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In this way the algebra of the evolution equations and the analysis methods of geometry are linked (Vector fields are often represented as arrows overlaid on the phase space but more technically they are defined on a related space called the tangent bundle). An orbit or trajectory is a connected path through phase space which is always tangent to the vector field. Hence an orbit traces the time-dependent solution to a dynamical system through a succession of instantaneous states. It captures the manner in which a system will change according to the evolution equation. The starting point of such an orbit is called its initial condition. Examples are given in Fig. 3.

dz1 dz2 h 2\ /on\ Ht = Z2' d" = j ^ — ^ Z2 — zi • (20)

A time series of this system for j = 2, showing the periodic nature of the oscillations, is given in Fig. 4(a). A single orbit, commencing with an initial condition in close proximity to the origin is shown in the planar phase space spanned by z1 and z2 in Fig. 4(b). It can be seen that this orbit diverges rapidly away from the origin and towards a closed loop in phase space, corresponding to the appearance of periodic oscillations in the time series. The appearance of periodic oscillations in the system motivates us to consider an alternative phase space representation, achieved by a change of coordinates to amplitude A and phase 0,

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