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HarmonicBalance.jl

The Harmonic Balance Method

The Harmonic Balance Method is a powerful analytical technique for determining the steady-state behavior of nonlinear, periodically driven systems. HarmonicBalance.jl is a Julia package that implements this method efficiently and flexibly. By approximating the system’s response as a sum of harmonic components with slowly varying amplitudes, this method transforms the original nonlinear, non-autonomous system into an autonomous set of differential equations. This transformation is achieved through a generalized Van der Pol transformation, which remaps the time-dependent system into a rotating reference frame where the equations become time-invariant.

Solving via Homotopy Continuation

To solve these autonomous equations and identify all possible steady-state solutions reliably, we employ the method of Homotopy Continuation. This numerical approach systematically deforms a simple system, for which solutions are known, into the complex target system, tracking solutions throughout the deformation process. This ensures that all steady states are found in a guaranteed manner, providing a comprehensive understanding of the system’s behavior.

Key features

• Simple interface to scan phase diagrams across multiple parameters.
• Built-in tools for detecting and analyzing Limit Cycles.
• Ongoing integration with Quantum Algebra packages for quantum systems in the mean field limit.
• Actively maintained with frequent feature updates.

The package is open-source and open to contributions.