By D. V. Anosov, Samuel Kh Aranson, V. I. Arnold, I. U. Bronshtejn, V. Z. Grines

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**Extra resources for Dynamical Systems I **

**Example text**

8. Singular points of an implicit c) a folded node, d) a folded focus. 4. Normal Forms of Folded Singular Points. A diffeomorphism whose square (under composition) is the identity transformation is called an inuolution. Fix a field on the plane with a singular point at the origin. An involution of the plane is said to be admissible for this vector field if the fixed points of the involution form a curve going through the origin, if the involution carries the field on this curve into the opposite field, and if neither the invariant nor anti-invariant eigenvector of the linearization of the involution at the origin are eigenvectors of the linear part of the field at the origin.

Similar theorems hold for differential equations on a torus. They follow from the preceding theorems and the following remark. Remark. , c’, analytically) equivalent to the standard and only if the corresponding monodromy mapping is topologiC’, analytically) equivalent to a rotation. 2. Diffeomorphisms of a Circle and Vector Fields on S3. Denjoy constructed an example of a Cl-diffeomorphism of the circle having an irrational rotation number which is not equivalent to a rotation of the circle (see [109]).

At such a point, the direction field on the 36 I. Ordinary Differential Equations surface of the equation is not singular, but is tangential to the criminant. The criminant is not singular, but is tangential to the kernel of the projection, and the discriminant curve has a cusp. There are infinitely many topologically different singularities of this type, but all are essentially one or other of only two kinds: a) and b) in Fig. 9. a c b Fig. 9. Jleat point, as sketched in Fig. 9, can be described as-follows (J.