By Tristan Needham

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**Example text**

Denoting the evolution maps corresponding to Fµ , and F˜µ , by µ , respectively ˜ µ , whenever µ ({x} × [0, t]) ⊂ U , the homeomorphism H , or the homeomorphisms H1 and h, satisfy: ˜ h(µ) (H1 (x, µ), t) = H1 ( µ (x, t), µ). A homeomorphisms like H in the above theorem is called a local topological conjugacy between the (parametrized) flows defined by Fµ and F˜µ . Preliminaries of dynamical systems theory 33 Fig. 2. Hopf bifurcation in the positive case: dynamics for µ < 0, µ = 0, respectively µ > 0.

E. homeomorphisms. In this case we have indeed something like ‘local structural stability’ at hyperbolic stationary points. 1. This is the reason that we mainly consider C 0 conjugacies, also called topological conjugacies. As already mentioned before (see the discussion on the Hopf bifurcation), a conjugacy maps periodic evolutions to periodic evolutions with exactly the same period. In the case when the time set is R, the period can be any (positive) real number. Such a period can be changed by a perturbation which is arbitrarily small in the C k sense.

45 46 49 51 55 59 60 66 68 69 70 72 74 78 78 79 80 81 1 The first author was partially supported by NSF Grant No. DMS0616585.