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As a motivation for this chapter we refer to the historical remarks of §2.2.5 and the references given there. The main question is to what extent a quasi-periodic subsystem, as defined in §2.2.3, is persistent under a small perturbation of the whole system, or say, under variation of parameters; see §3.2.
Till now we have dealt mainly with the properties of individual evolutions of dynamical systems. In this chapter we consider aspects that in principle concern al evolutions of a given dynamical system. We already saw some of this when dealing with structural stability (see 3.4.2) which means that under C-small perturbations the ‘topology’ of the collection of all evolutions does not change.
A dynamical system can be any mechanism that evolves deterministically in time. Simple examples can be found in mechanics, one may think of the pendulum or the solar system. Similar systems occur in chemistry and meteorology.We should note that in biological and economic systems it is less clear when we are dealing with determinism1. We are interested in evolutions, that is, functions that describe...
Until now the emphasis has been on the analysis of the evolutions and their organisation in state space, in particular their geometric (topological) and measure-theoretic structure, given the system. In this chapter, the question is which properties of thesystem one can reconstruct, given a time series of an evolution, just assuming that ithas a deterministic evolution law.We note that this is exactly...
In the previous chapter we already met various types of evolutions, including stationary and periodic ones. In the present chapter we investigate these evolutions more in detail and we also consider more general types. Here qualitative properties are our main interest. The precise meaning of ‘qualitative’ in general is hard to give; to some extent it is the contrary or opposite of ‘quantitative’....
When interpreting observed data of a dynamical system, it is of the utmost importance that both the initial state and the evolution law are only known to an acceptable approximation. For this reason it is also good to know how the type of the evolution changes under variation of both.When the type does not change under small variations of the initial state, we speak of persistence under such variation...
Over the last four decades there has been extensive development in the theory of dynamical systems. This book starts from the phenomenological point of view reviewing examples. Hence the authors discuss oscillators, like the pendulum in many variation including damping and periodic forcing , the Van der Pol System, the Henon and Logistic families, the Newton algorithm seen as a dynamical system and...
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