Show that a trajectory corresponds to a periodic solution if and only if it is a

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Periodic Solutions Periodic solutions of equations are solutions that describe regularly repeating processes. In such branches of science as the theory of oscillations and celestial mechanics, periodic solutions of systems of differential equations are of special interest. A periodic solution yi = Fi (t) of (1) consists of periodic functions of t that have the same period. In other words, Fi (t + t) = Fi (t) for all t and for some t > 0; t is called the period of the solution. If the system (1) is autonomous—that is, the functions fi = Fi (y1, …, yn), i = 1, …, n, do not explicitly depend on t —then to the periodic solutions there correspond closed trajectories in the phase space (y1, …, yn). A degenerate form of such trajectories are the equilibrium, or critical, points where The critical points correspond to trivial (constant) periodic solutions. The problem of finding nontrivial periodic solutions has been solved only for special types of differential equations.

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