Singular KAM Theory and Its Extensions

• Singular KAM Theory is an extension of classical KAM that rigorously addresses systematic singularities due to resonance, degeneracy, and weak regularity.
• It employs techniques like separatrix analysis, parameter translation, and quasi-linear Newton iterations to preserve quasi-periodic invariant tori despite standard obstructions.
• The framework provides sharp measure estimates and has practical applications in celestial mechanics, Hamiltonian PDEs, and systems with singular geometric structures.

Searching arXiv for relevant papers on Singular KAM Theory to ground the article in published work. Singular KAM Theory denotes a family of extensions of Kolmogorov–Arnold–Moser theory for situations in which the standard hypotheses fail in a structured way. In the literature, the singularity may arise from secular separatrices generated by simple resonances in nearly-integrable Hamiltonians, from arbitrarily weak regularity of a parameter-dependent frequency map, from singular perturbation regimes in Hamiltonian partial differential equations, from resonance in quasi-periodic media, from singular normal matrices in reversible systems, from proper degeneracy, or from hypersurface singularities in $b^m$-Poisson and folded symplectic manifolds (Biasco et al., 2023, Tong et al., 2023, Eliasson et al., 2016, Miranda et al., 2022). The common objective is the persistence, reconstruction, or measure-theoretic control of quasi-periodic invariant tori when the classical action–angle, twist, or reducibility mechanisms are obstructed.

1. Singular mechanisms and the departure from classical KAM

Classical KAM theory treats analytic nearly-integrable Hamiltonians in non-resonant regions, using a twist or nondegeneracy condition on the frequency map and excluding a small Cantor family of resonant parameters or actions. Singular KAM theory begins where this mechanism breaks down. In the simple-resonant setting, the averaged Hamiltonian becomes effectively one-degree-of-freedom with a pendulum-like separatrix, the action–angle map blows up, and the twist determinant vanishes at separatrices; classical estimates then fail (Biasco et al., 2023).

A different singular mechanism appears in parameterized Hamiltonian systems when the unperturbed frequency mapping $\omega(\xi)$ is only continuous in the external parameter and may be arbitrarily singular near a base point. In that setting, neither full-rank nondegeneracy of $\partial \omega/\partial \xi$ nor uniform weak convexity is available, and the basic problem is to preserve a prescribed Diophantine frequency by a small parameter translation despite irregular continuity (Tong et al., 2023).

In Hamiltonian PDE, the singularity is not a loss of analyticity in the finite-dimensional variables but the coexistence of infinitely many small divisors, shrinking spectral gaps, and only finitely many adjustable parameters. Eliasson–Grébert–Kuksin describe singular KAM in this sense as the extension of classical finite-dimensional KAM to a setting where there are infinitely many degrees of freedom, part of the spectrum accumulates, one has only a finite number of adjustable external parameters, and the nonlinear term must smooth out in order to close the estimates (Eliasson et al., 2015).

Other works use the same label for further obstructions. In resonant Frenkel–Kontorova theory, resonance with the substratum prevents the use of symplectic transformations and Ward identities, and solvability is restored by an extra equation and a counterterm (Llave et al., 2015). In reversible context 2, the obstruction is a singular normal matrix in the Floquet directions (Sevryuk, 2016). In $b^m$-Poisson geometry, the phase space itself carries a hypersurface singularity, so both action–angle coordinates and KAM estimates must be reformulated in singular symplectic terms (Miranda et al., 2022). This suggests that “singular” in singular KAM theory is not a single condition but a class of failures of the standard KAM template.

2. Separatrix singularities and nearly-integrable Hamiltonians

In the separatrix-based theory of analytic nearly-integrable Hamiltonians, the model problem is the mechanical Hamiltonian

$$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$

with $f$ in a generic real-analytic Morse-type class. The phase space is covered by a non-resonant region, simple-resonant tubes, and a double-resonant tube. In the non-resonant region one performs high-order averaging and applies classical KAM. In each simple-resonant zone one carries out a second averaging and reduces the Hamiltonian to a one-degree-of-freedom secular problem with an exponentially small remainder (Biasco et al., 2023).

