Smooth diffeomorphism

Within this context, ordinary differential equations are viewed as vector fields on manifolds or configuration spaces [2]. For example, Newton s equations are second-order differential equations describing smooth curves on Riemannian manifolds. Noether s theorem [4] states that a diffeomorphism,3 < ), of a Riemannian manifold, C, indices a diffeomorphism, D< >, of its tangent4 bundle,5 TC. If 4> is a symmetry of Newton s equations, then Dt(> preserves the Lagrangian o /Jc ) = jSf. As opposed to equations of motion in conventional... 

If the sets U and V are open sets both defined over the space that is, U ( R" is open and U C R" is open, where open means nonoverlapping, then the mapping t i U > V is an infinitely differentiable map with an infinitely differential inverse, and objects defined in U will have equivalent counterparts in V. The mapping i / is a diffeomorphism and it is a smooth and infinitely differentiable function. The important point is conservation rules apply to diffeomorphisms, because of their infinite differentiability. Therefore diffeomorphisms constitute fundamental characterizations of differential equations. 

If is a left-invariant vector field on 0, then generates a certain globally defined group of diffeomorphisms. A smooth homomorphism ( 0 is called... 

Theorem 2.2.3 ( Brailov, Fomenko). Let be a smooth compact closed orientable manifold obtained by gluing an arbitrary number of elementary manifolds of types I, Ilf HI (that is, full tori, cylinders, and trousers) through any diffeomor-phisms of their boundary tori T. Then there always exists a smooth compact symplectic manifold with a boundary diffeomorphic to a disconnected union... 

Let C7 (ai,e) be such a cylinder that its right end-face lies within Oi and the point Di within C (a, e) (Fig. 42(3)). Let us join the cylinder C (62,C2) by a tube T with the image of the cylinder C,(ai, c) via the diffeomorphism P in such a way that the tube T be a smooth continuation of both the cylinders and have open non-empty intersections with them. Let... 

Bifurcations of this type in diffeomorphisms have been the object of nuioericaJ. (8) amd theoretical (9-10) work. We learn from these studies that in addition to a smooth torus, a saddle-node bifurcation may yield other equally robust situations. In particular two of them appeax to be relevant to our work Either (i) the limit cycle (in the Poincare map) is destroyed when the saddle-node bifurcation occurs, and the disappearance of the saddle-node corresponds to the transition from a regulau , periodic, state to a chaotic one, or (ii) the limit cycle is destroyed before the disappearance of the saddle-node and there exints chaos concurrently with the initial periodic orbit (see fig 5c)... 

For the former type of boundary system, we consider an example of a homoclinic tangency. Let a C -smooth family of diffeomorphisms T(/i) have, at... 

Another example is a family of two-dimensional C -smooth diffeomor-phisms whose non-wandering set does not change until the boundary of Morse-Smale diffeomorphisms is reached. The situation is illustrated in Fig. 8.2.3. The two fixed points 0 and O2 have positive multipliers, and Wq contacts Wq along a heteroclinic trajectory, and so do Wq and. This example... 

Ilyashenko, Yu. S. and Yakovenko, S. Yu. [1991] Finite-smooth normal forms of local families of diffeomorphisms and vector fields, Uspechi Mat Nauk, Vol. 46, 1(277), 1-39. 

---