2 LAWRENCE MARKUS dynamics was too general and too rough for many investigations oF differential Sys- tems and a more specialized differentiable dynamics was required. There were two scientific developments which served to focus attention on differ- entiable dynamics during the past few decades. The first trend was the rise of interest in terrestrial as distinct from celestial mechanics. That is , dynamical Systems with frictional damping and resistance were studied in engineering problems of nonlinear elasticity and electronics. These problems in nonlinear oscillations theory [22,40] | e c j to differential equations, as typified by the Duffing and van der Pol equations, which are often structurally stable, and hence easier to analyse and classify than the Hamil- tonian Systems of celestial mechanics. Á second trend was the creation of the mathematical theory of differentiable mani- folds [7] and differential topology [ 6 2 ] . The study of qualitative dynamics [12,21,28,43] was more and more drawn towards differential topology until current research [61] treats the qualitative theory of differential equations as a branch, differentiable dy- namics, of differential topology. In the next section we shall illustrate these concepts of global dynamical Systems by a number of examples drawn from physical and engineering sciences. Also we illus- trate the basic concepts of topological equivalence and structural stability, as they apply to dynamical Systems on manifolds. For the time being we introduce these con- cepts with only intuitive or rough definitions. The precise definitions and analyses follow in the subsequent sections. Roughly speaking, two dynamical Systems are topologically equivalent in case there exists a homeomorphism between their phase Spaces that carries each sensed trajectory of one System onto a sensed trajectory of the other System. Á dynamical System is structurally stable in case every suitably small perturbation leads to a topologically equivalent dynamical System. 2. Differentiable Dynamics, Definitions and Examples Á topological dynamical System consists of a continuous action of the real line R on a topological Space M. That is, there is a continuous function ö: RlxM--M: U, x0)-*fU, x0) = xt such that ft: M-^M: xQ-+f(t, x0) is a homeomorphism of Ì onto itself for each t C R , and we have a flow, http://dx.doi.org/10.1090/cbms/003/02
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