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This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

We deal with Hamiltonian bifurcations associated with the reversible umbilic in two degrees of freedom systems defined by 0:1 resonance, i.e. the unperturbed equilibrium has two purely imaginary eigenvalues and a semisimple double-zero one. The Hamiltonian is written as the sum of integrable Hamiltonian $$N\big (\frac{1}{2}(x^{2}+y^{2}),q,p;\lambda \big )$$ and a small perturbation $$P(x,y,q,p;\lambda )$$ by the normalization procedure. The phase portrait of N on each level of the integral $$I_{1}=\frac{1}{2}(x^{2}+y^{2})$$ is then studied in detail, obtaining an unfolding related to the reversible hyperbolic umbilic catastrophe. The persistence of 2-tori (i.e. two-dimensional tori) for the full system is analysed via KAM theory pointed out just as in Broer et al. (Z Angew Math Phys 44: 389–432, 1993), (in: Langford, Nagata (eds) Normal forms and homoclinic chaos, Waterloo, (1992), Fields Institute Communications 4 (1995)). In a sense, our results can be seen as a four-dimensional extension of a planar problem studied by Hansmann (Phys D 112: 81–94, 1998).

In this paper, we prove a reducibility result for a relativistic Schrodinger equation on torus with time quasi-periodic unbounded perturbations of order 1/2. As far as we known, this is the first reducibility result for the relativistic Schrodinger equation.

The plane motion of a solid in a uniform gravity field is studied. The solid is suspended on a weightless inextensible thread, which remains stretched during the motion. It is assumed that the length of the thread is large ( $$\sim {{\varepsilon }^{{ - 1/2}}}$$ ) and the distance from the point of solid suspension to its center of gravity is small (~e). The equations of motion are presented as equations of a system with one rapidly rotating phase. This system is analyzed using classical perturbation theory and KAM theory. It is shown that for all values of time, the movement differs little (by ~e) from the slow oscillations of the thread in the vicinity of the descending vertical and the solid rotation relative to the suspension point with an almost constant angular velocity. The measure of the set of motions, different from the above motions, is estimated from above by the value of the order of $$\exp \left( { - c{\text{/}}\varepsilon } \right)$$ ( $$c > 0$$ = const).

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Russmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra- differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

we investigate the existences of KAM tori for the infinite dimensional Hamiltonian system with finite number of zeros among normal frequencies. By constructing a constant vector we show that, for “most” tangent frequencies in the sense of Lebesgue measure, either if the vector is zero, there is a KAM torus or if the vector is not zero, there is no KAM torus in some domain. As an application, we show that the nonlinear Schrodinger equation with a zero among normal frequencies possesses many quasi-periodic solutions.

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ : R n × T d → R n × T d : φ ( z θ ) = ( z + ϕ ( z ) + h ( z , θ ) + ϵ f ( z , θ ) θ + ω ) , where ϕ : R n → R n is a homogeneous function of degree l with l ≥ 2 and h = O ( | z | l + 1 ) . We obtain the following results: (a) For n = 1 , l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Russmann's non-resonant condition. (b) For n = 1 , and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition. (ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition; (iii) l = 2 , first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition; (iv) l > 2 , first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition. (c) In the case n > 1 , if first order average belongs to the interior of the range of ϕ, S p e c ( D ϕ ) ∩ i R = ∅ , first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus)., 34 pages. The material of Section 4 (included to achieve a self-contained presentation) almost coincides with that of Section 4 in arXiv:1612.07653

We study a standard two-parameter family of area-preserving torus diffeomorphisms, known in theoretical physics as the kicked Harper model, by a combination of topological arguments and KAM-theory. We concentrate on the structure of the parameter sets where the rotation set has empty and non-empty interior, respectively, and describe their qualitative properties and scaling behaviour both for small and large parameters. This confirms numerical observations about the onset of diffusion in the physics literature. As a byproduct, we obtain the continuity of the rotation set within the class of Hamiltonian torus homeomorphisms., New version, incorporating referee's remarks and adding a preliminary section to improve readability

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrodinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrodinger systems on two dimensional torus.

