A hyperjerk system pertains to a dynamical system regulated by an ordinary differential equation of $n$ th order, where $n \geq 4$ . The main contribution of this work is the finding of a new autonomous hyperjerk system with a half line equilibrium. The mathematical framework of the proposed hyperjerk system contains eight terms with an absolute function nonlinearity. The essential dynamic characteristics of the model are explored, encompassing analysis of equilibrium points and their stability, depiction of the phase trajectories, illustration of bifurcation patterns, and visualization of Lyapunov exponent graphs. Our finding shows that the new 4D hyperjerk system exhibits special behavior like multistability, period doubling reversals and antimonotonocity. The proposed hyperjerk system has been implemented with an electronic circuit using MultiSim 14.0. Moreover, the FPGA implementation of the proposed hyperjerk system is performed by applying two numerical methods: Forward Euler and Trapezoidal. Experimental attractors are given from an oscilloscope by using the Zybo Z7-20 FPGA development board, which are in good agreement with the MATLAB and MultiSim 14.0 simulations. Finally, based on the chaotic dynamical behavior of the proposed chaotic hyperjerk system, a new image encryption approach is proposed. The experimental outcomes of the presented encryption algorithm prove its efficiency and security.

Many biological systems and natural phenomena exhibit chaotic behaviors that are saved in time series data. This article uses time series that are generated by chaotic oscillators with different values of the maximum Lyapunov exponent (MLE) to predict their future behavior. Three prediction techniques are compared, namely: artificial neural networks (ANNs), the adaptive neuro-fuzzy inference system (ANFIS) and least-squares support vector machines (SVM). The experimental results show that ANNs provide the lowest root mean squared error. That way, we introduce a multilayer perceptron that is implemented using a field-programmable gate array (FPGA) to predict experimental chaotic time series.

This paper introduces a new chaotic jerk system with three cubic nonlinear terms. The stability properties of the three equilibrium points of the proposed jerk system are analyzed in detail. We show that the three equilibrium points of the new chaotic jerk system are unstable and deduce that the jerk system exhibits self-excited chaotic attractors. The bifurcation structures of the proposed jerk system are investigated numerically, showing period-doubling, periodic windows and coexisting bifurcations. An electronic circuit design of the proposed jerk system is designed using PSPICE. As an engineering application, a new image-encryption approach based on the new chaotic jerk system is presented in this research work. Experimental results demonstrate that the suggested encryption mechanism is effective with high plain-image sensitivity and the reliability of the proposed chaotic jerk system for various cryptographic purposes.

Reservoir computing has shown promising results in predicting chaotic time series. However, the main challenges of time-series predictions are associated with reducing computational costs and increasing the prediction horizon. In this sense, we propose the optimization of Echo State Networks (ESN), where the main goal is to increase the prediction horizon using a lower count number of neurons compared with state-of-the-art models. In addition, we show that the application of the decimation technique allows us to emulate an increase in the prediction of up to 10,000 steps ahead. The optimization is performed by applying particle swarm optimization and considering two chaotic systems as case studies, namely the chaotic Hindmarsh–Rose neuron with slow dynamic behavior and the well-known Lorenz system. The results show that although similar works used from 200 to 5000 neurons in the reservoir of the ESN to predict from 120 to 700 steps ahead, our optimized ESN including decimation used 100 neurons in the reservoir, with a capability of predicting up to 10,000 steps ahead. The main conclusion is that we ensured larger prediction horizons compared to recent works, achieving an improvement of more than one order of magnitude, and the computational costs were greatly reduced.

Mechanical jerk systems have applications in several areas, such as oscillators, microcontrollers, circuits, memristors, encryption, etc. This research manuscript reports a new 3-D chaotic jerk system with two unstable balance points. It is shown that the proposed mechanical jerk system exhibits multistability with coexisting chaotic attractors for the same set of system constants but for different initial states. A bifurcation analysis of the proposed mechanical jerk system is presented to highlight the special properties of the system with respect to the variation of system constants. A field-programmable gate array (FPGA) implementation of the proposed mechanical jerk system is given by synthesizing the discrete equations that are obtained by applying one-step numerical methods. The hardware resources are reduced by performing pipeline operations, and, finally, the paper concludes that the experimental results of the proposed mechanical jerk system using FPGA-based design show good agreement with the MATLAB simulations of the same system.