One-Dimensional Control Systems
The structure of one-dimensional control systems is particularly easy to analyze. One obtains a complete picture of the system behavior by studying the dynamics for constant control values, using some calculus. But the class of one-dimensional systems is sufficiently rich to furnish interesting examples for much of the theory developed in the previous chapters, including some not so obvious counterexamples. The concepts of global theory and linearization theory can be characterized explicitly for one-dimensional systems, which allows us to compute control sets, chain control sets, domains of attraction, spectral intervals, and invariant manifolds directly in terms of the system dynamics. With these characterizations we can go one step further and analyze the (local and global) bifurcation structure completely-something that is only partially possible in higher dimensions (see Sections 9.2 and 9.4).
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