International Encyclopedia of Systems and Cybernetics

"The capacity of a system to maintain the same essential patterns of behavior despite some modification in the structural interrelations between the systems parts" (T.F.H. ALLEN &.T.B. STARR, 1982, p.278).

The concept of structural stability was introduced in 1937 by the Russian mathematicians A. ANDRONOV and L. PONTRIAGIN. It led and is still leading to considerable developments in the topological study of the complex systems stabilility, as for example catastrophe theory (THOM and ZEEMAN), the various types of attractors, and deterministic chaos.

As stated by L. LÖFGREN: "A system is structurally stable if the topology of its trajectories is preserved under sufficiently small perturbations" (1974, p.78).

For K.B.DE GREENE: "A function or system is structurally stable if such qualitative properties as numbers and kind of critical points and basins of attraction remain unchanged after a sufficiently small perturbation" (1990, p.54).

Small modifications correspond mostly to small perturbations. Structural stability is compatible with more or less periodic changes of state, i.e. with ergodicity.

According to S. DINER: "A structurally stable dynamic system is… a topologically stable system, i.e. a system whose phase portrait undergoes few modifications and remains topologically self-similar under the influence of lesser perturbations in its movement equations… ; (it) maintains its essential qualitative properties when affected by a slight perturbation" (1992, p.360).

This is the case for merely local instabilities.

To the ALLEN and STARR's definition and LÖFGREN's comment, which correspond to confirmed stable systems, one should aggregate the following comment by K.DE GREENE: "… structural stability in dissipative-structure theory refers to the response of the system to the addition of new elements" (1988, p. 291).

And: "The structural stability of any existing deterministic system, potentially capable of evolution, is continually tested by both internal and external (random noise) fluctuations" (DE GREENE, 1994, p.14). On the contrary: "The system is structurally unstable if mutations (genetic, scientific, technological, "great persons" – my terms) can multiply and take over and modify system processes" (Ibid).

R. ROSEN sees a practical use for the concept: "Structural stability is a general framework for comparing one description of a class of objects against another description, usually in terms of measures of closeness" (1991b, p.480).