Mathematical Background

Model equations, analytical equilibria derivations, regional Jacobians, and fractional Matignon stability criteria.

To formally study the chaotic behavior and hiddenness of the fractional piecewise-linear Chua system, the package separates the algebraic vector field from the numerical Caputo contract.

1. Algebraic Model vs. Numerical Contract

The library distinguishes two separate layers of calculation:

Model Layer

Defines solely the algebraic vector field $F(X)$ for the state vector $X = (x,y,z)^T$ :

F(X) = \begin{bmatrix} \alpha(y - x - f(x)) \\ x - y + z \\ -\beta y - \gamma z \end{bmatrix}

Where the piecewise-linear (PWL) characteristic is defined as:

f(x) = m_1 x + \frac{m_0 - m_1}{2} (|x+1| - |x-1|)

Simulation Layer (Caputo Contract)

Defines the fractional-order integration under the Caputo derivative operator $^{C}D_t^q$ of order $q$ :

^{C}D_t^q X(t) = F(X(t))

with step size $h$ , history length $Lm$ , transient cutoff $t_{\rm burn}$ , and classification tolerances.

2. Analytical Equilibria Derivation

Equilibria points satisfy the algebraic system $F(X) = 0$ :

x - y + z = 0, \qquad -\beta y - \gamma z = 0

From the third equation:

z = -\frac{\beta}{\gamma} y

Substituting this into the second equation:

x - y - \frac{\beta}{\gamma} y = 0 \quad\implies\quad x = \frac{\beta + \gamma}{\gamma} y

Thus, we obtain the state coordinate relations:

y = \frac{\gamma}{\beta + \gamma} x, \qquad z = -\frac{\beta}{\beta + \gamma} x

Substituting these relations into the first equation $\alpha(y - x - f(x)) = 0$ yields:

f(x) = y - x = \left( \frac{\gamma}{\beta + \gamma} - 1 \right) x = -\frac{\beta}{\beta + \gamma} x

Defining the closing slope $s_c$ as:

s_c = -\frac{\beta}{\beta + \gamma}

The coordinate $x$ of all equilibria are the exact roots of the scalar equation:

f(x) = s_c x

Central Equilibrium

In the region $|x| \le 1$ , the function is linear $f(x) = m_0 x$ . Thus:

m_0 x = s_c x \quad\implies\quad (m_0 - s_c)x = 0

Since $m_0 \neq s_c$ , the central equilibrium is the origin:

E_0 = (0, 0, 0)^T

External Equilibria

For $|x| > 1$ , the function satisfies $f(x) = m_1 x \pm (m_0 - m_1)$ . This yields the outer coordinate roots:

x_\pm = \pm \frac{m_0 - m_1}{s_c - m_1}

Representing the two external equilibria $E_+ = (x_+, y_+, z_+)^T$ and $E_- = (x_-, y_-, z_-)^T$ :

E_\pm = \begin{bmatrix} x_\pm \\ x_\pm + f(x_\pm) \\ f(x_\pm) \end{bmatrix}

3. Regional Jacobians & Matignon Stability

Away from the switching planes $|x| = 1$ , the regional Jacobian matrix for slope $a \in \{m_0, m_1\}$ is:

J(a) = \begin{bmatrix} -\alpha(1+a) & \alpha & 0 \\ 1 & -1 & 1 \\ 0 & -\beta & -\gamma \end{bmatrix}

Under a commensurate Caputo fractional system $^{C}D_t^q \xi = J(a) \xi$ , the local asymptotic stability of each region is governed by the Matignon Criterion.

An equilibrium point is asymptotically stable if all eigenvalues $\lambda_i$ of $J(a)$ satisfy:

|\arg(\lambda_i)| > \frac{q \pi}{2}

If any eigenvalue enters the unstable sector $|\arg(\lambda)| \le q \pi / 2$ , the equilibrium is unstable.

4. Extended Mathematical Proofs & Audits

For a complete and highly formal mathematical derivation of all coordinate relations, regional Jacobians, transfer minor determinants $N(\lambda)$ , and details on the six basin classification labels (CLASS_LABELS), please see the Repository LaTeX Notes page, which catalogs the LaTeX publications and didactic guides available in the docs/ folder of the repository.