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The diffusion mechanism in Hamiltonian systems, close to completely integrable, is usually connected with the existence of the so-called "transition chains". In this case slow diffusion occurs in a neighborhood of intersecting separatrices of hyperbolic periodic solutions (or, more generally, lower-dimensional invariant tori) of the perturbed system. In this note we discuss another diffusion mechanism that uses destruction of invariant tori of the unperturbed system with an almost resonant set of frequencies. We demonstrate this mechanism on a particular isoenergetically nondegenerate Hamiltonian system with three degrees of freedom. The same phenomenon also occurs for general higher-dimensional Hamiltonian systems. Drift of slow variables is shown using analysis of integrals of quasi-periodic functions of the time variable (possibly unbounded) with zero mean value. In addition, the proof uses the conditions of topological transitivity for cylindrical cascades. [ABSTRACT FROM AUTHOR]

The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness. [ABSTRACT FROM AUTHOR]

Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the existence of orbits converging as time tends to infinity to the quasiperiodic solutions associated with the unperturbed system. The proof of this result relies on a new strategy based on a refined analysis of the Banach spaces and the functionals involved in the resolution of suitable nonlinear invariant equations. This result is proved for finite differentiable and real-analytic Hamiltonians. Analogous statements for time-dependent vector fields on the torus are also obtained as a corollary. These results extend a previous work of Canadell and de la Llave, where only exponential decay in time is considered. The relaxation of the decay in time makes the results in the present paper suited for applications in many physical problems, such as celestial dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hamiltonian systems H(y, x, ξ, ε) = 〈ω(ξ),y〉+εP(y, x, ξ, ε), where ω and P are continuous about ξ. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under a certain transversality condition and a weak convexity condition for the frequency mapping ω. As a direct application, we prove a Kolmogorov-Arnold-Moser (KAM) theorem when the perturbation P holds arbitrary Hölder continuity with respect to the parameter ξ. The infinite-dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in the parameter beyond Hölder's type. [ABSTRACT FROM AUTHOR]

In this article, we define a circular restricted (N + 1) -body problem on the surfaces M κ 3 , with κ = ± 1 . The motion of the primaries corresponds to an elliptic relative equilibria studied in Diacu (Relative equilibria of the curved N-body problem. Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012), where N identical mass particles are rotating uniformly at the vertices of a regular polygon placed at a fixed parallel of a maximal sphere. By introducing rotating coordinates, this problem gives rise to a 3 d.o.f. Hamiltonian system. This problem has an equilibrium point placed at the poles of S 3 and the vertex of H 3 , for any value of the parameters. We give information about the linear and nonlinear stability of these equilibria. Finally, we carry out a study about the existence of periodic solutions and KAM tori. [ABSTRACT FROM AUTHOR]

In this paper we consider the persistence of degenerate lower-dimensional tori in reversible systems with a degenerate normal equilibrium point, including hyperbolic and elliptic types. Based on the method of introducing external parameters, KAM iteration and implicit function theorem, we prove that if the perturbations are sufficiently small and frequency ω 0 satisfies the Diophantine condition, the reversible system still has a lower-dimensional torus with frequency ω 0 . [ABSTRACT FROM AUTHOR]

This article is concerned with the geometry of germs of real analytic surfaces in (C 2 , 0) having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in (C 2 , 0) having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions { τ 1 , τ 2 } at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to { z 1 z 2 = c o n s t } (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets. [ABSTRACT FROM AUTHOR]

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → w + r2n+1 is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of almost periodically forced harmonic oscillators x ¨ + τ 2 x = ϵ f (t , x) where τ ∈ A with A being a closed interval not containing zero, the forcing term f is real analytic almost periodic functions in t with the infinite frequency ω = (⋯ , ω λ , ⋯) λ ∈ Z . Using the modified Kolmogorov–Arnold–Moser (or KAM Arnold (Uspehi Mat. Nauk 18(5 (113)):13–40, 1963), Moser (Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962:1–20 1962), Kolmogorov (Dokl. Akad. Nauk SSSR (N.S.) 98:527–530 1954)) theory about the lower dimensional tori, we show that there exists a positive Lebesgue measure set of τ contained in A such that the harmonic oscillators has almost periodic solutions with the same frequencies as f. The result extends the earlier research results with the forcing term f being quasi-periodic. [ABSTRACT FROM AUTHOR]

For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G , we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n -colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n -coloring if and only if $n=3$. We finally suggest generalizations of the latter results to n -colorings of ascending HNN-extensions of finitely generated abelian groups. [ABSTRACT FROM AUTHOR]

This paper focuses on the vector fields on the quasi-periodically forced circle flow φ ˙ = ρ + f (φ , θ) , θ ˙ = α where the forcing term f is real analytic in its arguments and small enough, the frequency vector α ∈ R d (d ≥ 2) is beyond Brjuno frequency. We prove that the flow above is reducible analytically provided fibered rotation number ρ f : = ρ (ρ + f (φ , θ)) is Diophantine with respect to the base frequency α. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite-dimensional systems with multi-dimensional frequency weaker than Brjuno frequency. [ABSTRACT FROM AUTHOR]

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings. [ABSTRACT FROM AUTHOR]