bel said:Chaos and Fractals: New Frontiers of Science (0387979034/ 978-0387979038)
by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe is a pretty interesting introductory text. The classic paper by Edward Lorenz is, well, classic.
Chaos theory is a branch of mathematics that studies the behavior of dynamic systems that are highly sensitive to initial conditions, meaning that small changes in the starting conditions can result in vastly different outcomes. It seeks to understand the underlying patterns and principles behind seemingly random and unpredictable phenomena.
Chaos theory has applications in a wide range of scientific fields, including physics, biology, economics, and meteorology. It can help to explain complex systems and phenomena, such as weather patterns, population growth, and the behavior of subatomic particles. It also has practical applications in chaos-based cryptography and signal processing.
Some key concepts in introductory chaos theory include the butterfly effect, strange attractors, and sensitive dependence on initial conditions. The butterfly effect refers to the idea that small changes in initial conditions can lead to significantly different outcomes over time. Strange attractors are patterns or regions in phase space that dynamic systems tend to converge towards. Sensitive dependence on initial conditions refers to the idea that small differences in initial conditions can lead to vastly different outcomes in chaotic systems.
Chaos theory challenges the traditional notion of determinism, which states that if the starting conditions of a system are known, its future behavior can be predicted with certainty. In chaotic systems, however, infinitesimal differences in starting conditions can lead to vastly different outcomes, making long-term prediction impossible. This has important implications for understanding the predictability and stability of natural systems.
Some examples of chaotic systems in the real world include weather patterns, stock market fluctuations, population growth, and the movement of celestial bodies. These systems exhibit complex, unpredictable behavior that is influenced by small changes in initial conditions. Other examples include chemical reactions, fluid dynamics, and biological systems such as the human heart.