Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators

Attempts to produce a mathematical model of the electrical activity in certain sections of the digestive tract in animals have led to complex systems of interconnected nonlinear oscillators. The analytical solution of mutually coupled Van der Pol oscillators using the method of harmonic balance and based on the assumption of nearly sinusoidal oscillations is presented. Algebraic equations are derived from the system dynamic equations, which can be easily solved in the case of identical oscillators to reveal the effect of coupling on the overall system frequency, amplitudes, and phases. A simple hillclimbing method is used for the solution of the algebraic equations for the nonidentical oscillator case. Using this technique, a jump transition is found in the unstable limit cycle of the overall system that can be predicted from a manipulation of the algebraic equations.