Nonlinear dynamics from physics to biology

S elf-organization has been a hot topic in the second half of last century when physicists and chemists discovered a variety of nonequilibrium phenomena that could be subsumed under a common heading. Self-organizing systems form ordered states in space and time spontaneously and without an external template. The patterns are characterized as dissipative structures because their maintenance requires a flow of energy or matter. After introducing a flow of increasing strength into a system at equilibrium, patterns form instantaneously at certain critical values of the flux. In the language of dynamical systems theory the patterns emerge at bifurcation points corresponding to some critical intensity of the flow. At present we know many well-studied examples of self-organizing systems at many time scales and largely different spatial extensions. Examples are the gigantic red spot on Jupiter, cloud patterns in the atmosphere, the Bénard phenomenon in the coffee cup, the Taylor-Cuvette flow, the Belusov-Zhabotinskii reaction, Liesegang rings, and many other nonlinear phenomena. Recent progress in all fields where self-organization is important confirmed the original concepts and, in addition, gave rise to a new formulation of the old paradigms that allows for a distinction of different forms of self-organizing dynamics in physics, chemistry, and biology. We distinguish here three cases that involve different levels of complexity: self-organization of (i) structure, (ii) function, and (iii) intention or seeming purpose. Structural self-organization became a central issue of nonequilibrium dynamics ever since Alan Turing published his seminal work on chemical morphogenesis [1]. Turing suggested a chemical mechanism based on slow diffusion of an activator and fast diffusion of an inhibitor that can lead to spontaneous formation of stable stationary nonequilibrium patterns through diffusion of some key compounds and argued that such a mechanism could be responsible for the formation of biological patterns. It took 20 years before the Turing mechanism was incorporated into a conceptual framework for pattern formation in early embryonic development that results eventually in the patterns we find in adult organisms [2–6]. Activator and inhibitor are thought to represent two “morphogens,” leading to short-range activation and long-range inhibition. For a long time no diffusing morphogen was known in developmental biology and, moreAfter introducing a flow of increasing strength into a system at equilibrium, patterns form instantaneously at certain critical values of the flux. PETER SCHUSTER