Information Transfer in the Trojan Universality Class I: From a Nonlinear Circuit to an Elastic Pendulum

This paper examines how information about energy exchange can transfer between two unrelated physical systems that belong to the same universality class near a two-to-one internal resonance. One system is a balanced electrical circuit built from a pair of identical nonlinear capacitors, while the other is the planar elastic pendulum. Although the systems share no physical variables, units, or parameters, each admits a resonant Birkhoff normal form of degree four that can be computed directly from its Hamiltonian. For the particular family of initial conditions in which all the action begins in the faster mode, the normal form influences the subsequent exchange only through two dimensionless numbers: a scaled detuning and a resonant twist formed from the resonance vector and the matrix of quartic coefficients. The resulting reduced problem is integrable in closed form. Exchange of energy from the faster mode occurs precisely when the absolute value of the scaled detuning lies below four, a threshold that is independent of the twist. Complete transfer of the conserved action occurs exactly when the detuning equals twice the twist. Explicit expressions are obtained for the largest fraction of action that can be exchanged, for the initial growth rate of the slower mode, and for the asymptotic time of the first exchange. A finite-amplitude estimate and a transfer bound follow from the same reduction. In a concrete test, one free parameter of the circuit was adjusted so that its resonant twist matched that of the pendulum. The circuit’s measured exchange curve was then frozen as a prediction before the pendulum equations were integrated. The frozen curve agrees with the subsequent pendulum results to a root-mean-square error of roughly three thousandths, far better than predictions based on raw detuning or on the scaled detuning alone. The residual discrepancy scales as the square root of the initial action, consistent with the finite-amplitude estimate. All claims are local, conservative, and restricted to this resonance and this observable; nothing is asserted about other resonances, damped or forced systems, or global phase-space structure. The work therefore isolates the precise numerical content that must be carried from one member of the class to another in order to predict energy exchange.