Abstract
The dynamical systems' point of view is becoming increasingly important in a broad area of science and technology. Taken in its broadest sense, dynamical systems involve many branches of mathematics such as differential equations, differential topology, general topology, ergodic theory, complex analysis, and others. This chapter approaches the subject on a generic level, omitting differential equations and complex analysis and minimizing differential topology and ergodic theory, to introduce the basic concepts in dynamical systems. A dynamical system with continuous time, or flow on a metric space X is a family {ϕt : t ∈ ℝ } of homeomorphisms of <u>X</u>, such that the map (t, x) ↦ϕt (x) is continuous and φt+s(x) is equal to ϕt ◦ φs(x) for all x ∈ <u>X</u> and all t, s ∈ ℝ. It follows that ϕ0 is the identity map on <u>X</u> and ϕ−t is the inverse map of ϕt. The iterations of a map of an interval into itself present an example of nonlinear dynamical systems. The dynamics on an interval has many applications to physical systems.
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