A central step is the reduction of each single-resonance Hamiltonian to a standard form

$$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$

where $G(\cdot,q_1)$ is a Morse function with distinct critical values. The theory then constructs analytic action–angle variables in complex neighborhoods of the separatrices and proves that, near a critical energy $E_{\mathrm{crit}}$, the action has the singular expansion

$$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$

This logarithmic singularity is not treated as a defect to be avoided; it is the analytic datum from which one regularizes the twist determinant and proves that its vanishing is confined to a tiny sub-area (Biasco et al., 2023).

The resulting measure estimates are tailored to the Arnol'd–Kozlov–Neishtadt conjectures. For $\omega(\xi)$0 one obtains

$\omega(\xi)$1

while for $\omega(\xi)$2, outside a small disk $\omega(\xi)$3, the complement of invariant tori is exponentially small:

$\omega(\xi)$4

These bounds agree with the conjectured orders up to a polylogarithmic correction (Biasco et al., 2023).

The 2025 convex extension replaces the mechanical integrable part by a real-analytic $\omega(\xi)$5-convex Hamiltonian $\omega(\xi)$6 and studies

$\omega(\xi)$7

with $\omega(\xi)$8 generic real-analytic. The conclusion is that, if $\omega(\xi)$9 denotes the union of all primary or secondary Lagrangian invariant tori, then

$\partial \omega/\partial \xi$0

In particular, up to a logarithmic correction, the relative measure of surviving invariant tori is $\partial \omega/\partial \xi$1, again in agreement with the Arnol'd–Kozlov–Neishtadt conjecture (Barbieri et al., 16 Jul 2025). The extension shows that the singular separatrix analysis is not confined to the quadratic kinetic energy $\partial \omega/\partial \xi$2.

3. Parameter singularity and frequency-preserving persistence

Tong–Li and collaborators formulate singular KAM theory for parameterized Hamiltonians of the form

$\partial \omega/\partial \xi$3

where $\partial \omega/\partial \xi$4 is real-analytic in $\partial \omega/\partial \xi$5 but only continuous in the parameter $\partial \omega/\partial \xi$6, and $\partial \omega/\partial \xi$7 may have arbitrarily weak regularity. The aim is not merely persistence of some nearby torus, but persistence of a torus carrying exactly the unperturbed Diophantine frequency $\partial \omega/\partial \xi$8 (Tong et al., 2023).

The theory replaces Kolmogorov nondegeneracy by three hypotheses. First, $\partial \omega/\partial \xi$9 must be Diophantine:

$b^m$0

Second, the perturbation must satisfy relative singularity: the modulus of continuity of $b^m$1 in $b^m$2 is bounded by that of $b^m$3 in $b^m$4, uniformly in $b^m$5. Third, the map $b^m$6 must satisfy controllability, encoded by

$b^m$7

together with continuity, strict monotonicity, $b^m$8, and the integrability condition

$b^m$9

These conditions allow $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$0 to be arbitrarily singular, for example $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$1 near $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$2, provided the singularity is comparable to that of $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$3 (Tong et al., 2023).

The main theorem states that for sufficiently small $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$4 there exists a translated parameter $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$5 and an analytic $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$6-small canonical change of variables such that the perturbed Hamiltonian has an invariant $H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$7-torus carrying exactly the quasi-periodic flow with frequency $$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$8. The size of the translation is quantified by

$$H(y,x;\epsilon)=\frac12|y|^2+\epsilon f(x), \qquad \|f\|_s=1,$$9

If $f$0 is only $f$1-Hölder in $f$2 and $f$3 is injective, then $f$4 and the same conclusion follows. If $f$5 is nondegenerate linear and $f$6 is Lipschitz in $f$7, then for Lebesgue-almost every $f$8 one can choose a parameter so that a torus of frequency $f$9 persists (Tong et al., 2023).

The proof uses a quasi-linear Newton iteration on shrinking complex domains. At the $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$0th step one solves a truncated homological equation, removes low Fourier modes, and then performs a parameter translation

$$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$1

chosen so that the new normal form keeps exactly the same frequency $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$2. Relative singularity controls the parameter dependence needed in differentiating the homological equation, and controllability yields

$$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$3

together with convergence of $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$4 despite only weak continuity of $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$5 (Tong et al., 2023).

