An Improved Gumowski–Mira Map with Symmetric Lyapunov Exponents

The symmetric Lyapunov exponents (LEs) are known to be an inherent property of continuous-time conservative systems. However, the research on this interesting phenomenon in a discrete-time chaotic map has not been reported. Thus, this paper presents an improved 2D chaotic map based on Gumowski–Mira (GM) transformation, which has a stable fixed point or an unstable fixed point depending on its control parameters. Furthermore, it can display symmetric LEs and an infinite number of coexisting attractors with different amplitudes and different shapes. To demonstrate the complex dynamics of the 2D chaotic map, this paper studies its control parameters related to dynamical behaviors employing numerical analysis methods. Then, the hardware implementation based on STM32 platform is established for illustrating the numerical simulation results. Next, the random performance of the 2D chaotic map is tested by NIST FIPS140-2 suite. Finally, an image encryption algorithm based on the 2D chaotic map is designed, and the results obtained reveal that the proposed chaotic map has excellent randomness and is more suitable for many chaos-based image encryptions.