The method of harmonic balance
HarmonicBalance.jl focuses on systems with equations of motion that are subject to time-dependent terms of harmonic type.
Harmonic generation in oscillating nonlinear systems
Let us take a general nonlinear system of
The vector
As an example, let us first solve the harmonic oscillator in frequency space. The equation of motion is
where
Evidently,
The situation becomes more complex if nonlinear terms are present, as these cause frequency conversion. Suppose we add a quadratic nonlinearity
which couples all harmonics
Harmonic ansatz & harmonic equations
Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. The strategy of harmonic balance is to describe the motion of any variable
Within this space, the system is described by a finite-dimensional vector
Under the assumption that
which we call the harmonic equations. The main purpose of HarmonicBalance.jl is to obtain and solve these Harmonic equations. We are primarily interested in steady states
The process of obtaining the harmonic equations is best shown using the example below:
Example: the Duffing oscillator
Here, we derive the harmonic equations for a single Duffing resonator, governed by the equation
As explained above, for a periodic driving at frequency
Single-frequency ansatz
We first attempt to describe the steady states of the Duffing equations of motion using only a single harmonic,
with the harmonic variables
We see that the
Steady states can now be found by setting the l.h.s. to zero, i.e., assuming
Sidenote: perturbative approach
The steady states describe a response that may be recast as
describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of
Two-frequency ansatz
An approach in the spirit of harmonic balance is to use both harmonics
with
In contrast to the single-frequency ansatz, we now have 4 equations of order 3, allowing up to