Chua Non-Smooth

The non-smooth fractional Chua model with a linear-by-pieces characteristic: equations, parameters, equilibria, Matignon criterion, algebra validation, harmonic branches and EFORK status.

The Chua Non-Smooth model is the Danca benchmark used by this library. Its diode characteristic is linear by pieces; this is the mathematical form of the non-smooth model, not a separate piecewise system.

The order-one baseline is recorded separately in Chua Integer q=1 Reference.

Equations

The non-smooth characteristic can be written equivalently with an explicit absolute-value form (standard in textbooks) or with a saturation function:

\begin{aligned} {}^C D_t^q x &= \alpha (y-x-f(x)),\\ {}^C D_t^q y &= x-y+z,\\ {}^C D_t^q z &= -\beta y-\gamma z,\\ f(x) &= m_1 x+\tfrac{1}{2}(m_0-m_1)(|x+1|-|x-1|). \end{aligned}

The saturation form used in the algebraic derivation is identical:

f(x) = m_1x+(m_0-m_1)\operatorname{sat}(x).

Parameters

Parameter	Symbol	Value	Description
`alpha`	$\alpha$	8.4562	coupling coefficient
`beta`	$\beta$	12.0732	third-equation coefficient
`gamma`	$\gamma$	0.0052	third-equation damping
`m0`	$m_0$	-0.1768	inner regional slope
`m1`	$m_1$	-1.1468	outer regional slope
`q`	$q$	0.9998	fractional order (Caputo)

from hidden_attractors import chua_nonsmooth_parameters

params = chua_nonsmooth_parameters()

Equilibria

Equilibrium	$x$	$y$	$z$
`E0`	0	0	0
`E+`	6.588307886539	0.002836402256	-6.585471484283
`E-`	-6.588307886539	-0.002836402256	6.585471484283

Python, MATLAB and Wolfram reproduce these values. The maximum Python equilibrium residual is 1.10e-15.

from hidden_attractors.models import equilibria_nonsmooth, jacobian_nonsmooth, rhs_nonsmooth

eq = equilibria_nonsmooth(params)
dx = rhs_nonsmooth(eq["E+"], params)
J = jacobian_nonsmooth(eq["E+"], params)

Matignon Criterion

For $q=0.9998$ , the threshold is $q\pi/2=1.570482167530$ radians.

Region	Eigenvalues	Classification
Inner, `E0`	$-7.9587261113$ , $-0.0038088643 \pm 3.2494460858i$	stable
Outer, `E+`, `E-`	$2.2193492642$ , $-0.9915895521 \pm 2.4067596392i$	unstable

$Matignon complex-plane criterion for the non-smooth fractional Chua equilibria, generated by the Python validation script.$

Python artifact: validation/01_algebra/matignon_complex_plane.png, generated by tools/validation/validate_chua_fractional_nonsmooth_algebra.py.

Algebra Validation at `q=0.9998`

This section records the first completed validation stage for the non-smooth fractional Chua case reported by Danca (2017). It validates equations, equilibria, regional Jacobians, the Lur’e split, and harmonic branches. It does not claim that numerical integration, chaos, or hiddenness have already been validated.

Check	Acceptance tolerance	Recorded maximum error	Result
Equilibrium substitution in Python, MATLAB and Wolfram	$10^{-10}$	$1.10 \times 10^{-15}$	passed
Symbolic/analytic Jacobian comparison	$10^{-10}$	$0$	passed
Analytic Jacobian vs central differences	$10^{-6}$	$2.47 \times 10^{-9}$	passed
Eigenvalues across tools	$10^{-8}$	$<10^{-15}$	passed
Matignon classification	Same stable/unstable decision	same result	passed

Transfer Sign Convention

MATLAB and the algebraic report use

W_{\mathrm{report}}(z)=r^T(zI-P)^{-1}q_v,

while the Python seed API historically computes

W_{\mathrm{code}}(\omega)=r^T(P-zI)^{-1}q_v=-W_{\mathrm{report}}(z), \qquad z=(i\omega)^q.

Therefore these checks are equivalent:

1-kW_{\mathrm{report}}=0, \qquad 1+kW_{\mathrm{code}}=0.

Harmonic Branches

Branch	$\omega_0$	$k$	$a_0$	Seed $(x_0,y_0,z_0)$
1	2.040286051079	0.210022792962	5.851767785486	$(5.8517677855,\ 0.3704086003,\ -8.3609729344)$
2	3.244926730975	0.956945404928	1.053016610257	$(1.0530166103,\ 0.8532234830,\ -1.5095086807)$

After sign normalization, Python and MATLAB agree with a maximum closure residual of 2.01e-13 and a maximum amplitude difference of 1.48e-12. The supplied Wolfram Language source also verifies the symbolic identities after renaming its protected identifier Tr.

Report Correction

The fractional non-smooth derivation in 170526.pdf correctly reaches $1-kW_q=0$ and $k=1/\operatorname{Re}(W_q)$ for its stated transfer. However, preceding general text and the later arctan balance section also write $1+W_qN=0$ without changing the transfer sign. Those occurrences must be normalized to the minus convention, or the document must explicitly redefine the transfer with the opposite sign.

Source Roles

Source	Role in this validation
Danca (2017), Hidden Chaotic Attractors in Fractional-Order Systems	Direct published reference for this exact non-smooth model, parameter set, $q=0.9998$ , and neighborhood test context.
Petras (2008), A Note on the Fractional-Order Chua’s System	Confirms the physical fractional PWL Chua model family; its parameter set and orders differ.
Sene (2021), Mathematical Views of the Fractional Chua’s Electrical Circuit Described by the Caputo-Liouville Derivative	Supports fractional Chua stability and dynamical reporting methods; its equations omit the $-\gamma z$ damping term and are not direct numeric confirmation.

Corrected EFORK Rerun

The former attractor image has been removed. It was generated before the native EFORK stages were aligned with the published K3/known-history formula. A new line-only rerun with the corrected EFORK backend, using the Danca-replication seed, q=0.9998, h=0.05, and retained history over T=500, converges toward E0:

Diagnostic	Recorded value
Final distance to `E0`	$2.51 \times 10^{-7}$
Post-transient range norm	$2.50 \times 10^{-7}$
Classification	collapsed toward `E0`

Therefore no EFORK attractor is published on this page until a corrected integration locates a bounded, non-trivial orbit and the dynamic validation stage is repeated.

Validation Status

The algebra stage is complete. The next required evidence is integrator convergence and independent-solver comparison. In particular, after correcting the EFORK stage-history evaluation, the previously published dynamic figure is not retained: the corrected full-history rerun from the Danca-replication seed collapses toward E0. Bounded chaotic dynamics and hiddenness must be revalidated before any attractor figure is promoted.