Couldn't get result for complicated equations

[Dan-Kechik]

Hello. I tried example from the paper "HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance" with third-order nonlinearity and single harmonic ansatz. Then I tried to complicate the system: enhanced power of nonlinearity, replaced it by other non-linear functions (exp, sin). It seems to work till the 10-th power of x, but if nonlinear component was x^10, x^12, exp(x), sin(x), harmonic equations, that were derived from differential equation, remained the same:

Is there some limts on equations complexity?

[Dan-Kechik]

Code to reproduce screenshot:
using HarmonicBalance
@variables α, ωd , ω0 , F , t , γ , x(t) # declare constant variables and a function x(t)
diff_eq = DifferentialEquation(d(d(x , t) ,t) + ω0^2 * x + γ*d(x , t ) + α * x^10 ~ F * cos(ωd * t ) , x )
add_harmonic!( diff_eq , x , ωd ) # specify the ansatz x = u ( T ) cos (ωdt ) + v ( T ) sin (ωdt ) # implement ansatz to get harmonic equations
harmonic_eq = get_harmonic_equations(diff_eq)

The same when I tried:
using HarmonicBalance
@variables α, ωd , ω0 , F , t , γ , x(t) # declare constant variables and a function x(t)
diff_eq = DifferentialEquation(d(d(x , t) ,t) + ω0^2 * x + γ*d(x , t ) + α * sin(x) ~ F * cos(ωd * t ) , x )
add_harmonic!( diff_eq , x , ωd ) # specify the ansatz x = u ( T ) cos (ωdt ) + v ( T ) sin (ωdt ) # implement ansatz to get harmonic equations
harmonic_eq = get_harmonic_equations(diff_eq)

[oameye]

Hey @Dan-Kechik, we have never tried components higher than $x^6$ or so. In general, the code will only work with algebraic equations of motion, so $exp(x)$ and $sin(x)$ will not work properly. However, in theory $x^10$ should work. The package should have errored for $exp(x)$ and $sin(x)$, so I will consider this as a bug and will investigate further.

[Dan-Kechik]

I've found that the problem is reproduced for even powers of x. When I tried polynomial of high degree (the 25th for example), that works fine. But the bug reproduced when I tried any even power, e.g. x^4.
When I tried polynomials containing components of even power, these components did not effected the result. For example:

using HarmonicBalance
@variables α, ωd , ω0 , F , t , γ , x(t) # declare constant variables and a function x(t)
diff_eq = DifferentialEquation(d(d(x , t) ,t) + ω0^2 * x + γ*d(x , t ) + α * (x^5 + x^4) ~ F * cos(ωd * t ) , x )
add_harmonic!( diff_eq , x , ωd ) # specify the ansatz x = u ( T ) cos (ωdt ) + v ( T ) sin (ωdt ) # implement ansatz to get harmonic equations
harmonic_eq = get_harmonic_equations(diff_eq)

and

using HarmonicBalance
@variables α, ωd , ω0 , F , t , γ , x(t) # declare constant variables and a function x(t)
diff_eq = DifferentialEquation(d(d(x , t) ,t) + ω0^2 * x + γ*d(x , t ) + α * (x^5 ) ~ F * cos(ωd * t ) , x )
add_harmonic!( diff_eq , x , ωd ) # specify the ansatz x = u ( T ) cos (ωdt ) + v ( T ) sin (ωdt ) # implement ansatz to get harmonic equations
harmonic_eq = get_harmonic_equations(diff_eq)

produde the same result:

[Dan-Kechik]

I supposed that components of even power do not effect on the first harmonic: squared x = ucos(wt)+vsin(wt) do not contain the components of the first power! I got the same result for the 4th power. And this is a reason why equations for the first harmonic did not changed. Then, to capture desired effects, I should track even harmonics.
I tried even powers with more harmonics in ansatz and got the result. As I understand now, one should put in ansatz so many harmonics as one wants to track, together with harmonics of less power and harmonics of drive force. For example, if I want to track the second harmonic, I should put in ansatz wd and 2wd, but should I put in the 4wd and so on, if they are known to be prominent in spectrum?