This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models.

A novel dynamic vibration absorber(DVA) model with negative stiffness and inerter-mass is presented and analytically studied in this paper. The research shows there are still two fixed points independent of the absorber damping in the amplitude frequency curve of the primary system when the system contains negative stiffness and inerter-mass. The optimum frequency ratio is obtained based on the fixed-point theory. In order to ensure the stability of the system, it is found that inappropriate inerter coefficient will cause the system instable when screening optimal negative stiffness ratio. Accordingly, the best working range of inerter is determined and optimal negative stiffness ratio and approximate optimal damping ratio are also obtained. At last the control performance of the presented DVA is compared with three existing typical DVAs. The comparison results in harmonic and random excitation show that the presented DVA could not only reduce the peak value of the amplitude-frequency curve of the primary system significantly, but also broaden the efficient frequency range of vibration mitigation.

Purpose—This study aims to perform an in-depth analysis of double-diffusive natural convection (DDNC) in an irregularly shaped porous cavity. We investigate the convective heat transfer process induced by the lower wall treated as a heat source while the side walls of the enclosure are maintained at a lower temperature and concentration, and the remaining wall is adiabatic. Various factors, such as the Rayleigh number, Darcy effects, Hartmann number, Lewis number and effects of magnetic inclination are evaluated for their influence on flow dynamics and heat distribution. Design/methodology/approach—After validating the results, the FEM (finite element method) is used to simulate the flow pattern, temperature variations, and concentration by solving the nonlinear partial differential equations with the modified Rayleigh number (10⁴ ≤ Ra ≤ 10⁷), Darcy number (10⁻⁴ ≤ Da ≤ 10⁻¹), Lewis number $(0.1\le Le\le 10)$, and Hartmann number $\left(0\le Ha\le 40\right)$ as the dimensionless operating parameters. Findings—The finding shows that the patterns of convection and the shape of the isotherms within porous enclosures are notably affected by the angle of the applied magnetic field. This study enhances our understanding of how double-diffusive natural convection (DDNC) operates in these enclosures, which helps improve heating and cooling technologies in various engineering fields. Research limitations/implications—Numerical and experimental extensions of the present study make it possible to investigate differences in thermal performance as a result of various curvatures, orientations, boundary conditions, and the use of three-dimensional analysis and other working fluids. Practical implications—The geometry configurations used in this study have wide-ranging applications in engineering fields, such as in heat exchangers, crystallization, microelectronics, energy storage, mixing, food processing, and biomedical systems. Originality/value—This study shows how an inclined magnetic field affects double-diffusive natural convection (DDNC) within a porous system featuring an irregularly shaped cavity, considering various multiphysical conditions.