Singular Bifurcation Approach

• Singular bifurcation approach is a framework addressing nonanalytic bifurcations by employing geometric desingularization and alternative regularization techniques.
• It provides explicit scaling laws and diagnostic methods to analyze singularities where classical analytic expansions fail.
• Its applications span from microswimmer models to nonlinear PDEs, highlighting mechanistic differences from classical, analytic bifurcation theory.

A singular bifurcation approach refers to analytical and geometric techniques developed to paper bifurcations in dynamical systems, PDEs, and variational problems where the standard (analytic) theory fails due to the presence of singularities. Singularities in this context may appear either in the dependence of the equations on small parameters (singular perturbations), nonanalytic dependence of the bifurcation normal form, or in degeneracies such as nonisolated critical points or loss of normal hyperbolicity. Unlike classical bifurcation theory, which typically relies on local analytic expansions and centers around nondegenerate (transverse) branches, singular bifurcation theory must address breakdowns of standard hypotheses, often requiring alternative methods such as geometric desingularization (blow-up), variational Morse index jumps, regularization mappings, or specialized normal forms involving nonanalytic terms.

1. Distinction Between Classical and Singular Bifurcations

In classical bifurcation theory, the normal forms describing the emergence of new solution branches are analytic in the relevant amplitude(s) and parameter(s). Canonical normal forms include the supercritical pitchfork $\dot A = \mu A - A^3$ (with $A \sim \pm\sqrt{\mu}$ for $\mu>0$) and the saddle-node (fold) $\dot A = \mu - A^2$. The bifurcation diagrams can be built using convergent Taylor expansions in $A$ and $\mu$.

Singular bifurcations, by contrast, are characterized by the breakdown of analyticity in the amplitude $A$. This breakdown is often associated with nonanalytic dependence such as $|A|$, fractional or logarithmic exponents, or non-polynomial behavior. A prototypical singular pitchfork takes the form

$\dot A = \mu A - A|A|,$

with steady states $A \sim \pm\mu$. Here, the mapping $A \mapsto |A|$ is nonanalytic at $A=0$, and the associated bifurcation diagram cannot be captured by power-series expansions. Singular bifurcation phenomena arise in various contexts: e.g., infinite system size limits in hydrodynamic models, PDEs with singular nonlinearities, and slow-fast systems near folded singularities (Farutin et al., 2021).

2. Geometric and Analytic Regularization of Singular Bifurcations

To analyze nonanalytic or singularly perturbed bifurcations, a regularization method is essential. The central idea introduced by Farutin & Misbah is to perform a variable change that "moves" the singularity to infinity, allowing for a uniformly convergent expansion (Farutin et al., 2021). Consider an implicit equation $f(x,\epsilon) = 1$ with a parameter $\epsilon$ and singularity at $x^2 = -\epsilon^2$. The mapping

$\epsilon = x_0(1-s),\quad x^2 = x_0^2(2s - s^2)$

transforms the problem so that $x^2 + \epsilon^2 = x_0^2$ is constant along curves of constant $x_0$. A series expansion in $s$ becomes convergent for $|s| \leq 1$, and the bifurcation equation is solved for $x_0$ at $s=1$. This approach directly yields the correct, typically nonanalytic, scaling for the amplitude as $\epsilon \to 0$.

This framework encompasses not only the "angular" singularity (as in the $|A|$ law) but also bifurcations with general exponents $|A|^\gamma$: $$\dot A = \mu A - A|A|^{2\alpha} \implies A \sim \pm \mu^{1/(2\alpha)}.$$ Here, $\alpha=1$ recovers the $|A|$ nonlinearity with $A \sim \mu$, while other $\alpha$ yield "fold-type" or "cusp-type" diagrams (Farutin et al., 2021).

3. Singular Bifurcation in Model Systems: The Swimmer Problem

The swimmer model analyzed in (Farutin et al., 2021) exemplifies a singular bifurcation:

• A spherical microswimmer emits or absorbs solute, generating self-propulsion via a phoretic mechanism in an infinite 3D domain.
• The dimensionless velocity $\bar V_0$ is determined by an implicit equation reflecting the isotropic drift instability:

$$f(\bar V_0, 0) = Pe\left[\frac{1}{3} - \frac{1}{6}|\bar V_0| + \frac{1}{20}\bar V_0^2 + \ldots\right]$$

leading to the singular normal form $\dot x = \mu x - 2x|x|$ $(\mu = Pe/3 - 1)$, with the nontrivial branch $|x| \sim \mu$.

• Regularization by finite system size or bulk solute absorption turns the bifurcation into a regular supercritical pitchfork, $x \sim \sqrt{\mu}$, demonstrating how singular bifurcation behavior is intrinsically linked to the idealized (singular) limit.

Generalization to other models (e.g., 2D swimmers with exponentially weak amplitude response, Taylor sheet swimming with algebraic divergence of series coefficients, vesicle deformation series, etc.) demonstrates that the regularization approach provides not only leading-order scaling but also analytic control over the full normal forms in both regular and singular regimes (Farutin et al., 2021).

4. Classification and Scaling Laws for Singular Bifurcations

The regularization procedure reveals a classification of possible singular bifurcation types based on the exponent $\alpha$:

• $\alpha > 1/2$: "fold-type," infinite slope at onset,
• $0 < \alpha < 1/2$: "cusp-type," vanishing slope,
• $\alpha = 1/2$: "angular,"
• $\alpha < 0$: "unbounded" (blow-up).

The critical amplitude exponent is $1/(2\alpha)$, in contrast to the classical universal $1/2$. Singular pitchfork diagrams arising from these exponents possess fundamentally different local geometric features than regular bifurcations, indicating a broader family of structurally stable but nonanalytic local behaviors.

5. Broader Implications and Theoretical Significance

Many nonlinear and active systems previously modeled as having classical (analytic) bifurcations have been shown to display singular bifurcation structure in the limit of vanishing regularization (e.g., infinite domain size, zero bulk absorption, unchanged by symmetries incompatible with standard Lyapunov-Schmidt analysis). The regularization theory

$$(\epsilon,x) \longmapsto (x_0,s), \quad \epsilon = x_0(1-s),\ x^2 = x_0^2(2s - s^2)$$

systematically unifies the entire taxonomy of pitchfork, cusp, and general nonanalytic bifurcation diagrams under a single analytic framework (Farutin et al., 2021). This fills a longstanding gap in bifurcation theory and facilitates the detection, classification, and analytic continuation of singular bifurcations in both theoretical studies and practical computation.

6. Methodological Impact and Scope of Application

The singular bifurcation approach and its regularization mapping can be directly applied to:

• Nonlinear PDEs where the bifurcation problem exhibits loss of analyticity in the amplitude near solution branches;
• Coupled ODE-PDE systems (e.g., active matter, hydrodynamic modes in unbounded geometries);
• Biological and physical models exhibiting threshold phenomena, abrupt pattern onset, or subcritical/singular transitions beyond classical categorization.

The method provides:

• Explicit scaling laws for the amplitude and bifurcation diagrams even in the presence of essential singularities;
• A diagnostic approach to extract the singular exponent from low-order series expansions or data;
• A constructive regularization that yields a well-posed analytic bifurcation problem for both theoretical and numerical analysis (e.g., continuation methods).

This suggests that proper recognition and handling of singular bifurcations is essential for accurate modeling of nonanalytic transitions in a wide range of nonlinear systems. The approach is extensible to multicomponent and spatially extended systems wherever singularities hinder direct analytic or numerical bifurcation analysis.