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators \begin{document}$ \begin{equation*} \ddot{x}+\lambda^{2}x = \epsilon f(\omega t,x), \end{equation*} $\end{document} where \begin{document}$ \lambda \in \mathcal{O} $\end{document} , a closed interval not containing zero, the forcing term \begin{document}$ f $\end{document} is real analytic, and the frequency vector \begin{document}$ \omega \in \mathbb{R}^d \, (d \geq 2) $\end{document} is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency \begin{document}$ \omega\in\mathbb{R}^{d}, $\end{document} which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ≥ m ≥ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and * sufficiently small, there exists a Cantor set with almost full Lebesgue measure such that responsive tori exist for each if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.

Given l > 2 ν > 2 d ≥ 4 , we prove the persistence of a Cantor–family of KAM tori of measure O ( e 1 / 2 − ν / l ) for any non–degenerate nearly integrable Hamiltonian system of class C l ( D × T d ) , where ν − 1 is the Diophantine power of the frequencies of the persistent KAM tori and D ⊂ R d is a bounded domain, provided that the size e of the perturbation is sufficiently small. This extends a result by D. Salamon in [25] according to which we do have the persistence of a single KAM torus in the same framework. Moreover, as for the persistence of a single torus, the regularity assumption is essentially optimal.

In this paper, one dimensional nonlinear wave equations $$\begin{aligned} u_{tt}-u_{xx}+M_{\xi _{2}}u +\varepsilon f(\omega _{1} t, x, u)=0 \end{aligned}$$ with Dirichlet boundery conditions are considered, where $$M_{\xi _{2}}$$ is a real Fourier multiplier, f is a real analytic function with $$f(\omega _{1} t, -x, -u)=-f(\omega _{1} t, x, u)$$ and the forced frequencies $$\omega _1=(1,\alpha )$$ are Liouvillean. We obtain a family of $$C^{\infty }$$ smooth, bounded non-response solutions with Liouvillean forced frequencies. This is based on an infinite dimensional KAM theorem with angle-dependent normal form.

By constructing an infinite dimensional KAM theorem of the normal frequencies being dense at finite-point, we show that some shallow water equations such as Benjamin-Bona-Mahony equation and the generalized $d$-Dim. Pochhammer-Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions which are quasi-periodic in time.

In this paper, the fractional nonlinear Schrodinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractio...

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations.

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange--d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given., Comment: 27 pages, 9 figures

We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive.

In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency.

In the present paper, without Russmann's nondegenderate condition we show the persistence of KAM invariant tori. Using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three-body problems, especially, for Lagrange equilibrium position L2 with certain degeneracy.

We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

In this paper, two dimensional modified Boussinesq equation $$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$ under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.

We prove quantitative statistical stability results for a large class of small \begin{document}$ C^{0} $\end{document} perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof., 33 pages, 1 figure

In this chapter we investigate the dynamics of classical nonlinear Hamiltonian systems, which—a priori—are examples of continuous dynamical systems. As in the discrete case (see examples in Chap. 2), we are interested in the classification of their dynamics. After a short review of the basic concepts of Hamiltonian mechanics, we define integrability (and therewith regular motion) in Sect. 3.4. The non-integrability property is then discussed in Sect. 3.5. The addition of small non-integrable parts to the Hamiltonian function (Sects. 3.6.1 and 3.7) leads us to the formal theory of canonical perturbations, which turns out to be a highly valuable technique for the treatment of systems with one degree of freedom and shows profound difficulties when applied to realistic systems with more degrees of freedom. We will interpret these problems as the seeds of chaotic motion in general. A key result for the understanding of the transition from regular to chaotic motion is the KAM theorem (Sect. 3.7.4), which assures the stability in nonlinear systems that are not integrable but behave approximately like them. Within the framework of the surface of section technique, chaotic motion is discussed from a phenomenological point of view in Sect. 3.8. More quantitative measures of local and global chaos are finally presented in Sect. 3.9.

