Extracting harmonic equations

Harmonic Balance method

Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.

The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.

# HarmonicBalance.get_harmonic_equations — Function.

get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)

Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.

The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.

Example

julia> @variables t, x(t), ω0, ω, F;

# enter the simple harmonic oscillator
julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);

# expand x in the harmonic ω
julia> add_harmonic!(diff_eom, x, ω);

# get equations for the harmonics evolving in the slow time T
julia> harmonic_eom = get_harmonic_equations(diff_eom)

A set of 2 harmonic equations
Variables: u1(T), v1(T)
Parameters: ω0, ω, F

Harmonic ansatz:
x(t) = u1*cos(ωt) + v1*sin(ωt)

Harmonic equations:

(ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F

(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

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# HarmonicBalance.harmonic_ansatz — Function.

harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

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# HarmonicBalance.slow_flow — Function.

slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

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# HarmonicBalance.fourier_transform — Function.

fourier_transform(
    eom::HarmonicEquation,
    time::Num
) -> HarmonicEquation

Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

Underlying assumption: all time-dependences are harmonic.

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# HarmonicBalance.drop_powers — Function.

drop_powers(expr, vars, deg)

Remove parts of expr where the combined power of vars is => deg.

Example

julia> @variables x,y;
julia>drop_powers((x+y)^2, x, 2)
y^2 + 2*x*y
julia>drop_powers((x+y)^2, [x,y], 2)
0
julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
x^2 + y^2 + 2*x*y

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HarmonicVariable and HarmonicEquation types

The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz $x_i(t)=\sum _{j=1}^M{u}_{i,j}(T)\cos ({\omega }_{i,j}t)+v_{i,j}(T)\sin ({\omega }_{i,j}t)$ is used. Internally, each pair $(u_{i,j},v_{i,j})$ is stored as a HarmonicVariable. This includes the identification of ${\omega }_{i,j}$ and $x_i(t)$, which is needed to later reconstruct $x_i(t)$.

# HarmonicBalance.HarmonicVariable — Type.

mutable struct HarmonicVariable

Holds a variable stored under symbol describing the harmonic ω of natural_variable.

Fields

• symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

• name::String: Human-readable labels of the variable, used for plotting.

• type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

• ω::Num: The harmonic being described.

• natural_variable::Num: The natural variable whose harmonic is being described.

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When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a HarmonicEquation. For an initial equation of motion consisting of $M$ variables, each expanded in $N$ harmonics, the resulting HarmonicEquation holds $2NM$ equations of $2NM$ variables. Each symbol not corresponding to a variable is identified as a parameter.

A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.

# HarmonicBalance.HarmonicEquation — Type.

mutable struct HarmonicEquation

Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

Fields

• equations::Vector{Equation}: A set of equations governing the harmonics.

• variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

• parameters::Vector{Num}: The parameters of the equation set.

• natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

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