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Extra info for Answers to a Selection Problems from Classical Electrodynamics John David Jackson
16, this has a signiﬁcant theoretical structure because it explicitly contains two elements necessary to maintain Chaotic Oscillators 19 sustained oscillations: besides the natural oscillation of the system generated by the potential function V (x), it includes explicitly a dissipation term, f (x, dx/dt), and a forcing term, A1 sin (ω1 t), whose competition is needed for that aim. For a generic ﬂow these elements may not appear so explicitly, but they will be implied in some form. An example of three-dimensional ﬂow which allows chaotic dynamics is the Lorenz model [Lorenz (1963)].
5(f)] shows the emergence of new details as it happened with H´enon map studied above. This example therefore, shows how the structure of the Poincar´e map can be useful to distinguish chaotic from non-chaotic motions. A non-trivial case of non-chaotic motion has been considered; if a limit cycle attractor was used, the distinction would have been easier. This is because a periodic orbit, which repeats itself periodically, cuts the surface of section in a well deﬁned ﬁnite number of points; therefore, the plot of the map in the Poincar´e section is a discrete set made usually of few points.
The ﬂow F (x) given by Eqs. e. three points which verify F (x) = 0. One of them is the origin x0 = (0, 0, 0), and the other two are located symmetrically around the origin at x+ = [(b − a) / (b + 1) , 0, (a − b) / (b + 1)], and at x− = −x+ . The dynamics induced by the ﬂow is such that the motion around x0 is unstable, while stable rotations around x+ and x− are allowed. For small values of α these rotations, chaotic or not, occur in relatively low amplitude orbits around one or another of these two points.