Chaos is a paradigm shift of all science, which provides a collection of concepts and methods to analyze a novel behavior that can arise in a wide range of disciplines. However, most of researches in simulations and applications of chaos are performed on finite-state automata, which inevitably causes chaos to collapse. Here we present a hybrid model by controlling digital system with continuous chaotic system to construct chaos on finite-state automata. A new concept and method named Generalized Symbolic Dynamics (GSD) is proposed to target the hybrid system. Based on GSD, a rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney. Moreover, analog-digital hybrid circuit is built for the digital chaotic system. Finally, a simple pseudorandom number generator is designed as a proof of concept. Results show that the proposed generator has good performance for cryptography. Such digital chaotic systems, which are not subject to degradation, could pave the way for widespread applications of chaos.

混沌是所有科学的范式转变，它提供了一系列概念和方法来分析在各种学科中可能出现的新颖行为。然而，大多数关于混沌的模拟和应用的研究都是在有限状态自动机上进行的，这不可避免地导致混沌崩溃。在这里，我们提出了一种通过控制数字系统和连续混沌系统来构造有限状态自动机混沌的混合模型。针对混合系统，提出了一种新的概念和方法，称为通用符号动力学（GSD）。基于GSD，给出了严格的证明，即从Devaney的角度来看，受控数字系统是混沌的。此外，为数字混沌系统建立了模数混合电路。最后，设计了一个简单的伪随机数生成器作为概念证明。结果表明，所提出的生成器具有良好的密码学性能。这样的数字混沌系统不会退化，可以为混沌的广泛应用铺平道路。