This paper constructs a six-term new simple 3D jerk system modeled by chaotic model memory oscillators with four parameters that control the behavior. The suitable choice of one of these parameters helps the system describe behavior and attractors. This means that the choice is a parameter of the associated behavior (dissipative or conservative) and attractors (self-excited or hidden). Some features of the equilibrium are observed that are related to the dependence on these parameters, such as saddle-foci, non-hyperbolic, and node-foci. This system is rich in dynamic features including chaotic, quasi-periodic (2-torus), and periodic via the utilization of bifurcation diagrams and Lyapunov spectrum. Finally, a new image encryption algorithm is introduced that utilizes the jerk system. The algorithm is assessed through statistical performance analysis, according to the results of the experiments and security tests, it has been verified that the suggested image encryption algorithm is highly secure and could be a viable option for real-world applications.

Coexisting self-excited and hidden attractors for the same set of parameters in dissipative dynamical systems are more interesting, important, and difficult compared to other classes of hidden attractors. By utilizing of nonlinear state feedback controller on the popular Sprott- S system to construct a new, unusual 4D system with only one nontrivial equilibrium point and two control parameters. These parameters affect system behavior and transformation from hidden attractors to self-excited attractors or vice versa. As compared to traditional similar kinds of systems with hidden attractors, this system is distinguished considering it has (��-2) positive Lyapunov exponents with maximal Lyapunov exponent. In addition, the coexistence of multi-attractors and chaotic with 2-torus are found in the system through analytical results and experimental simulations which include equilibrium points, stability, phase portraits, and Lyapunov spectrum. Furthermore, the anti-synchronization realization of two identical new systems is done relying on Lyapunov stability theory and nonlinear controllers strategy. lastly, the numerical simulation confirmed the validity of the theoretical results.