A geometrical treatment of singular trajectories. (English) Zbl 0464.58014
MSC:
37G99 | Local and nonlocal bifurcation theory for dynamical systems |
37N99 | Applications of dynamical systems |
53B20 | Local Riemannian geometry |
53B21 | Methods of local Riemannian geometry |
53B50 | Applications of local differential geometry to the sciences |
Keywords:
families of singular trajectories; limit tangent vectors defined at a common singularity; generalized tangent space; exponential map; phase space; geodesics; wedge-shaped set of geodesics; Christoffel symbols; structurally stable dynamical system; conjugate points along a maximal geodesic of finite lengths; analytic varieties; generalized Gauss-Bonnet formula on analytic and algebraic varieties; particles represented as Riemannian singularities which are structurally stable; geometrical field theoriesReferences:
[1] | Geroch, R., Local characterization of singularities in general relativity, J. Math. Phys., 9, 450-465 (1968) · Zbl 0172.27905 |
[2] | Carpenter, G., A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differential Equations, 23, 32-56 (1977) |
[3] | Bell, J.; Cook, L. P., On the solutions of a nerve equation, SIAM J. Appl. Math., 35, 4, 678-688 (1978) · Zbl 0449.35084 |
[4] | Anosov, D., Geodesic flows on a Riemannian manifold with negative curvature, (Proc. Steklov Inst. Math. (1962)), 61-78 |
[5] | Thom, R., Structural Stability and Morphogenesis (1975), Benjamin: Benjamin Reading, Mass · Zbl 0303.92002 |
[6] | Łojasiewicz, S., Sur le problème de la division. Studia Math., 8, 87-136 (1959) · Zbl 0115.10203 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.