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By D M Brink; G R Satchler

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The motivation behind this prescription is the expectation that the solution of the above equation, for small ε (but ε = 0), is very nearly a simple harmonic of the form A cos(t − θ ), which it would identically be, if ε ≡ 0. The perturbations induced by the terms with ε = 0 on the right hand side of the above equation are then expected to be reflected in, • slow changes in the amplitude A and phase θ of the near harmonic, • higher harmonics through the xk ’s. 152) leads to (2εAθt1 ) cos(t − θ ) + (−2εAt1 ) sin(t − θ ) + ε(x1tt + x1 ) = εA3 cos3 (t − θ ) + εF0 cos λt + · · · .

129) k 2 9k 4 2π 1+ + + ··· . 131) α 2 11α 4 2π 1+ + + ··· . 132) describes the amplitude-dependent corrections to the linear period. 3 Fully Nonlinear Evolution It is quite possible that the predictions made by the linearized stability analysis in the foregoing will not be borne out by the full nonlinear equations. Elliptic points, as predicted by a linearized stability analysis, are particularly susceptible to a breakdown under a nonlinear perturbation. 133) y = x + y x2 + y2 . 133) predicts that x = y = 0 is an elliptic point.

By contrast, in a three-dimensional phase space, the solution curves can avoid self-intersection by going off in the third dimension, thereby rendering the topological structure of the trajectories quite complex in this case. 4 Non-autonomous Systems A common example of non-autonomous systems is one in which a system is subjected to an external time-dependent force F (t). 149) where ε can be considered to be a “coupling parameter”. In the limit ε → 0, the system becomes autonomous. In this limit, the phase space is two-dimensional, but for ε = 0, the phase space becomes three-dimensional.

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