Intermittent Circle Maps in Dynamical Systems

• Intermittent circle maps are one-dimensional dynamical systems on S¹ characterized by alternating laminar phases and bursts of chaos due to neutral fixed points or flat intervals.
• Their analysis relies on transfer operators, spectral gap methods, and renormalization techniques to establish invariant measures, mixing rates, and critical phase transitions.
• Applications of these maps to Cherry flows and threshold-driven oscillators provide practical insights into slow mixing, heavy-tail statistics, and fractal geometry in dynamical systems.

Intermittent circle maps are a class of one-dimensional dynamical systems on the circle $S^1 = \mathbb{R}/\mathbb{Z}$ characterized by regions of regular dynamics ("laminar phases") interrupted by episodes of irregular, chaotic behavior. They exhibit prototypical intermittency as formalized in Pomeau–Manneville maps and their many variants, and arise as Poincaré return maps in systems such as Cherry flows and threshold-driven oscillators. Their dynamics are governed by the presence of neutral (indifferent) fixed points or flat intervals, leading to significant phenomena such as slow mixing, heavy-tail statistics, zero Lyapunov exponents, and sharp phase transitions in fractal geometry.

1. Model Classes and Defining Features

The canonical intermittent circle map %%%%1%%%% is typically $\mathcal{C}^2$, of algebraic degree $d \geq 2$, and possesses either (i) indifferent (neutral) branches—regions in which $f'_\alpha(x)$ approaches unity with a prescribed power law—or (ii) flat intervals, open arcs $U \subset S^1$ mapped to a point, with singularities governed by local power exponents. A representative model is given by $$f_\alpha(x) \approx x + \operatorname{sgn}(x) |x|^{\alpha+1}$$ near a neutral fixed point at $x=0$, with $f'_\alpha(x) - 1 \asymp |x|^{\alpha}$ for $x \neq 0$, and $f'_\alpha(x) > 1$ elsewhere (Etubi, 26 Jan 2026).

Maps with flat intervals are defined by pinching a segment $U$ onto a point, with power-law approach:

$$f(x) \sim h_{\pm}\bigl((x-b)^{\alpha_+}\bigr), \quad x \to b^-; \qquad f(x) \sim h_{-}\bigl((a-x)^{\alpha_-}\bigr), \quad x \to a^+,$$

for critical exponents $\alpha_+, \alpha_- > 1$ and suitable local diffeomorphisms $h_{\pm}$ (Palmisano, 2012, Palmisano et al., 2019). The precise scaling at the endpoints determines dynamical and geometrical properties.

2. Invariant Measures, Transfer Operators, and Mixing

Intermittent circle maps often admit a unique absolutely continuous invariant probability measure (SRB measure) $\mu_\alpha$ with a density $h_\alpha$ displaying singularities at neutral points; for example, $h_\alpha'(x) \sim -\alpha |x|^{-\alpha-1}$ near $x=0$ (Etubi, 26 Jan 2026). Correlation decays are polynomial: for Hölder observables $\psi \in L^\infty$, $\varphi \in C^1$ with $\int \varphi = 0$, the decay rate is

$$|\int \psi \cdot (\varphi \circ f_\alpha^n) - \int \varphi \int \psi| = O\left(n^{1-1/\alpha} (\log n)^{1/\alpha}\right),$$

contrasting with the exponential mixing of uniformly expanding maps.

The action of the Perron–Frobenius (transfer) operator $L_\alpha$ is

$$(L_\alpha \varphi)(x) = \sum_{y : f_\alpha(y) = x} \frac{\varphi(y)}{f_\alpha'(y)},$$

which can be analyzed via cone-type Banach spaces to establish spectral gaps and invariant cones $C_{*,1}, C$ controlling regularity and singularity structure (Etubi, 26 Jan 2026). Cone invariance is fundamental for proving existence and differentiability of SRB measures despite singular dynamics.

3. Intermittency, Laminar Phases, and Recurrence

The hallmark of intermittency in these systems is the alternation between prolonged "laminar" phases, where the orbit remains close to a neutral or flat region, and bursts of rapid, chaotic behavior elsewhere. In models with flat intervals, passage times near the flat spot display universal scaling laws governed by the critical exponents. The mean duration of laminar episodes diverges as a parameter (e.g., $\mu$ controlling gap formation) approaches a critical value—often as $O((\mu-\mu_c)^{-1/2})$, characteristic of type-I (saddle-node) intermittency (Derks et al., 2019). The scaling of recurrence times near the neutral region is

$\langle \ell \rangle \sim (\mu - \mu_c)^{-1/2},$

with heavy-tailed statistics dictated by the degree of the singularity.

In degenerate geometry (critical exponents $\alpha \leq 2$), laminar phases become arbitrarily long and typical recurrence sets shrink super-exponentially, resulting in vanishing Hausdorff dimension for the non-wandering set; in bounded geometry ($\alpha > 2$), laminar phases are uniformly bounded, leading to Cantor-like attractors with positive dimension (Palmisano, 2012, Palmisano et al., 2019).

