We present a simple new method to systematically identify all topological structures (e.g., positive feedback loops) potentially leading to locally unstable steady states: ICSA-The instability causing structure analysis. Systems without any instability causing structure (i.e., not fulfilling the necessary topological condition for instabilities) cannot have unstable steady states. It follows that common bistability or multistability and Hopf bifurcations are excluded and sustained oscillations and deterministic chaos are most unlikely. The ICSA leads to new insights into the topological organization of chemical and biochemical systems, such as metabolic, gene regulatory, and signal transduction networks.