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation, $${\rm{i}}u_t-u_{xx}+{\rm{i}}(|u|^4u)_x=0, \;\; x\in\mathbb{T}.$$ The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)].

In this paper, we consider the existence of quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs near the equilibrium point for most parameter values.

This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincare methods. An algorithm is presented for constructing them in the form of series (fractional degrees of a small parameter), which is implemented using classical perturbation theory based on the theory of canonical transformations of Hamiltonian systems. The problem of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The results obtained are applied to the problem of subharmonic oscillations of a pendulum placed on a moving platform in a homogeneous gravitational field. The platform rotates with constant angular velocity about a vertical passing through the suspension point of the pendulum, and simultaneously executes harmonic small-amplitude oscillations along the vertical. Families of subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability is solved.

In this paper, we consider a class of quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators, and study Stoker's problem (the existence of response...

In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [ 3 ] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr \begin{document}$ \ddot{\mbox{o}} $\end{document} dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent structures (LCSs). For the time-dependent perturbation problems, we analyze in detail the development of transport barriers that play an important role in the transport process. Firstly, the analytical study of KAM theory is adopted to explain the transport and mixing phenomenon in measured and simulated airfoil flow. Then, the flow topology of unsteady flow behind an airfoil is investigated for the low Reynolds number problem. Simulations are carried out based on a particular Finite-Time Lyapunov Exponent (FTLE) technique for the detection of invariant manifolds of the hyperbolic trajectories. Besides, the Characteristic Base Split (CBS) scheme combined with a dual time stepping technique is utilized to simulate such transient flow problems. Thus, in the course of the current research, the role of the velocity phase plot during vortex formation is explored that is highly periodic and resulted in the formation of a stable pattern of manifolds and invariant tori. Hence, the proposed study encouraged the new picture of the vortex shedding and flow separation process. As a conclusion, our results give a better understanding of invariant tori control transport phenomena that will lead to a new heuristic for unsteady flows.

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system \begin{document}$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 & = & 0 \\ &&\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 & = & 0 \end{array} \right. $\end{document} under periodic boundary conditions, where \begin{document}$ 0 \begin{document}$ 0 are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

We consider the equation \begin{document}$\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document} where \begin{document}$ f $\end{document} is sufficiently regular, radially symmetric in \begin{document}$ x $\end{document} , and \begin{document}$ f(\cdot,0)\equiv 0 $\end{document} . We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in \begin{document}$ y $\end{document} and decaying as \begin{document}$ |x|\to\infty $\end{document} uniformly in \begin{document}$ y $\end{document} . Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of \begin{document}$ f_u(x,0) $\end{document} and \begin{document}$ f_{uu}(x,0) $\end{document} , and is independent of higher-order terms in the Taylor expansion of \begin{document}$ f(x,\cdot) $\end{document} . In particular, our results apply to some quadratic nonlinearities.

This paper is concerned with the 1-dimensional quintic nonlinear Schrodinger equations with real valued \begin{document}$ C^{\infty} $\end{document} -smooth given potential \begin{document}$ \sqrt{-1}u_{t} = u_{xx}-V(x)u-|u|^4u $\end{document} subject to Dirichlet boundary conditions. By means of normal form theory and an infinite-dimensional Kolmogorov-Arnold-Moser (KAM, for short) theorem, it is proved that the above equation admits a family of elliptic tori where lies small amplitude quasi-periodic solutions with two frequencies of high modes.

We prove that a $$C^k$$, $$k\ge 2$$ pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

We consider the 2D Schrodinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasi-modes) of the Schrodinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp.