The sharpness is emphasized by explicit counterexamples. Violation of the Diophantine condition, of relative singularity, or of controllability destroys the frequency-preserving conclusion in distinct ways; discontinuity of $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$6 or $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$7 destroys the argument at once; and the convergence rate of $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$8 can be arbitrarily slow or arbitrarily fast. The same framework also yields a partial frequency-preserving theorem, where only a block $$H_{\mathrm{std}}(p_1,q_1;\hat p)= (1 + g(p_1,q_1;\hat p))\,p_1^2 + G(\hat p,q_1),$$9 is preserved and the remaining block drifts on a small Cantor tail, and an infinite-dimensional version without any spectral asymptotics (Tong et al., 2023).

4. Infinite-dimensional singular perturbations and Hamiltonian PDE

In the work of Eliasson–Grébert–Kuksin, singular KAM theory addresses Hamiltonian PDE with regularizing nonlinearities. The abstract normal form is

$G(\cdot,q_1)$0

where $G(\cdot,q_1)$1 is block-diagonal over Fourier shells and the spectrum splits into elliptic frequencies $G(\cdot,q_1)$2 and a finite hyperbolic part. The perturbation is analytic in the finite-dimensional variables and measured in norms that control the Hamiltonian, its differential, and its Hessian in an algebra of infinite matrices with exponential off-diagonal decay (Eliasson et al., 2015).

The small divisors are Melnikov combinations

$G(\cdot,q_1)$3

and the singular aspect is that the spectral gap defects shrink as $G(\cdot,q_1)$4. The analysis therefore requires spectral asymptotics, transversality in the external parameter, weighted sequence spaces, and smoothing of order $G(\cdot,q_1)$5 so that each Poisson bracket or homological solve gains decay in the mode index. This gain compensates the derivative loss generated by the small divisors (Eliasson et al., 2015).

The normal-form theorem yields, under a smallness condition of the form

$G(\cdot,q_1)$6

a Cantor set of parameters $G(\cdot,q_1)$7 with

$G(\cdot,q_1)$8

and a real-analytic symplectic map sending $G(\cdot,q_1)$9 to a new normal form plus a quadratically small remainder. The torus $E_{\mathrm{crit}}$0 persists as a reducible, linearly stable, $E_{\mathrm{crit}}$1-dimensional invariant torus (Eliasson et al., 2015).

The companion abstract theorem for space-multidimensional Hamiltonian PDE makes the singular aspect explicit by allowing hyperbolic normal directions and by treating genuine singular perturbation problems. In its abstract form, for

$E_{\mathrm{crit}}$2

one obtains a Cantor-like subset $E_{\mathrm{crit}}$3 with

$E_{\mathrm{crit}}$4

and symplectic changes $E_{\mathrm{crit}}$5 such that the conjugated Hamiltonian has an invariant torus $E_{\mathrm{crit}}$6 carrying the linear flow $E_{\mathrm{crit}}$7. The integrable part may contain a hyperbolic part, so the constructed invariant tori may be unstable (Eliasson et al., 2016).

The applications include the multidimensional beam equation with a convolutive potential, nonlinear Schrödinger equations with a $E_{\mathrm{crit}}$8-smoothing nonlinearity, and the beam equation without parameters. In the parameter-free beam problem one has the singular relation

$E_{\mathrm{crit}}$9

which links the size of the perturbation to the transversality constant and still permits a Cantor family of small tori (Eliasson et al., 2016). This is a paradigmatic singular perturbation regime: the smallness requirement depends on the rate at which the nonresonance constants degenerate.

5. Resonance, degeneracy, reversibility, and singular geometry

A distinct singular KAM mechanism appears in the quasi-periodic Frenkel–Kontorova model when the frequency of the solution resonates with the frequencies of the substratum. The equilibrium equation cannot be handled by transformations as in the Hamiltonian case, nor by Ward identities as in the nonresonant quasi-periodic-media case. The method therefore introduces an auxiliary unknown $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$0 and a scalar counterterm $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$1 so that the linearized operator factorizes as

$$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$2

The counterterm projects the potential into a codimension one manifold on which the obstructing small divisors are paired by the factorization. The resulting a-posteriori theorem yields locally unique analytic branches, and the Newton scheme leads to algorithms with complexity per step $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$3 and memory $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$4 (Llave et al., 2015).

In finite-dimensional bifurcation theory, Li–Shang develop a singular KAM theorem for finitely differentiable vector fields with multiple degeneracies, applicable to multiple Hopf and zero-multiple Hopf bifurcations. The method combines analytic smoothing of the finitely differentiable nonlinearities, a KAM iteration in Whitney spaces, and parameter elimination on nested Cantor sets. The theorem produces a Cantor family of parameters $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$5 with

$$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$6

and for each parameter in $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$7 a quasi-periodic invariant torus whose frequency remains close to the reference frequency and whose normal form is block-diagonal (Li et al., 2018).

