Quasi-Time-Periodic Solutions in Dynamics

• Quasi-time-periodic solutions are trajectories in dynamical systems characterized by incommensurate base frequencies that densely fill an n-torus in phase space.
• They generalize classical quasiperiodic behavior and require advanced methods like Nash–Moser techniques in weighted spaces to handle weak invariance under decaying perturbations.
• Applications in celestial mechanics illustrate how slowly decaying time-dependent perturbations yield weakly asymptotically quasiperiodic behavior, maintaining O(1/t) proximity to unperturbed tori.

A quasi-time-periodic solution is a trajectory of a dynamical system whose evolution exhibits incommensurate (non-commensurable, i.e., not rationally related) base frequencies, resulting in motion that densely fills an $n$-torus in phase space. These solutions generalize periodic and quasi-periodic (on $n$-tori) motions and are central in Hamiltonian systems, especially in perturbed integrable cases. An advanced generalization is the "weakly asymptotically quasiperiodic solution", pertinent for non-autonomous or dissipative perturbations, especially when conventional invariant tori are destroyed or approached asymptotically rather than realized exactly.

1. Definitions and Notions

The foundational concept is the (strong) quasiperiodic torus: for an integrable autonomous Hamiltonian $H_0(I,\phi)$ in action-angle variables $(I,\phi)\in\mathbb R^n\times\mathbb T^n$, the flow on an invariant torus $\phi_0:\mathbb T^n\to\mathbb R^n\times\mathbb T^n$ with frequency vector $\omega$ is given by $$X_0\circ\phi_0(\theta) = d\phi_0(\theta)\cdot\omega$$. Trajectories on these tori are strictly quasiperiodic.

For time-dependent Hamiltonians $H(I,\phi,t)=H_0(I,\phi) + P(I,\phi,t)$ with a time-dependent perturbation $P$ decaying as $t\to\infty$, the notion of quasi-time-periodicity must be adapted due to the breakdown of strict invariance.

Weakly Asymptotically Quasiperiodic Solution:

Given the full time-dependent vector field $X^t$ and the unperturbed torus $\phi_0$, a family of embeddings $\phi^t:\mathbb T^n\to$ phase space forms a $C^\sigma$-weakly asymptotic cylinder if

• $$\sup_{\theta} \|\phi^t(\theta) - \phi_0(\theta)\|_{C^\sigma} \to 0$$ as $t\to\infty$,
• $$X^t\circ\phi^t(\theta) = D\phi^t(\theta)\cdot[\omega + \Gamma^t(\theta)] + \partial_t\phi^t(\theta)$$ with $\Gamma^t(\theta)\to 0$.

Any trajectory of the full system starting on $\phi^{t_0}$ remains $O(1/t)$-close to the unperturbed quasiperiodic orbit, and is thus called a weakly asymptotically quasiperiodic solution (Scarcella, 2022).

2. Existence Theorems and Functional Framework

For non-autonomous Hamiltonians with decaying time-dependent perturbations, the existence of (weakly) asymptotically quasiperiodic solutions requires decoupling small denominator obstacles from standard KAM theory. The abstract setting is:

• $H(I, \phi, t) = H_0(I, \phi) + P(I, \phi, t)$, with $H_0$ integrable ($\omega$ possibly resonant).
• Perturbation $P$ satisfies $|P(\cdot, t)|_{C^r}\leq C t^{-\alpha}$, $\alpha>1$, $r$ sufficiently high.

The right spaces to track decay and regularity are weighted Hölder (or Sobolev) spaces:

$$S_{\sigma, \ell} = \left\{ f(\theta, I, t) : \sup_{t \geq 1} t^\ell |f(\cdot, t)|_{C^\sigma} < \infty \right\}.$$

No Diophantine or twist conditions are required since the time-dependent homological equation replaces the standard small-divisor-affected equation.

Main Existence Result (abstract form) (Scarcella, 2022):

• Under the above regularity, decay, and smallness hypotheses, there exists a family $\phi^t$ and disturbances $\Gamma^t\in S_{\rho,1}$ solving the invariance equation, with $\|\phi^t-\phi_0\|_{C^\sigma}=O(1/t)$ and $\|\Gamma^t\|=O(1/t)$. Any trajectory starting “on the cylinder” remains $O(1/t)$-close to a straight-line quasiperiodic trajectory.

This generalizes previous results with stronger decay (e.g., exponential), allowing for only polynomial decay as long as the exponent exceeds 1.

