This textbook offers a fundamental introduction to chaotic systems without compromising mathematical precision. The text covers the classical theoretical aspects of dynamic systems in appealing depth, thoroughly addressing all essential examples from the literature. Additionally, it deals with more modern questions, such as the nature of computability and the reliability of computers. The distinction from advanced methods is clearly defined by the exclusion of statistical methods. For the core phenomena, concise yet complete programs in the Mathematica programming language are provided, which are easy to understand and can be easily adopted and adapted for individual experimentation. Over 180 examples and exercises illustrate the material and deepen understanding.