In reversible context 2, Sevryuk studies invariant tori for systems reversible with respect to an involution $$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$8 under the constraint

$$I_1(E)=a(z)+b(z)\,z\log z,\qquad z=|E-E_{\mathrm{crit}}|.$$9

Here the main obstruction is a singular normal matrix $\omega(\xi)$00. The decisive nondegeneracy condition is

$\omega(\xi)$01

equivalently $\omega(\xi)$02. Persistence is obtained by adjoining an auxiliary block, embedding the family into a versal unfolding, and invoking the Broer–Ciocci–Hanßmann–Vanderbauwhede theorem on quasi-periodic stability with singular normal matrices. The output is a Whitney $\omega(\xi)$03 family of reducible invariant tori on a Cantor-like parameter set (Sevryuk, 2016).

On $\omega(\xi)$04-Poisson manifolds, singularity is geometric rather than perturbative. Near the hypersurface $\omega(\xi)$05, the local normal form is

$\omega(\xi)$06

Miranda–Planas prove an action–angle theorem near Liouville tori lying in $\omega(\xi)$07, and a KAM theorem for Hamiltonians with singular part

$\omega(\xi)$08

Under analytic nondegeneracy and Diophantine conditions, the perturbed system retains a real-analytic family of invariant tori, with measure of destroyed tori bounded by $\omega(\xi)$09. By desingularization, this yields KAM theorems for folded symplectic manifolds and for smooth symplectic systems with a distinguished hypersurface that the perturbation tracks to high order (Miranda et al., 2022).

6. Measure estimates, sharpness, and applications

One of the unifying themes of singular KAM theory is that the surviving tori occupy substantially more phase space than classical $\omega(\xi)$10-type bounds would suggest. In the separatrix-based theory, the measure of the complement of invariant tori matches the Arnol'd–Kozlov–Neishtadt conjectural order up to a logarithmic loss: $\omega(\xi)$11 for $\omega(\xi)$12 and exponential smallness outside a small disk for $\omega(\xi)$13 (Biasco et al., 2023). In the convex extension, the complement satisfies $\omega(\xi)$14, again emphasizing that the singular contribution is concentrated near resonant and separatrix structures rather than distributed throughout phase space (Barbieri et al., 16 Jul 2025).

Another recurring theme is sharpness. Tong–Li et al. provide counterexamples showing that each of the conditions of internal Diophantine nonresonance, relative singularity, and controllability is essentially indispensable in the frequency-preserving problem. They also show that the convergence rate of the translating parameter can be arbitrarily slow, for example when $\omega(\xi)$15, or arbitrarily fast (Tong et al., 2023). In this sense singular KAM theory is not merely constructive; it is also diagnostic about which singular behaviors are allowable.

Applications to celestial mechanics and properly-degenerate Hamiltonians illustrate how these abstract theories enter concrete models. Pinzari–Liu prove quantitative KAM theorems for maximal Lagrangian tori and for whiskered tori in properly-degenerate systems, with the measure bounds

$\omega(\xi)$16

for maximal tori and, in the codimension-two whiskered case,

$\omega(\xi)$17

Applied to the spatial three-body problem, this yields a positive-measure $\omega(\xi)$18-torus family of stable Lagrangian tori coexisting with a positive-measure $\omega(\xi)$19-torus family of whiskered tori near the co-planar, co-circular, outer-retrograde configuration (Pinzari et al., 2023).

Open directions are also explicit in the literature. For the separatrix-based theory, open problems include removing the logarithmic loss, extending beyond mechanical Hamiltonians, and proving matching lower bounds on the non-torus measure in the generic regime (Biasco et al., 2023). The convex extension already moves beyond the mechanical case, while noting that an extension from $\omega(\xi)$20 to $\omega(\xi)$21 would require a quantitative Morse–Sard analysis of zeros of $\omega(\xi)$22, presumably via Yomdin–Comte tame geometry (Barbieri et al., 16 Jul 2025). A plausible implication is that singular KAM theory has developed into a collection of specialized but interoperable frameworks, each isolating a different obstruction to torus persistence and replacing the missing classical mechanism by a comparably precise analytic substitute.