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 |
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