3. Nash–Moser Approach and Homological Equations

The proof mechanism leverages an abstract Nash–Moser theorem set within a scale of weighted spaces $S_{\sigma,\ell}$. The essential obstacle is the loss of regularity arising from derivatives with respect to the angle variable in the functional invariance equation.

• Near-identity coordinate change:

Solutions are sought in the form $\phi^t(\theta) = (\theta, v(\theta,t))$. Substituting into the invariance equation yields a nonlinear, time-dependent PDE for $v$.

• Linearization and smoothing:

The linearized operator $D_v F$ loses derivatives; standard Newton iteration fails. Nash–Moser smoothing is deployed, with smoothing operators $S_\tau$ and careful "tame" bounds linking high and low regularity norms.

• Time-dependent Homological Equation:

The inhomogeneous linear problem to invert is

$$\partial_\theta\kappa\cdot(\omega+f(\theta,t)) + \partial_t\kappa + g(\theta,t)\kappa = z(\theta,t).$$

Unlike classical KAM, no small denominators arise; inversion is via integration along characteristics $(\theta+\omega t, t)$.

• Polynomial decay exponent role:

The requirement $\alpha>1$ ensures the weighted norms are sufficient for control, as the integrability over $t$ becomes crucial.

4. Applications: Celestial Mechanics and Structural Asymptotics

An explicit application is to the planar three-body problem perturbed by a distant, time-dependent celestial body receding to infinity with asymptotic velocity. The time-dependent Hamiltonian is $H=H_0+H_c$, with $H_0$ the isolated three-body Hamiltonian and $H_c$ (the comet’s contribution) decaying like $O(1/t^2)$ in suitable coordinates.

After symplectic reduction and suitable charting (e.g., Jacobi and Weinstein), invariant KAM tori for $H_0$ are constructed. The result:

• There exist orbits in the perturbed system remaining $O(1/t)$-close to the unperturbed KAM torus for large $t$.
• The center-of-mass exhibits logarithmic drift, while planetary orbits approach the KAM torus as time progresses (Scarcella, 2022).

A serious dynamical difference appears for polynomial—but slow—decay; classic strong convergence to invariant tori fails, necessitating the weak (asymptotic) definition.

5. Broader Context and Related Classes

The notion of quasi-time-periodic solutions, as well as their weakly asymptotic analogs, generalizes essential features in dynamical stability and persistence for almost-integrable systems subject to decaying, time-dependent perturbations. The weak concept is vital where perturbations prevent exact tori or when the time decay is too slow for rapid convergence.

• In contrast, for exponentially decaying perturbations, "asymptotically quasiperiodic" solutions with convergence to tori in strict sense can be established, even with minimal smallness assumptions (Scarcella, 2022).
• In systems where the decay is integrable but not exponentially fast, the Nash–Moser framework with weighted norms ensures control without Diophantine or twist properties (Scarcella, 2022), broadening the class of Hamiltonians where asymptotic quasiperiodic behavior persists.
• If the time-dependent perturbation does not decay sufficiently, or does so too slowly, the weak (asymptotic) solution defines the optimal persistence notion.

6. Comparison to Quasiperiodic and Asymptotic Regimes

Notion	Decay Assumptions	Strict Invariance	Techniques	Limiting Dynamics
Quasiperiodic	None (autonomous)	Yes	KAM	Invariant torus
Asymptotically quasiperiodic	Exponential or polynomial, $\int_0^\infty$ decay	Yes (asymptotically)	Fixed point, Implicit Function Thm	Converges to torus
Weakly asymptotically quasiperiodic	Polynomial, $\alpha>1$	No (only $O(1/t)$ proximity)	Nash–Moser, weighted spaces	Remains close, not invariant

The weakly asymptotic regime captures limit behaviors where strong persistence is lost but solutions remain in a controlled, slowly vanishing dynamical shadow of the original torus.

7. Implications and Extensions

The theory of weakly asymptotically quasiperiodic solutions has broad potential for applications in celestial mechanics (e.g., stability analysis in systems with dissipative or slowly decaying perturbations), geophysical flows, and nearly integrable mechanical systems with time-dependent environmental effects. The Nash–Moser approach on weighted spaces developed in this context accommodates both the loss of regularity and weaker decay scenarios, providing a flexible framework adaptable beyond classical invariant-torus settings (Scarcella, 2022). The general concepts and proofs emerge as limiting (low-decay) extensions of KAM-type perturbation theory.