A classical limit-cycle system that mimics the quantum-mechanical harmonic oscillator

Highlights: • Classical nonlinear systems with infinite limit-cycle sequences are constructed. • "Limit-cycle energy" defined as average of linear oscillator energy over a cycle. • Sequence of "limit-cycle energies" mimics spectrum of a quantum-mechanical system. • Higher limit-cycle energies tend to energy levels of quantized systems. • Perturbation methods apply also to high-lying limit cycles. Classical harmonic oscillators affected by appropriately chosen nonlinear dissipative perturbations can exhibit infinite sequences of limit cycles, which mimic quantized systems. For properly chosen perturbations, the large-amplitude limit cycles approach circles. The higher the amplitude of the limit cycle is, the smaller are the dissipative deviations from energy conservation. The weaker the perturbation is, the earlier on does the asymptotic behavior show up already in low-lying limit cycles. Simple modifications of the Rayleigh and van der Pol oscillators yield infinite sequences of limit cycles such that the energy spectrum of the higher-amplitude limit cycles tends to that of the quantum-mechanical particle in a box. For another judiciously chosen dissipative perturbation, the energy spectrum of the higher-amplitude limit cycles tends to that of the quantum-mechanical harmonic oscillator. In all cases, one first finds the limit-cycle solutions for dissipation strength, $\epsilon \ne 0$. The "energy of each limit cycle" then oscillates around an average value. In the limit $\epsilon \to 0$ these oscillations vanish, and the limit cycles in the infinite sequence attain constant values for their energies, a characteristic that is required for such classical systems to mimic Hamiltonian quantum-mechanical systems.