-dense Pgi*(A') is found in Pw*(A'). Additionally, we demonstrated that Pgi(A) is τk-dense in Hwvk(V) and Pgi*(A') is τk*-dense in Hw*vk(U) for a Banach space with a decreasing Schauder basis A, a polynomially convex weakly open subset V of A, and a polynomially convex weak-star open subset U of A.

In this work, it is considered that a certain quasilinear elliptic equation in an open bounded domain in Rn over a vector space, and we derive gradient estimates for weak solutions of p-Laplacian type elliptic equations with tiny bounded mean oscillation coefficients locally Lp, p ≥ q. In addition, we provide the key findings.