1. History of Differentiable Dynamics In 1762 Lagrange [ 27 ] developed a general theory of small vibrations of a dynami- cal System near an equilibrium State, and in 1883 Floquet [17J treated the vibrations about a periodic orbit. While these two classical theories were strictly valid only for linear dynamical Systems, they were completely general within this mathematical frame- work and they furnished the foundations for all further qualitative theory of ordinary differential equations. Late in the nineteenth Century Poincare [48J and Liapunov L30] proved that these earlier methods were also applicable to small vibrations of nonlinear Systems. More- over they related their analyses to general theories of stability. One basic concept of stability concerned the stability of an invariant set, usually an equilibrium State or a periodic orbit, upon small perturbations of the initial data of nearby trajectories. An- other type of stability referred to the permanence of periodic orbits upon small pertur- bations of the parameters or coefficients of the dynamical System. The first concept of stability led to the modern theory of Liapunov stability L28J, and the second to structural stability [28J. The treatise of Poincare on celestial mechanics [48J also dealt with general ideas of recurrence and transitivity, especially for incompressible, that is conservative, dy- namical Systems. However these new methods in dynamics involved a confusing mix- ture of topology, analysis, and geometry. There was a serious need for an abstract or axiomatic presentation that could clarify and systematize the foundations of the quali- tative theory of dynamical Systems. In 1927 Birkhoff [8] published a polished form of his abstract theory of dynamical Systems treated as continuous flows in metric spaces. He defined limit sets, minimal sets and other invariant sets, stability, recurrence, and transitivity in purely topologi- cal terms. This theory of topological dynamics was later set forth in extensive form by Hedlund and Gottschalk [ 1 8 ] . While the theory of topological dynamics provided a very valuable framework for the understanding of qualitative dynamics, it missed many of the features that relate to the differentiable structures of dynamics. For instance, the relations between the eigenvalues of the variational equation about an equilibrium point, and the Liapunov stability of that point are foreign to topological dynamics. In other words, topological Research partially supported by NONR 3776(00). 1 http://dx.doi.org/10.1090/cbms/003/01

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