Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling

Abstract

SUMMARYKaplan and Yardeni have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non-linear coupling in first order. In contrast with such solutions in other non-linear systems, each of these limit cycles is a normal mode of the unperturbed equations, with no change in frequency. The sources of this simple behaviour are studied here with the equations expressed in terms of the normal mode co-ordinates of the unperturbed system rather than in the original co-ordinates. It is found that the simplicity of the Kaplan-Yardeni solutions arises partly from an additional symmetry of the perturbation terms beyond the cyclic symmetry and partly from the specific choices of the perturbations. Extension of their systems to arbitrary N leads to the result that all such sets of equations have similar simple limit cycles. More general cyclically symmetric sets of equations are also discussed, with limit cycle solutions whose frequencies are shifted from the zero-order values by easily calculated amounts or with solutions which are linear combinations of zero-order normal modes.The results are exact, obtained without the use of small-perturbation theory.

The Analytic Determination of the PPV for Second-Order Oscillators Via Time-Varying Eigenvalues

“…It is difficult to find a closed-form solution for (9) for a given initial condition, whereas for (10), its complete solution at is given by (12) where the natural solution of (10) is given by (13) and the second part of expression (12) is nothing more than the particular solution obtained for (10) for . After applying transformations (7) and (8) to (6), the following system of two decoupled first-order differential equations is obtained: (14) where quantities and are the time-varying eigenvalues associated to system (6).…”

Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling. Part II: Coupling with phase shift

“…(8) Equation (7) shows that in order for there to be a solution of the desired type, d, must be positive. If it is required that R, = w , , then, as seen from (8), d must be real; however, c and g can be complex since they do not appear in (7). (Thus D , symmetry of the perturbation terms, which makes all perturbation coefficients real, is sufficient but not necessary for R, = w,.…”