Codimension-2 Grazing Bifurcations

• Codimension-2 grazing bifurcations are organizing centers in piecewise-smooth and hybrid systems where simultaneous degeneracy conditions trigger complex interactions of periodic orbits and sliding motions.
• Their analysis employs local return maps, normal forms, and unfolding parameters to predict the creation, stability, and annihilation of invariant cycles.
• Applications span mechanical oscillators and power electronics, where precise computation and numerical continuation methods reveal bifurcation diagrams with multiple transition curves.

Codimension-2 grazing bifurcations are organizing centers for the onset, destruction, and interplay of periodic orbits and sliding motions in piecewise-smooth and hybrid dynamical systems. They arise when two independent degeneracy conditions—typically a grazing contact and a second constraint such as resonance, fold coincidence, or tangency multiplicity—are simultaneously satisfied. This results in a bifurcation point in two-parameter families, from which emanate several codimension-1 transition curves that together structure local phase portraits and the emergence and annihilation of various invariant objects, such as crossing cycles and sliding loops.

1. Definitions and General Principles

A grazing bifurcation refers to the scenario in which a limit cycle or periodic orbit of a dynamical system becomes tangent to a discontinuity boundary (switching or impact surface), so that contact occurs nontransversally at a single point, typically with quadratic tangency. In a codimension-1 grazing, this requires a single scalar constraint on system parameters. In a codimension-2 grazing bifurcation, an additional independent constraint is imposed, such as a resonance (e.g., Floquet multiplier equal to ±1), the coincidence of two folds, or the coalescence of tangency points. Codimension-2 cases include:

• Tangency in both vector fields at a point (e.g., double visible two-fold in Filippov systems (Novaes et al., 2017)).
• Simultaneous occurrence of grazing and resonance (e.g., resonant grazing in hybrid oscillators (Ghosh et al., 17 Oct 2025)).
• Higher-multiplicity tangency points (e.g., double tangency in return maps (Fang et al., 30 Apr 2024)).
• Symmetry-induced degeneracy (e.g., in $\mathbb{Z}_2$-symmetric Filippov systems (Chen et al., 1 Jun 2025)).

In piecewise-smooth systems, the switching manifold $\Sigma$ (e.g., $\Sigma = \{y=0\}$ in planar systems) partitions the phase space into regions where different smooth flows apply. The local bifurcation analysis centers on the return map or Poincaré map near the grazing cycle, quantifying intersections, tangencies, and their unfoldings in parameter space.

2. Local Normal Forms and Parameter Unfolding

Two-Fold and Grazing Scenarios in Filippov Systems

In planar Filippov systems, visible two-folds occur at points where both vector fields are tangent to the switching surface, and both tangencies are "visible" (the corresponding orbits emerge into their respective crossing regions). The generic local forms for $Z = (X, Y)$ near a visible two-fold at $\Sigma: y=0$, after time and coordinate changes, are:

• $X(x, y) = (1, x)$ for $y > 0$,
• $Y(x, y) = (-1, x-\alpha)$ for $y < 0$, where the unfolding parameter $\alpha$ encodes the horizontal offset between folds. A second parameter $\beta$ measures the mismatch in re-entry points on $\Sigma$ for the forward and backward flows (Novaes et al., 2017).

Grazing-Sliding with Symmetry

In $\mathbb{Z}_2$-symmetric systems, e.g., those invariant under $(x, y) \mapsto (-x, -y)$, the local normal form exploits symmetry and leads to the characterization of families $Z(x, y; \alpha)$, with coupled unfoldings $(\alpha_1, \alpha_2)$ affecting both the location of tangency and the offset of the limit cycle from $\Sigma$ (Chen et al., 1 Jun 2025).

High-Multiplicity Tangency

When grazing occurs with higher tangency multiplicity, as in double tangency ($m=2$), the leading order normal form for the return map incorporates quadratic terms in both $x$ (the local coordinate on $\Sigma$) and the unfolding parameters $(\alpha, \beta)$, with the latter controlling, e.g., the vertical position of the cycle and the splitting of a double tangency into simple ones (Fang et al., 30 Apr 2024).

3. First Return Map Analysis and Bifurcation Structure

The dynamics near grazing are governed by the Taylor expansion of the local return (Poincaré) map: $$P(x; \alpha, \beta) = x + a_{10}\alpha + a_{01}\beta + a_{20}x^2 + a_{11}x\alpha + a_{02}\beta^2 + O(3)$$ where coefficients such as $a_{10}$ and $a_{01}$ stem from derivatives of flow times and depend on the underlying vector fields and geometry of tangency. These expansions are typically derived by explicit time-of-flight integration and matching across the switching manifold.

For the classic codimension-2 grazing with $m=2$ (Fang et al., 30 Apr 2024):

• Fixed points of $P$ satisfy $a_{20}x^2 + a_{10}\alpha + a_{01}\beta = 0$,
• The fold curve $a_{10}\alpha + a_{01}\beta = 0$ in $(\alpha, \beta)$-space delineates simple and double roots,
• Two crossing cycles exist for $a_{10}\alpha + a_{01}\beta < 0$,
• A single sliding loop exists for $a_{10}\alpha + a_{01}\beta > 0$.

In planar Filippov systems with elementary simple two-fold connection (Novaes et al., 2017), the displacement function $f(x; \alpha, \beta)$ and its zeros organize the bifurcation diagram. Five codimension-1 curves, corresponding to grazing, saddle-node, critical crossing, and fold connections, emanate from the codimension-2 organizing center, structuring nine regions with distinct minimal invariant sets.

