# Local structural stability

of a compact invariant set $F$ of a smooth dynamical system

The preservation of all topological properties of the system in some neighbourhood of $F$ under any sufficiently small (in the $C^1$ sense) perturbation of the system. More precisely, local structural stability consists in the following: There are neighbourhoods $U\supset V\supset F$ and for any $\epsilon>0$ there is a $\delta>0$ such that under a perturbation of the original system in $U$ at a distance at most $\delta$ from it in the $C^1$-metric there is a homeomorphic imbedding $V\to U$ that shifts points by at most $\epsilon$ and takes segments of trajectories of the original system lying in $V$ to segments of trajectories of the perturbed system. (Thus, strictly speaking, local structural stability is a property not of the set $F$ itself but of the system considered in a neighbourhood of $F$.)

If $F$ is an equilibrium position of a flow (continuous-time dynamical system) (or a fixed point of a cascade, that is, a dynamical system with discrete time), then the local structural stability of $F$ implies the preservation of the topological properties of the system under linearization at the point $F$. It is almost obvious that in this case a necessary condition for local structural stability is that the eigen values of the linearized system are outside the imaginary axis (respectively, do not lie on the unit circle). The condition is also sufficient (the Grobman–Hartman theorem, see [1], Chapt. IX). From this it is easy to derive a necessary and sufficient condition for the local structural stability of a periodic trajectory of a flow: Only one multiplier (cf. Multipliers) of the variational equation lies on the unit circle. There are also results about the local structural stability of certain hyperbolic sets (see Hyperbolic set, and [2], ).

#### References

 [1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [2] D.V. Anosov, "On a class of invariant sets in smooth dynamical systems" , Proc. Fifth Internat. Conf. Non-Linear Oscillations , 2 , Kiev (1970) pp. 39–45 (In Russian) [3a] C. Robinson, "Structural stability of vector fields" Ann. of Math. (2) , 99 (1974) pp. 154–175 [3b] C. Robinson, "Correction to "Structural stability of vector fields" " Ann. of Math. (2) , 101 (1975) pp. 368