4. Phase Transitions and Fractal Geometry

A central phenomenon is the sharp phase transition as the critical exponents at neutral points or flat interval boundaries cross a threshold. For symmetric power-law maps with a flat interval and exponent $\alpha$:

• If $1 < \alpha \leq 2$, the non-wandering set $\Omega$ has $\dim_H(\Omega) = 0$ (degenerate geometry).
• If $\alpha > 2$, $\dim_H(\Omega) > 0$ (bounded geometry, Cantor attractor) (Palmisano, 2012).

For the general asymmetric case with exponents $(\ell_1, \ell_2)$ at each boundary, the renormalization operator produces a universal phase boundary curve $\Gamma$ in the $(\ell_1, \ell_2)$-plane, determined by the unstable eigenvalue $\lambda_u(\ell_1, \ell_2) = 1$ (Palmisano et al., 2019). Systems with $(\ell_1, \ell_2)$ in the "regular" region ($\lambda_u < 1$) exhibit bounded geometry and positive Hausdorff dimension; in the "intermittent" region ($\lambda_u > 1$), orbits concentrate near the flat spot with degenerate zero-dimensional attractors.

Regime	Exponent(s)	Geometry	Dim$_H$ of Non-Wandering Set
Degenerate	$1 < \alpha \leq 2$	Shrinking	$0$
Bounded/Cantor	$\alpha > 2$	Bounded	$>0$
General case	$\lambda_u(\ell_1,\ell_2)$	See above	$0$ or $>0$

This transition carries over to Cherry flows and threshold systems, with local exponents derived from linearization.

5. Linear Response and Statistical Stability

Intermittent circle maps exhibit linear response: under suitable regularity and technical assumptions, the SRB measure $\mu_\alpha$ depends differentiably (in a weak sense) on the parameter $\alpha$, and the map

$$\alpha \mapsto \mathcal{R}_\psi(\alpha) = \int_{S^1} \psi\, d\mu_\alpha$$

is differentiable for observables $\psi \in L^q$, $q > (1-\alpha)^{-1}$. The derivative obeys the linear response formula:

$$\frac{d}{d\alpha}\Big|_{\alpha} \int \psi\, d\mu_\alpha = \int_{S^1} \psi(x) \left( I - L_\alpha \right)^{-1} \left[ \sum_{i=1,d} (X_{\alpha,i} N_{\alpha,i} h_\alpha)' \right] (x)\, dx,$$

with $(I - L_\alpha)^{-1}$ realized as a Neumann series on zero-mean cone functions; $N_{\alpha,i}$ are branch operators, $X_{\alpha,i}$ vector fields vanishing at endpoints (Etubi, 26 Jan 2026). Statistical stability (weak-* convergence of SRB measures under perturbation) arises in skew-product extensions (e.g., solenoid maps), where stability of the base intermittent circle map lifts to higher dimensions.

6. Creation of Discontinuities and Bifurcation Structures

In smooth threshold or Cherry-type models, small parameter changes near criticality can cause the birth of gaps (discontinuities) in the circle map:

• A tangency induces a gap of size $O(\sqrt{\varepsilon})$ at the threshold parameter $\varepsilon$.
• The map develops square-root singularities at the gap edge(s), altering local dynamics (Derks et al., 2019).
• Bifurcation structures split: traditional Arnold tongues bounded by saddle-node curves now interleave border-collision (BC) curves—Type I at singular edge ($F'\to \infty$), Type II with finite slope—modifying rotation-locked regions.
• Codimension-2 bifurcations occur as secondary tangencies produce multiple gaps, creating novel intermittent regimes with complex switching among intervals of continuity.

The generic scaling for laminar phase duration adjacent to BC boundaries is $O((\mu-\mu_c)^{-1/2})$, characteristic of type-I intermittency. Loss of injectivity leads to multiple gaps, with local normal forms containing multiple square-root singularities organizing an infinite cascade of BC and SN curves.

7. Dynamical Invariants and Observable Statistics

Dynamical invariants of intermittent circle maps reflect their singular geometry:

• The Lyapunov exponent is typically zero in both regular and degenerate regimes, reflecting non-expanding dynamics; in the regular regime, typical orbits equidistribute on the Cantor repeller, while in the degenerate intermittent regime measure concentrates near the flat spot.
• SRB-type invariant measures, when defined, may be non-atomic (regular regime) or singular, supported on vanishing neighborhoods of the flat region (intermittent regime) (Palmisano et al., 2019).
• Poincaré recurrence statistics near flat intervals display heavy tails, with exponents determined by critical exponents $(\ell_1, \ell_2)$.

This synthesis details the mechanisms, outcomes, and mathematical infrastructure underlying intermittent circle maps, as well as their role in the theory of phase transitions, statistical stability, bifurcations, and observable fractal statistics. The critical scaling exponents and the spectral properties of associated operators define sharp boundaries between drastically different dynamical and geometrical behaviors, with universal implications across low-dimensional dynamics and applications in threshold-driven physical and biological systems.