Structural stability

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The structural stability of systems is a system property that is related to the topological equivalence of rivers and solutions of differential equations . This means that systems are compared, the descriptive equations of which depend on one or more parameters. If a completely different behavior results from a slight change in the value of the parameters, it is said that the system is not structurally stable for this parameter value, or that it experiences a bifurcation .

The definition is:

A system is called structurally stable in if there is one such that and are topologically equivalent for all with . ${\displaystyle {\dot {x}}=f(x,{\bar {\mu }})}$ ${\displaystyle {\bar {\mu }}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle {\dot {x}}=f(x,\mu )}$ ${\displaystyle {\dot {x}}=f(x,{\bar {\mu }})}$ ${\displaystyle \mu }$ ${\displaystyle \left\|\mu -{\bar {\mu }}\right\|<\varepsilon }$

In other words: There is a homeomorphism which converts the trajectories of the first system into those of the second.

If a system is not structurally stable for a given value , this is called a bifurcation of the system in . ${\displaystyle \mu }$ ${\displaystyle \mu }$

Andronov – Pontryagin's theorem

This theorem says that a system is structurally stable in an environment (des ) only if: ${\displaystyle \mathbb {R} ^{2}}$

• There are only a finite number of rest positions and cycles in this environment , and they are all hyperbolic.
• There is no solution that goes back to the same saddle or connects two different saddles.