In this work, we explore the superposition of two well-known chaotic oscillators, namely, the Duffing double-well and the forced van der Pol with the fractional order derivative. The proportional fractional derivative has been taken for numerical simulations and highly chaotic solution to improve some information of security systems has been found. The existence and the uniqueness of a super system are stated in the form of theorems using the Lipschitz condition locally. The qualitative properties of chaotic dynamics are described by mean of Lyapunov exponent (LE), eigenvalues, bifurcation and Poincaré maps. The analog circuit is also intended, with the help of different physical instruments, to validate the superposition of chaotic systems. The randomness level of a superposition chaotic system is tested via standard test suite, and the qualified set of a 32-bit array with high haphazardness is used for encryption as well as decryption. Furthermore, a security analysis is performed using different parameters, such as the uncertainty, similarity etc. The outcomes for the properties, time evolution, phase portrait, and oscilloscopic views are presented in tabulated and graphical forms.

中文翻译：

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分数阶非线性混沌模型的定性研究：电子实现和安全数据增强

在这项工作中，我们探索了两个著名的混沌振荡器的叠加，即 Duffing 双阱和具有分数阶导数的强制 van der Pol。数值模拟采用了比例分数阶导数，找到了改善安全系统某些信息的高度混沌解。超级系统的存在性和唯一性在局部使用 Lipschitz 条件以定理的形式表示。混沌动力学的定性性质通过李雅普诺夫指数 (LE)、特征值、分岔和庞加莱映射来描述。模拟电路还旨在借助不同的物理仪器来验证混沌系统的叠加。通过标准测试套件测试叠加混沌系统的随机性水平，并使用具有高度随机性的32位数组的合格集合进行加密和解密。此外，使用不同参数（例如不确定性、相似性等）执行安全分析。属性、时间演变、相图和示波器视图的结果以表格和图形形式呈现。