article · IEEE Access
A newly identified four-dimensional autonomous hyperjerk system features an equilibrium along a half line and incorporates an absolute function nonlinearity within an eight-term structure. Analysis of the model reveals complex dynamical phenomena, including multistability, antimonotonicity, and reversals in period doubling, verified through bifurcation patterns and Lyapunov exponent trajectories. The mathematical system has been realised in hardware, first through simulated electronic circuitry and subsequently via an FPGA implementation utilising Forward Euler and Trapezoidal numerical techniques. Physical measurements from an FPGA board confirmed consistent attractor behaviours matching numerical simulations. Utilising the chaotic properties of this system, an image encryption algorithm was developed and tested experimentally, demonstrating high operational security and efficiency.
Chaotic systems are valuable tools for generating complex, unpredictable signals necessary for secure data transmission. By proving that this novel mathematical model can be realised physically on standard FPGA microchips and electronic circuits, the research bridges abstract non-linear dynamics and practical computing. It offers an adaptable foundation for hardware-level data protection and secure multimedia communication.
This work demonstrates applied feasibility for digital security, particularly in image encryption and secure communications. Developers of embedded cryptographic systems or FPGA-based security hardware could adapt the design for robust data scrambling. Because the system has already been tested on an FPGA development board alongside circuit simulations, the technology sits at an applied and tested stage, though it requires further integration into commercial communication protocols.
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A hyperjerk system pertains to a dynamical system regulated by an ordinary differential equation of nth order, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≥ 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.
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DOI: 10.1109/access.2024.3351693
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