Contents IJPAM: Volume 94, No. 2 (2014)

In this paper, we answer the question that many researchers did ask us about: "what is the geometrical meaning of the conformable derivative?". We answer the question using the concept of fractional cords. Fractional orthogonal trajectories are also introduced. Some examples illustrating the concepts of fractional cords and fractional orthogonal trajectories are given.

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using ${\vartheta }_i$-th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent $\left(L{E}_{max}\right)$ are used to demonstrate their impact on the system’s dynamics. Furthermore, we employ the sample entropy approach (SampEn), $C_0$ complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes.

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents ($L{E}_s$) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings.

In this paper, a numerical examination of propagating nonlinear dissipative dust-acoustic breathers and rogue waves (RWs) in electron depleted dusty plasmas having two superthermal ions of different temperature has been made. An important ingredient in this study is the inclusion of the dissipative effect due to the viscosity of the dust grains in the evolution wave equation. Accordingly, a damped/modified nonlinear Schrödinger equation (DNLSE), i.e. the standard nonlinear Schrödinger equation (NLSE) in addition to the damping term, is obtained using a reductive perturbation (the derivative expansion) method. Without taking into account the effect of dust viscosity, the standard NLSE is also examined, and the effect of relevant physical parameters on the breathers and rogue waves is examined. Moreover, the impact of dust kinetic viscosity on both breather structures and RWs is investigated by solving DNLSE numerically with the Dirichlet boundary conditions. This model may be useful to understand the excitation of dust-acoustic waves in the Earth's magnetotail.

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders, revealing their impact on dynamics. Through the maximum Lyapunov exponent $(L{E}_{max})$, phase portraits, and bifurcation charts. In addition, we assess the complexity and confirm the chaotic features in the map using the approximation entropy $ApEn$ and $C_0$ complexity. Studies show that the commensurate and incommensurate derivative values influence the fractional chaotic map-based three functions, which exhibit a variety of dynamical behaviors, such as hidden attractors, asymmetry, and symmetry. Moreover, the new system’s stabilizing employing a 3D nonlinear controller is introduced. Finally, our study validates the research results using the simulation MATLAB R2024a.