Smooth Sigmoid Surrogate (SSS) Approach

• The SSS approach is a modeling technique that replaces abrupt switches with smooth sigmoid functions to capture both linear and hidden nonlinear dynamics.
• It systematically encodes nonlinear correction terms, enabling accurate simulation and analysis of systems with threshold or switching behaviors.
• It outperforms classical linear smoothing methods by retaining critical nonlinearities that affect stability, bifurcation, and attractor behavior.

The Smooth Sigmoid Surrogate (SSS) Approach constitutes a family of modeling and computational strategies in which discontinuous or non-differentiable elements—most prominently indicator functions and step-like switches—are systematically replaced with parametric sigmoid functions. This transformation is notable for its wide applicability in systems exhibiting switching or thresholding behavior (such as piecewise smooth dynamical systems, electronic circuits, survival analysis, and statistical learning) and for its principled capacity to retain and analyze “hidden” nonlinear dynamics that are otherwise lost in classical regularization procedures.

1. Mathematical Formulation and Core Principles

The foundation of the SSS approach is the formal replacement of discontinuous switch mechanisms with smooth sigmoid expansions. For a system characterized by a piecewise vector field,

$$\frac{dx}{dt} = \begin{cases} f_+(x) & \text{if } v(x) > 0,\ f_-(x) & \text{if } v(x) < 0, \end{cases}$$

the Filippov convex combination represents the dynamics near a discontinuity as

$$\frac{dx}{dt} = \frac{f_+(x)+f_-(x)}{2} + \frac{f_+(x)-f_-(x)}{2}\,\lambda,$$

with $\lambda = \operatorname{sign}(v(x))$ for $v \ne 0$.

Under the SSS framework, the step function is substituted by a differentiable sigmoid ${\varphi}_{\epsilon}(v)$, satisfying $$\varphi_{\epsilon}(v) = \operatorname{sign}(v) + \mathcal{O}(e^{-{|v|}/{\epsilon}})$$ for $|v|>\epsilon$ and strictly monotonic derivative ($\varphi'_\epsilon(v) > 0$). This induces a power series expansion: $$\frac{dx}{dt} = f(x; \lambda) = \sum_{n=0}^{\infty} \alpha_n(x)\,[\lambda(v(x))]^n,$$ with $\lambda = \varphi_\epsilon(v(x))$ in the smooth case.

Matching coefficients for $v>0$ and $v<0$ yields

$$\sum_{n=0}^{\infty}\alpha_n(x) = f_+(x), \quad \sum_{n=0}^{\infty}\alpha_n(x)\,(-1)^n = f_-(x).$$

A canonical rearrangement leads to

$$\frac{dx}{dt} = \frac{f_+(x) + f_-(x)}{2} + \frac{f_+(x) - f_-(x)}{2}\,\lambda + \underbrace{(\lambda^2 - 1)\,g(x,\lambda)}_{E(x; \lambda)}.$$

Here, $E(x; \lambda)$ represents “hidden” nonlinear contributions, which vanish away from the switch ($\lambda = \pm 1$) but may crucially influence dynamics within the switching layer.

2. Encoding and Impact of Hidden Nonlinear Terms

A central tenet of SSS is the systematic retention and quantification of nonlinear "hidden" terms originating from the smoothing process—terms that classical Filippov regularization neglects. These contributions, $E(x; \lambda)$, are negligible except in the vicinity of the threshold, where the smooth sigmoid transitions between its plateau values. In physical systems, for example, DC-DC converters, these terms can dramatically alter qualitative behaviors such as stability, location of pseudo-equilibria in sliding modes, or lead to the emergence of previously undetected attractors.

A variant formulation,

$$\frac{dx}{dt} = \frac{f_+(x) + f_-(x)}{2} + \frac{f_+(x) - f_-(x)}{2}\mu + p(\mu),$$

with $\mu = \varphi_{\epsilon}(V_b - V)$ and $p(\mu)$ a nonlinear function (e.g., $p(\mu) = \mu - \sigma(1-\mu)\mu$), illustrates how even small nonlinear perturbations (parameterized by $\sigma$) can shift saddle equilibria and impact global behavior, distinguishing the SSS approach from purely linear smoothings.

3. Generalization and Sigmoid Series Expansion

Unlike "tautological smoothing"—whereby a discontinuity is replaced with a sigmoid but only the linear combination is preserved—the SSS approach employs the full formal series in the switching variable. The resulting general model is

$$\frac{dx}{dt} = \frac{f_+(x) + f_-(x)}{2} + \frac{f_+(x) - f_-(x)}{2}\,\varphi_{\epsilon}(v(x)) + [\varphi_{\epsilon}(v(x))^2 - 1]\,g(x, \varphi_{\epsilon}(v(x))).$$

This construction not only recovers classical sliding mode behavior in the leading order but, via the nonlinear correction term, captures subtle and experimentally observed phenomena absent in convex-combination models.

4. Application Domains and Representative Examples

The SSS technique has found application in:

• Piecewise Smooth Dynamical Systems: Modeling switch-like transitions (e.g., impacts in mechanical systems, relay and relay-like phenomena in electronics, biological switches during mitosis). SSS allows simulation and analysis of fast transitions while preserving crucial nonlinear effects within switch surfaces.
• Electronic Control Circuits: For DC-DC converters or systems controlled by relay behavior, the SSS model accurately predicts changes in attractor structure, stability domains, and displacement of pseudo-equilibria as a function of hidden terms. Parameter choices in $p(\mu)$ can be tuned for design or control objectives.
• General Systems with Threshold Dynamics: Wherever instantaneous switching is an approximation (such as in neural models with sigmoidal firing rates), SSS enables high-fidelity modeling of smooth but rapid transitions.

5. Implications for Model Construction and System Analysis

The SSS approach enables:

• Robust Modeling: By retaining all orders in the sigmoid expansion, modelers can avoid loss of relevant nonlinearities and hidden terms that affect system response, bifurcations, and long-term behavior.
• Experimental Alignment: The framework is particularly effective in recovering and explaining experimentally observed sliding-mode phenomena, pseudo-attractors, and nuanced behaviors otherwise missed by tautological smoothing approaches.
• Controller Design: Inclusion of small but significant nonlinear corrections informs control strategies especially in systems operating near critical thresholds, such as electronic circuits at switching boundaries.

6. Significance Relative to Classical Approaches

Conventional Filippov regularization, rooted in convex combinations and linear smoothing, can replicate only gross features of the transition, missing critical nonlinearities. The SSS approach, by comparison, systematically encodes both linear and nonlinear aspects, demonstrating—using explicit series expansions and rigorous matching arguments—that smooth regularizations can, depending on nonlinear contributions, impart markedly different hidden dynamics. The effect is not merely cosmetic; structural stability, bifurcation diagrams, sliding attractors, and global system behavior all depend sensitively on the retention and explicit computation of these terms.

7. Conclusion and Practical Guidance

The Smooth Sigmoid Surrogate (SSS) approach provides a principled, mathematically exact methodology to model real-world systems with switch-like behavior, corrects the deficiencies of linear smoothing, and clearly identifies the dynamical consequences of “hidden” nonlinear terms. Its adoption enables more nuanced, robust, and predictive models for piecewise smooth systems, electronic circuits, and any domain in which idealized discontinuities are only approximations to underlying, fast but finite transitions. For practitioners, careful selection of the sigmoid function and explicit tracking of series expansion terms are essential for accurate modeling, control, and analysis in the presence of near-discontinuous phenomena.