In continuous dynamical systems, a hidden attractor occurs when its basin of attraction does not connect with small neighborhoods of equilibria. This research aims to investigate the presence of hidden-like attractors in a class of discontinuous systems that lack equilibria. The nature of non-smoothness in Filippov systems is critical for producing a wide variety of interesting dynamical behaviors and abrupt transient responses to dynamic processes. To show the effects of non-smoothness on dynamic behaviors, we provide a simple discontinuous system made of linear subsystems with no equilibria. The explicit closed-form solutions for each subsystem have been derived, and the generalized Poincaré maps have been established. Our results show that the periodic orbit can be completely established within a sliding region. We then carry out a mathematical investigation of hidden-like attractors that exhibit sliding-mode characteristics, particularly those associated with grazing-sliding behaviors. The proposed system evolves by adding a nonlinear function to one of the vector fields while still preserving the condition that equilibrium points do not exist in the whole system. The results of the linear system are useful for investigating the hidden-like attractors of flow behavior across a sliding surface in a nonlinear system using numerical simulation. The discontinuous behaviors are depicted as motion in a phase space governed by various hidden attractors, such as period doubling, period-m segments, and chaotic behavior, with varying interactions with the sliding mode.

The main objective of this paper is to investigate the deformation behavior in a two-dimensional homogeneous, isotropic thermo-elastic half-space subjected to an external heat source applied at its boundary and electro-magnetic field, within the framework of the Lord-Shulman theory. The novelty of this work lies in formulating a new coupled dynamic model that integrates mechanical stresses with temperature interactions in an elastic medium. The Lame’s potential method and normal mode analysis method is applied to solve the resulting coupled differential equations, obtaining both deterministic and solutions. The temperature, displacement component and stress components of these waves are obtained analytically with the use of suitable boundary conditions. Focusing on aluminium epoxy-like material, graphical representations of numerically simulated findings using MATHEMATICA software show how time, magnetic field and permeability of electric field. The work includes detailed graphical representations of crucial discoveries such as temperature distributions, stress components, and displacement components which provide amazing visual insights into the complex interactions that occur within thermo-elastic systems. From the distributions, it can be found the wave type heat propagation in the medium. A comparison is made with the results obtained in the presence and absence of the electro-magnetic field. By forging a link between theoretical insights and practical applications, this study stands to make a significant contribution to the field of materials science, providing a solid foundation for subsequent inquiries within this rapidly progressing arena. A graphic comparison of our physical quantity accuracy results with earlier studies shows the significant influence of electro-magnetic field on thermoelastic phenomena. The analysis of thermoelastic problems with electro-magnetic field benefits from the study.

Significance: This study addresses critical industrial and biomedical applications including glass blowing (thermal management of shrinking sheets), polymer sheet extrusion (controlled cooling), magnetic drug delivery (nanoparticle targeting), and nuclear reactor cooling (enhanced heat transfer). Aim: We present a novel nonlinear analysis of magnetohydrodynamic (MHD) boundary layer flow of a Jeffery Al₂O₃ nanofluid over a shrinking permeable plate with convective boundary conditions, uniquely integrating mixed convection, Ohmic dissipation, heat generation, Brownian motion, and thermophoresis within a non-Newtonian nanofluid framework. Methodology: The governing partial differential equations are transformed using similarity transformations and solved via the Adomian decomposition method (ADM). Comprehensive validation against RK4, RK45, and bvp4c demonstrates excellent agreement with maximum relative errors below $5\times {10}^{-4}$. Key Contribution: (i) Normal velocity decreases by 15–25% as the Biot number increases from $Bi=0.4$ to $0.6$; (ii) tangential velocity decreases by 20–30% as the magnetic parameter increases from $M=5$ to 15; (iii) temperature increases by 30–40% as the Eckert number increases from $Ec=0.5$ to $2.5$; (iv) ADM converges within 12–15 terms with $L_2$ errors $<{10}^{-5}$; (v) skin friction coefficient increases from $C_f=3.02713$ to $3.90082$ as $Q_0$ increases from 1 to 4; (vi) Nusselt number values: $Nu/\sqrt{Re}=0.4621$ at $Pr=0.7$, $0.8954$ at $Pr=2$, $3.2890$ at $Pr=20$. These quantitative findings provide design guidelines for engineers in thermal management and biomedical applications.

In this paper, we present a novel approach to characterize the flow and heat transfer characteristics of a tri-hybrid nanofluid (Blood + TiO$ _2 $ + Al$ _2 $O$ _3 $ + Cu) through a curved, thermally heated channel using analytical and numerical frameworks based on a dynamical system. Dynamical system theory enables systematic identification and analysis of fundamental flow phenomena, including equilibrium states, stability, and bifurcations, which are often difficult to capture with traditional methods. Long-wavelength and low-Reynolds reductions yield analytical flow fields, from which the nonlinear dynamical system was constructed. The analytical determination of the general equilibrium points was complicated by the channel geometry and the strong nonlinearity of the model; therefore, a numerical approach based on domain decomposition techniques was adopted to overcome these difficulties. Saddle-node bifurcations, periodic orbits, and heteroclinic orbits provide a framework to characterize recirculating flows and the transitions between different flow states. The impact of physical properties and nanoparticle characteristics, which affect both global and local bifurcation behavior, was also examined. As a novel result, we revealed multiple equilibrium points with distinct bifurcation behaviors, capturing key fluid motion phenomena, such as trapping, governed by the channel geometry and tri-hybrid nanofluid properties. Our results, based on a dynamical systems analysis, refine and extend previous studies by clarifying the stability of equilibrium solutions, identifying key bifurcation regimes, and revealing the underlying nonlinear flow behavior.