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由于忆阻器具有独特的非线性特性和记忆效应,基于忆阻器的混沌系统成为目前研究的热点.然而,关于忆阻混沌系统的不稳定周期轨道研究目前较少.本文通过引入三角函数忆阻器改进三维混沌系统,构建了一个新型四维忆阻混沌系统.通过Lyapunov指数、庞加莱截面、相图、时域波动图对系统进行动力学行为分析.针对变分法在寻找可靠圈猜想受限的问题,创新性地提出了一种基于三角函数物理特性的优化策略.通过该优化策略,结合符号动力学对新系统的不稳定周期轨道进行了系统分析,并进一步采用自适应反步法控制已知轨道的稳定性.对新型忆阻混沌系统序列进行NIST测试,发现该序列具有良好的伪随机性,适用于图像加密算法的应用.设计了基于新系统的图像加密算法,忆阻混沌系统的应用显著提高了密钥空间和密钥敏感度,增强了图像加密的安全性.算法首先通过RGB三个通道之间的跨平面置乱对彩色图像像素首次置乱,随后进行单平面的二次置乱,充分改变图像像素.算法采用异或运算进行扩散,其NPCR和UACI的平均值表明其具有强大的差分攻击能力.此外,通过直方图、相关性、抗剪切攻击及运行时间等测试验证其可靠性.最后,在DPS平台上验证了实验结果与数值分析结果的一致性.Due to their unique nonlinear characteristics and memory effects, memristor-based chaotic systems have become a significant focus of research. However, studies on unstable periodic orbits in memristive chaotic systems remain relatively scarce. In this paper, a novel four-dimensional memristive chaotic system is constructed by introducing a trigonometric-function-based memristor to enhance a three-dimensional chaotic system. The dynamical behaviors of the system are analyzed using Lyapunov exponents, Poincaré sections, phase portraits, and time-domain plots. The proposed memristive chaotic system exhibits rich dynamical characteristics, including transient behavior, intermittent chaos, and diverse attractor dynamics under parameter variations. To overcome the limitations of the variational method in finding reliable initial guesses for unstable periodic orbits, an innovative optimization strategy leveraging the physical characteristics of trigonometric functions is proposed. Integrated with symbolic dynamics, this strategy enables the rapid acquisition of robust initial guesses for unstable periodic orbits within specific intervals. Furthermore, it allows for the migration of these guesses to other regions of the attractor, ultimately achieving full coverage of the attractor's unstable periodic orbits. Following a systematic analysis of the unstable periodic orbits in the new system, the adaptive backstepping method is employed to control the stability of the known unstable periodic orbits, namely 320 and 013. The pseudorandom sequences generated by the novel memristive chaotic system successfully passed the NIST suite, with all test items yielding P-values greater than 0.01, which confirms their excellent pseudo-random characteristics. The application of this system in image encryption achieves a key space of 10120, significantly enhancing the key space and key sensitivity of the algorithm. The encryption process begins with cross-plane scrambling operations among the RGB color channels for initial pixel processing, followed by intra-plane scrambling to further disrupt the pixel arrangement. XOR operations are then employed for pixel value diffusion. The algorithm demonstrates outstanding resistance to differential attacks, with average NPCR and UACI values reaching 99.6041% and 33.4933%, respectively. Comprehensive security analyses, including histogram analysis, correlation analysis, resistance to cropping attacks, and runtime evaluation, verify that the proposed encryption scheme not only possesses strong security capabilities but also maintains high computational efficiency, making it highly suitable for practical image encryption applications. Finally, the realizability of the system is verified by utilizing a DSP circuit.
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Keywords:
- Chaotic System /
- Memristor /
- Periodic Orbit /
- Image Encryption
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