In systems with symmetry (Chen et al., 1 Jun 2025), the displacement (or return) map in two parameters produces a similar structure, but bifurcation curves are subject to symmetry constraints and explicit Melnikov-like conditions, dividing parameter space into regions with two standard cycles, sliding cycles, homoclinic sliding, etc.

4. Bifurcation Diagrams and Organizing Role

Codimension-2 grazing bifurcations act as central organizing points in parameter space. The full bifurcation diagram is characterized by:

• Five smooth codimension-1 curves (e.g., grazing, saddle-node of cycles, critical crossings, sliding/escaping transitions),
• Their pairwise intersections at the codimension-2 point organize a network of nine open regions with qualitatively distinct phase portraits, such as:
  • Two crossing limit cycles,
  • One crossing plus one sliding/escaping cycle,
  • Pseudo-equilibrium connections,
  • Homoclinic orbits involving sliding or escaping dynamics.

A table summarizing key organizing features:

Codimension-2 Scenario	Main Codim-1 Curves	Cycle Types Created/Destroyed
Two-fold in Filippov (visible)	Grazing, saddle-node, etc.	Crossing, sliding, pseudo-equilibrium cycles
Grazing-sliding, $\mathbb{Z}_2$-symmetric	Grazing, double crossing	Standard, grazing, sliding, homoclinic cycles
m=2 grazing loop	Fold line ($a_{10}\alpha + a_{01}\beta=0$)	Crossing, sliding loops
Resonant grazing (hybrid impact)	Grazing, resonant (SN, PD)	Bistability, period-doubling, chaos

Each region and transition corresponds to a physical scenario: birth/death of limit cycles, creation of sliding/shattering loops, or sudden onset of complex/mixed mode oscillations as parameters are varied.

5. Analytical and Numerical Methods

The standard approach to codimension-2 grazing bifurcations is via local return maps (Poincaré or displacement maps), Taylor expansion near the grazing point, and bifurcation equation analysis. Explicit coefficient calculation often involves integrating along the grazing limit cycle and evaluating derivatives of the vector field components.

In hybrid or impact systems, square-root singularities in Poincaré maps near grazing pose challenges for standard root-finding and continuation methods. The VIVID function approach (Ghosh et al., 17 Oct 2025) circumvents these by casting the periodicity (p-loop) condition as the zero of a smooth (differentiable) function, avoiding invalid excursions beyond the physical domain and ensuring robust numerical continuation even at the singular limit.

Main computational distinctions:

• Return map approach: Analytically calculates bifurcation curves and cycle stability; local geometric and dynamical insight.
• VIVID/continuation algorithms: Numerically trace periodic orbits and bifurcation loci directly, suitable for codim-2 grazing especially where root singularities and degenerate eigenvalues arise.

Alternative methods such as collocation (e.g., in AUTO or coco) are more general but require careful mesh control near grazing and can lose accuracy due to non-smooth map behavior.

6. Applications and Distinctions Among Scenarios

Codimension-2 grazing bifurcations are observed in systems where piecewise-smoothness, hybrid impacts, or switching play a central role: mechanical impact oscillators, relay feedback systems, power electronics, and other engineered or biological systems with discontinuity-induced phenomena.

Characteristic phenomena include:

• Bistability and coexistence of non-impacting and impacting periodic orbits in hybrid impact oscillators near resonance points (Ghosh et al., 17 Oct 2025).
• Sliding and escaping cycles in Filippov systems, with transitions governed by the location and interaction of folds (Novaes et al., 2017, Chen et al., 1 Jun 2025).
• Rich interplay of crossing and sliding loops in higher-multiplicity tangency scenarios, with explicit connection to the algebraic properties of the return map (Fang et al., 30 Apr 2024).

The precise nature of cycle creation, stability, and annihilation, as well as the arrangement of bifurcation curves, depends on the geometric and analytic structure of the vector fields, the types of symmetry present, and the construction of the switching or impact manifold.

7. Summary of Main Results and Theorems

The codimension-2 grazing bifurcation acts as an organizing center for all possible transitions involving crossing, sliding, and pseudo-equilibrium cycles in piecewise-smooth and hybrid systems. Main results from recent literature include:

• The set of systems admitting an elementary simple two-fold cycle forms a codimension-2 submanifold in function space, with explicit parametrization and a bifurcation diagram resolved up to third order in unfolding parameters (Novaes et al., 2017).
• In symmetric Filippov systems, Melnikov-type integrals supply explicit nondegeneracy and bifurcation conditions, and the phase diagram is exhaustively described in terms of displacement maps and parameter-dependent dividing curves (Chen et al., 1 Jun 2025).
• For high-multiplicity grazing (e.g., $m=2$), the bifurcation structure reduces to explicit conditions on map coefficients, classifying all local cycle configurations in the neighborhood of the degenerate grazing (Fang et al., 30 Apr 2024).
• In hybrid impact systems with resonance, codimension-2 points involve the collision of grazing and resonance loci, predictably organizing the birth and death of periodic orbits, their stability, and routes to chaos. New numerical approaches such as the VIVID function allow robust computation through such organizing centers (Ghosh et al., 17 Oct 2025).

These results provide a unified and explicit framework for both qualitative and quantitative analysis of grazing-induced phenomena in piecewise-smooth dynamical systems, bridging rigorous geometric theory and practical computational methodology.