Sustained oscillations of the Lotka-Volterra type

The Lotka-Volterra type of equations provides a model for sustained oscillations in chemical systems with an overall affinity approaching infinity. Perturbations at finite distances from the steady state are also periodic in time. Within the phase space (Xvs. Y), the system produces an infinite number of continuous closed orbits surrounding the steady state 

After dividing one of these equations by the other, we have the following equation for the trajectories in the X, Y space 

Only the orbits infinitesimally close to the steady state may be considered stable, according to Liyapunov s theory of stability. However, at a finite distance from the steady state, two neighboring points belonging to two distinct cycles tend to be far apart from each other because of differences in the period. Such motions are called stable in the extended sense of orbital stability. The average concentrations of X and Y over an arbitrary cycle are equal to their steady-state values (Xs = 1 and Y.. A 1). Under these conditions, the average entropy production over one period remains equal to the steady-state entropy production. 

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Example 13.10 Sustained oscillations of the Lotka—Volterra type... 

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