Poincare-Hopf theorem

O Equation (60.10) states that both J and B must be perpendicular to VP. This means that J and B must lie on a surface with no components perpendicular to it. Then, the question arises What can the shape of the surface to which the J and B are bound be. The answer is a consequence of the Poincare-Hopf theorem, namely, that the surface must have the form of a topological torus, where the field lines lie on nested surfaces, and there are no field lines connecting one surface with the other. 

We begin the analysis of (4), (5) by constructing a trapping region and applying the Poincare-Bendixson theorem. Then we ll show that the chemical oscillations arise from a supercritical Hopf bifurcation. 

This was used to derive Eq. (15). A special case of equation (56) was previously used to classify the ways to build a box The Poincare index theorem was extended by Hopf to vector fields on arbitrary manifolds. For vector fields with m isolated hyperbolic critical points, the Poincare-Hopf index theorem is " ... 

Method 4. Index theory approach.. This method is based on the Poincare-Hopf index theorem found in differential topology, see, e.g., Gillemin and Pollack (1974). Similarly to the univalence mapping approach, it requires a certain sign from the Hessian, but this requirement need hold only at the equilibrium point. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

In Section 1, the local analysis of the stability properties of nonlinear equations was discussed. In the 1880s Poincare discovered a theorem that makes a statement about the entire set of critical points of dynamical systems on two-dimensional manifolds. This result was later extended by Hopf to arbitrary manifolds (see Appendix for greater detail and references). Here the theorem is applied to dynamics in chemical systems. 

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