Lectures on Dynamical Systems, Structural Stability, and by Kotik K Lee

By Kotik K Lee

This ebook relies on a graduate direction for scientists and engineers.

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Extra info for Lectures on Dynamical Systems, Structural Stability, and their Applications

Example text

Each encounter decreases the prey (goats) population and increases the predator (wolves) population. The effects of these encounters are accounted for by the second terms in the differential equations. Of course, these equations are highly simplified and do not take into account a number of external factors such as general environment conditions, supply of other food for both predator and prey, migration of the populations, disease, and crowding. An important application of a model of this type is the study and control of pests and feed on agricultural crops.

Remark: There are two warnings to be made concerning the limits of an arbitrary topological space. First, the cauchy criterion for the convergence of a sequence of real numbers has no analogue in a topological space in general. This is because in general topological space there is no uniform standard of nearness which can be applied to a variable pair of points. , the metric). Second, there is no guarantee that the limit of a sequence of points, if it exists, is unique. It is desirable to restrict our attention to topological spaces satisfying a condition wich will enable the uniqueness of the limit of a sequence of points to be proven.

So the Heine-Borel theorem says that a set A in R" is compact iff it is closed and bounded. We have just said that because compactness (or a set is closed and bounded) is defined entirely in terms of open sets, thus we refer it to be a topological property. But we want to check whether if a space is homeomorphic to a given compact space is also compact. 9 Let f be a continuous mapping of a topological space X onto a topological space Y. Then if X is compact, and Y is Hausdorff, Y is compact. Proof: Let F be a given open covering of Y.

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