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.

This paper introduces a novel Jerk-like system characterized by discontinuous vector fields along a codimension-1 switching surface. This system is capable of exhibiting both the existence and non-existence of equilibria in response to defined arbitrary functions. Non-smooth properties are investigated, which are essential for identifying and allowing rapid transient responses to dynamic events that influence the system's behaviors. The proposed discontinuous systems, which include both crossing and sliding solutions, are examined specifically for the detection of self-excited and hidden attractors. The dynamic characteristics of these systems are examined utilising the criteria for discontinuous solutions, the Poincarè return map, the spectrum of Lyapunov exponents, and bifurcation diagrams. The analysis reveals that incorporating the switching surface into Jerk-like subsystems alters the dimensions of the generated attractors. This alteration provides the necessary basis for the formation of period-doubling orbits that culminate in chaotic attractors. In the scenarios involving sliding motion within the proposed system, various attractors are identified that exhibit sensitive dependence not only on the initial conditions and parameter variations, but also on the interaction of the flow with the discontinuity surface. An interesting result indicates that the proposed novel Jerk system, which is distinguished by the absence of both real and virtual equilibria, reveals a family of periodic orbits in the sliding region surrounding the pseudo-equilibrium. The use of analytical solutions and numerical techniques successfully identifies a wide range of attractors, including periodic orbits, period-doubling phenomena, and chaotic behavior in both crossing and sliding modes.

The primary aims of this research is to reduce marijuana use within the general population, addressing its harmful effects and its status as an illegal substance that poses ongoing risks to public health in developing countries. To tackle this issue, we have developed a mathematical model that translates real-world marijuana consumption into mathematical terms using first-order nonlinear ordinary differential equations. A key objective of this study is to modify the NERAPH "Non-smokers, Experimental-smokers, Recreational-smokers, Addicts, Hospitalized individuals, and Prisoners" model by incorporating optimal control measures. This modified model is designed to estimate the initial rate of marijuana transmission within these groups. To implement effective control strategies, we conduct sensitivity analyses to identify the most impactful parameters. These findings enable us to recommend targeted control variables for the most sensitive parameters, leading to the development of an optimal control model aimed at reducing marijuana consumption as efficiently as possible. Finally, we compare the outcomes of the basic and optimal control models through numerical results to evaluate the effectiveness of each approach. This comparison provides a clear measure of how optimal control techniques enhance the model's ability to reduce marijuana use more effectively than the existing basic model.