We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.

We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal ‘‘fingerprint’’ of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.

A one-dimensional Schrödinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices. When the potential is periodic, the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength. This critical index is also evaluated as the solution to an escape-rate problem for the recursion relations.

The interactions between proteins, DNA, and RNA in living cells constitute molecular networks that govern various cellular functions. To investigate the global dynamical properties and stabilities of such networks, we studied the cell-cycle regulatory network of the budding yeast. With the use of a simple dynamical model, it was demonstrated that the cell-cycle network is extremely stable and robust for its function. The biological stationary state, the G₁state, is a global attractor of the dynamics. The biological pathway, the cell-cycle sequence of protein states, is a globally attracting trajectory of the dynamics. These properties are largely preserved with respect to small perturbations to the network. These results suggest that cellular regulatory networks are robustly designed for their functions.

The Gutenberg‐Richter power law distribution for energy released at earthquakes can be understood as a consequence of the earth crust being in a self‐organized critical state. A simple cellular automaton stick‐slip type model yields D(E) ≈ E^−τ with τ≈1.0 and τ≈1.35 in two and three dimensions, respectively. The size of earthquakes is unpredictable since the evolution of an earthquake depends crucially on minor details of the crust.