Evolving Tangent Hyperbolic memristor based 6D chaotic model with fractional order derivative: Analysis and applications

Abstract

The nonlinear dynamics of fractional order systems and memristor based circuits have attracted an escalating attention in recent years. This paper reports on a novel six dimensional nonlinear dynamical system with a tangent hyperbolic memristor circuit and amalgamated image encryption. The dynamical system is analyzed using standard tools, including phase portraits, equilibrium points, Eigenvalues, Routh–Hurwitz stability criterion, Lyapunov exponents and bifurcation diagrams. The analysis suggests that the developed system is chaotic in nature and has exciting new 2D and 3D trajectories that have not been reported before. The chaotic system is numerically solved for different values of the fractional order q, and the results of equilibrium points and Lyapunov exponents are mentioned in tables. Fractional order circuits are designed for q=1and q=0.99utilizing the approximation fractional order technique of the transfer function. The developed circuit is generalized in a way that the circuit can work for both the integer and non-integer fractional order values by just introducing the fractional block to make it work for q=0.99, and removing it with a conventional integrator capacitor to work for q=1. The circuit simulation and numerical results were compared, and signifying that they were both in good agreement with each other. Lastly, the random number generated from the chaotic system is utilized to scramble the image via the scheme of Amalgamated Image Encryption. The scrambled image was tested using different image security test algorithms to support the idea that the chaotic system and image can together form an advantageous key.

Keywords

• Memristor
• Fractional calculus
• Image encryption
• Tangent-hyperbolic