We prove an infinite-dimensional KAM theorem for a Hamiltonian system with sublinear growth frequencies at infinity. As an application, we prove the reducibility of the linear fractional Schr\"odinger equation with quasi-periodic time-dependent forcing., Comment: 35pages

In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: $$\begin{aligned} \begin{aligned} y_{tt}(t,x)=\mu y_{xxxx}+y_{xx}+\left( y^{3}+\varepsilon f(\omega t,x)\right) _{xx},\,\, x\in [0, \pi ],\,\, \mu >0, \end{aligned} \end{aligned}$$subject to the hinged boundary conditions $$\begin{aligned} \begin{aligned} y(t,0)=y(t,\pi )=y_{xx}(t,0)=y_{xx}(t,\pi )=0, \end{aligned} \end{aligned}$$where $$\omega =(1,\alpha )$$ with $$\alpha $$ being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allow $$\alpha $$ to be any irrational numbers, and thus the frequency $$\omega =(1,\alpha )$$ is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine.

Quasi-periodic high order degenerate bifurcation theories have been well established, but works which are related to almost-periodic bifurcations seem to be very few. In this paper, we consider the...

The Lagrange case in the rigid body dynamics is completely integrable, with a family of invariant tori supporting periodic or quasi-periodic motion. We study perturbations of this case. In the non-periodic case the KAM theory predicts no changes in the evolution. In the periodic cases one expects existence of isolated limit cycles, which can be studied using Melnikov functions. We find these cycles in the case when the invariant torus is close to so-called critical circle. The presented approach is analogous to our previous analysis of the Hess–Appelrot case. In particular, we show that the number of created limit cycles in the latter case is uniformly bounded.

This paper is concerned with the derivative nonlinear Schrodinger equation with periodic boundary conditions i u t + u x x + i ( f ( x , u , u ¯ ) ) x = 0 , x ∈ T : = R / 2 π Z , where f is an analytic function of the form f ( x , u , u ¯ ) = μ | u | 2 u + f ≥ 4 ( x , u , u ¯ ) , 0 ≠ μ ∈ R , and f ≥ 4 ( x , u , u ¯ ) denotes terms of order at least four in u , u ¯ . We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields.

This paper is concerned with a nonlinearly modified Kawahara equation with periodic boundary conditions $$ u_{t}+u_{xxx}-u_{5x}+\alpha u^{2}u_{x}=0. $$ Based on an abstract infinite-dimensional KAM theorem for unbounded perturbation vector fields and partial Birkhoff normal form, we prove the existence of n-dimensional invariant tori for the equation.

We consider a general canonical form of a multi-scale Hamiltonian system near a family of unperturbed lower dimensional, quasi-periodic, invariant tori, in which tangential and normal frequencies can admit equal or different scales. Extending works of Broer and Zhao (2017 Celest. Mech. Dyn. Astron. 127 95–119); Xu et al (2017 Ann. Henri Poincare 18 53–83), we show the persistence of the majority of these lower dimensional tori when the normal matrices are non-degenerate and a Melnikov non-resonant condition among certain tangential and normal frequencies are satisfied when they admit equal scales. Such a general persistence result of lower dimensional tori allows a broader range of applications to stability problems arising in celestial mechanics.

By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form $x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots $ , $f:(0,+\infty )\to (0,+\infty )$ , f is sufficiently smooth and the initial conditions are $x_{-1}, x _{0}\in (0,+\infty )$ . We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient $\alpha _{1}$ . Then we apply the results to several difference equations.

We give a survey on local and semi-local bifurcations of divergence-free vector fields. These differ for low dimensions from generic' bifurcations of structure-less dissipative' vector fields, up to a dimension-threshold that grows with the co-dimension of the bifurcation.

It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. In this paper we outline a new approach for the analysis of Hamiltonian systems with divided phase space. These systems are approximated by a sequence of Hamiltonian systems having an increasing (but finite) number of KAM islands. The islands in the approximating systems are sub-islands of the islands in the initial system with an infinite number of KAM-islands. We apply this approach to two-dimensional billiards and demonstrate that it works. In particular the statistical characteristics of the approximating systems tend to the ones for the whole system when the number of islands in the approximating systems grows. Therefore our approach opens up a new way for numerical and analytical studies of the dynamics of Hamiltonian systems with divided phase space., Comment: 8 pages (RevTex), 9 figures; version as accepted by Experimental Mathematics