This paper presents a generalization of chaotic systems using two-dimensional affine transformations with six introduced parameters to achieve scaling, reflection, rotation, translation and/or shearing. Hence, the location of the strange attractor in space can be controlled without changing its chaotic dynamics. In addition, the embedded parameters enhance the randomness and sensitivity of the system and control its response. This approach overpasses performing the transformations as post-processing stages by applying them on the resulting time series. Trajectory control through dynamic parameters is demonstrated. Simulation results validate the proposed analysis for both the simplest and Lorenz chaotic systems. An image encryption scheme is implemented using transformed Lorenz system resulting in a more secure encryption scheme in comparison to Lorenz and other recent related works. The scheme exhibits good performance when assessed using the PRNG properties, encrypted image histogram and its uniformity through chi square test, pixel correlation, Mean Squared Error (MSE), entropy, Peak Signal-to-Noise Ratio (PSNR), the National Institute of Standards & Technology (NIST) test, key space, key sensitivity, resistance to differential, ciphertext-only, known plaintext, and chosen plaintext attacks, robustness against noise and computation time.

本文介绍了使用带有六个引入参数的二维仿射变换来实现缩放、反射、旋转、平移和/或剪切的混沌系统的泛化。因此，可以在不改变其混沌动力学的情况下控制奇异吸引子在空间中的位置。此外，嵌入的参数增强了系统的随机性和灵敏度并控制了其响应。这种方法通过将转换应用于生成的时间序列来超越将转换作为后处理阶段的执行。演示了通过动态参数的轨迹控制。仿真结果验证了对最简单和洛伦兹混沌系统的建议分析。图像加密方案是使用转换的洛伦兹系统实现的，与洛伦兹和其他最近的相关工作相比，该加密方案更安全。当使用 PRNG 属性、加密图像直方图及其通过卡方检验、像素相关、均方误差 (MSE)、熵、峰值信噪比 (PSNR)、美国国立卫生研究院的均匀性进行评估时，该方案表现出良好的性能。标准与技术 (NIST) 测试、密钥空间、密钥敏感性、对差分、仅密文、已知明文和选择明文攻击的抵抗力、抗噪声和计算时间的鲁棒性。