Whereas a linear system has only one type of behaviour everywhere in the phase plane, a nonlinear system may exhibit qualitatively different types of behaviour in different regions. Let Σ be a nonlinear system having k qualitatively different types of behaviour in the phase plane. Suppose the system Σ to be linearised in each of the k regions, to yield linear approximations Σι, ..., Σk. It is a reasonable proposition that a knowledge of the behaviour of the system Σ in each of the k regions, can be obtained from a knowledge of the behaviour of each of the k linear approximations. The extent to which the proposition is valid was determined in a doctoral thesis in 1892 by A. M. Lyapunov. (Available in translation as Lyapunov (1966)). The approach outlined above and described below is sometimes called Lyapunov's first method.
Determination of the qualitative behaviour of a nonlinear second-order system by linearisation (Lyapunov's first method), Page 1 of 2
< Previous page Next page > /docserver/preview/fulltext/books/ce/pbsp006e/PBSP006E_ch6-1.gif /docserver/preview/fulltext/books/ce/pbsp006e/PBSP006E_ch6-2.gif