Describing Function

Lur'e split matrices, fractional transfer function derivation, harmonic balance equation, and the Weyl-Caputo memory bridge.

The describing-function (DF) method is used to construct high-precision periodic starting seeds. This is done by separating the piecewise-linear system into a linear feedback block and a static scalar nonlinearity (Lur’e Form).

1. Lur’e Split Matrices

The vector field is decomposed into a linear state operator PP and a scalar nonlinear feedback term ψ(x)\psi(x):

CDtqX=PX+b ψ(rTX)^{C}D_t^q X = P X + b \, \psi(r^T X)

where the scalar feedback σ=rTX=x\sigma = r^T X = x operates on:

ψ(x)=m0−m12(∣x+1∣−∣x−1∣),f(x)=m1x+ψ(x)\psi(x) = \frac{m_0 - m_1}{2} (|x+1| - |x-1|), \qquad f(x) = m_1 x + \psi(x)

The corresponding constant matrices are defined as:

P=[−α(1+m1)α01−110−β−γ],b=[−α00],r=[100]P = \begin{bmatrix} -\alpha(1+m_1) & \alpha & 0 \\ 1 & -1 & 1 \\ 0 & -\beta & -\gamma \end{bmatrix}, \qquad b = \begin{bmatrix} -\alpha \\ 0 \\ 0 \end{bmatrix}, \qquad r = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}

2. Fractional Transfer Function Derivation

The complex transfer function of the linear fractional-order block is:

Wq(s)=rT(sqI−P)−1bW_q(s) = r^T (s^q I - P)^{-1} b

To isolate the fractional exponent, we define λ=sq\lambda = s^q and evaluate the rational function:

W^q(λ)=rT(λI−P)−1b\widehat W_q(\lambda) = r^T (\lambda I - P)^{-1} b

Where the complex frequency variable under order qq is:

λ=(iω)q=ωq(cos⁡qπ2+isin⁡qπ2)\lambda = (i\omega)^q = \omega^q \left( \cos\frac{q\pi}{2} + i\sin\frac{q\pi}{2} \right)

The matrix (λI−P)(\lambda I - P) is:

λI−P=[λ+α(1+m1)−α0−1λ+1−10βλ+γ]\lambda I - P = \begin{bmatrix} \lambda + \alpha(1+m_1) & -\alpha & 0 \\ -1 & \lambda + 1 & -1 \\ 0 & \beta & \lambda + \gamma \end{bmatrix}

Let the minor determinant of the lower 2x2 block be N(λ)N(\lambda):

N(λ)=(λ+1)(λ+γ)+β=λ2+(1+γ)λ+(β+γ)N(\lambda) = (\lambda + 1)(\lambda + \gamma) + \beta = \lambda^2 + (1+\gamma)\lambda + (\beta + \gamma)

The complete matrix determinant is:

Δ(λ)=det⁡(λI−P)=(λ+α(1+m1))N(λ)−α(λ+γ)\Delta(\lambda) = \det(\lambda I - P) = (\lambda + \alpha(1 + m_1)) N(\lambda) - \alpha(\lambda + \gamma)

By cofactor expansion, the (1,1)(1,1) entry of the inverse matrix yields:

[(λI−P)−1]11=N(λ)Δ(λ)[(\lambda I - P)^{-1}]_{11} = \frac{N(\lambda)}{\Delta(\lambda)}

Thus, the exact rational expression for the fractional transfer function is:

W^q(λ)=−αN(λ)Δ(λ)\widehat W_q(\lambda) = -\alpha \frac{N(\lambda)}{\Delta(\lambda)}

3. Harmonic Balance & the Weyl-Caputo Bridge

The describing-function periodic limit cycle condition is:

1−NDF(A)W^q((iω)q)=01 - N_{\rm DF}(A) \widehat W_q\left((i\omega)^q\right) = 0

Where NDF(A)N_{\rm DF}(A) represents the amplitude-dependent describing function gain.

The Weyl-Caputo Bridge

Describing functions assume infinitely periodic stationary solutions. In fractional calculus, this corresponds to the Weyl/Liouville-Weyl fractional derivative over infinite history (−∞,t](-\infty, t].

However, physical integrations are conducted using the Caputo fractional derivative, which is causal and starts at t0t_0.

The discrepancy between the two definitions appears as an initial history memory term:

  • Describing functions generate highly accurate approximate starting seeds.
  • The Caputo causal solver initiates transient trajectories from these seeds to settle on the true underlying attractors.
  • Ocultedad (hiddenness) is ultimately evaluated through spherical sweeps and basins around equilibria, not by the harmonic balance algebraic equations alone.