KAM Theory Research Articles

Study on the finite element method of Hamiltonian system with chaos

By comparing with symplectic different methods, the quadratic element is an approximately symplectic method which can keep high accuracy approximate of symplectic structure for Hamiltonian chaos, and it is also energy conservative when there have chaos phenomenon. We use the quadratic finite element method to solve the H[Formula: see text]non–Heiles system, and this method was never used before. Combining with Poincar[Formula: see text] section, when we increase the energy of the systems, KAM tori are broken and the motion from regular to chaotic. Without chaos, three kinds of methods to calculate the Poincar[Formula: see text] section point numbers are the same, and the numbers are different with chaos. In long-term calculation, the finite element method can better keep dynamic characteristics of conservative system with chaotic motion.

Complex behavior of the density in composite quantum systems

In this paper, we study how the probability of presence of a particle is distributed between the two parts of a composite fermionic system. We uncover that the difference of probability depends on the energy in a striking way and show the pattern of this distribution. We discuss the main features of the latter and explain analytically those that we understand. In particular, we prove that it is a non-perturbative property and we find out a large/small coupling constant duality. We also find and study features that may connect our problem with certain aspects of non linear classical dynamics, like the existence of resonances and sensitive dependence on the state of the system. We show that the latter has indeed a similar origin than in classical mechanics: the appearance of small denominators in the perturbative series. Inspired by the proof of KAM theorem, we are able to deal with this problem by introducing a cut-off in energies that eliminates these small denominators. We also formulate some conjectures that we are not able to prove at present but can be supported by numerical experiments.

On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range

In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighbourhood of a given I0 for nearly integrable and analytic Hamiltonian system H(I,θ)=H0(I)+ϵH1(I,θ) of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending only on initial I0.

Positive measure of KAM tori for finitely differentiable Hamiltonians

Consider an integer $n \geq 2$ and real numbers $\tau>n-1$ and $l>2(\tau+1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^l$. Under the stronger assumption that the system is a $C^{l+\tau}$ perturbation of an analytic integrable system, P{\o}schel proved the persistence of a set of positive measure of Diophantine tori. We improve the last result by showing it is sufficient for the perturbation to be of class $C^{l}$ and the integrable part to be of class $C^{l+2}$.

Quasi-Periodic Solution of Nonlinear Beam Equation on $${\mathbb T}^d$$ with Forced Frequencies

In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition: utt+Δx2u+Mξu+f(ω¯t,u)=0,u=u(t,x),t∈R,x∈Td,d≥2where Mξ is a real Fourier multiplier, f=f(θ¯,u) is a real analytic function with respect to (θ¯,u), and f(θ¯,u)=O(|u|2). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori.

The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain [9].

Normal resonances in a double Hopf bifurcation

We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and kam theory are being used.

Kolmogorov–Arnold–Moser theorem for nonlinear beam equations with almost-periodic forcing

In this study, we construct a Kolmogorov–Arnold–Moser theorem regarding the existence of almost-periodic solutions for some infinitely dimensional Hamiltonian systems with almost-periodic forcing. This theorem is applied to an almost-periodically forced nonlinear beam equation with periodic boundary conditions to obtain the almost-periodic solutionsutt+(−∂xx+μ)2u+ψ(ωt)f(u)=0,μ>0,t∈R,x∈R, where ψ(ωt) is real analytic and almost periodic on t and the nonlinearity f is a real-analytic function near u=0 with f(0)=f′(0)=0.

Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

In this paper, one dimensional nonlinear wave equation \begin{document}$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $\end{document} with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations

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.

Long time stability of KAM tori for the generalized Boussinesq equation

In this paper, we prove that the KAM tori for the generalized Boussinesq equation with the external parameters, which generates an infinite dimensional Hamiltonian system, are long-time stable. Precisely, we prove that the solutions with initial datum in the δ-neighborhood of KAM torus still stay close to the KAM torus for |t|≤δ−M with M≥1. The proof is based on constructing a partial normal form of higher order and showing that the p-tame property for the Hamiltonian vector field persists after the changes of variables of the KAM scheme and under normal form iterative procedure.

A Window into the World of KAM Theory

SummaryWhat happens when you periodically force a nonlinear oscillator in the absence of damping? For linear oscillators, such as the mass-on-a-spring model typically encountered in that first differential equations course, the behavior of the forced system is easily and well understood. In this article, a simple mechanical model is used to illuminate the power and beauty of the theory of Kolmogorov, Arnol’d, and Moser. Known as KAM theory, this profound 20th century mathematical achievement answers the question posed above. Model simulations illustrating the complex coexistence of regular and chaotic motions are presented. Additionally, KAM theory is placed within its historical context, namely, the quest to determine the stability of the solar system.

KAMTheorem with Normal Frequencies of FiniteLimit‐Pointsfor Some Shallow Water Equations

By constructing an infinite‐dimensional KAM theorem of the normal frequencies being dense at a finite point, we show that some shallow water equations such as the Benjamin‐Bona‐Mahony equation and the generalizedd‐dimensional Pochhammer‐Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions that are quasi‐periodic in time . © 2020 Wiley Periodicals LLC

Quasi-periodic solutions of fractional nonlinear Schrödinger equation systems

In this paper, the fractional nonlinear Schrödinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractional nonlinear Schrödinger equation system with real Fourier multiplier. By establishing a block diagonal normal form, we prove the existence of a class of Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori of an associated infinite-dimensional dynamic system.

Applications of the Moser’s Twist Theorem to Some Impulsive Differential Equations

By KAM theorem, we prove that all solutions of Duffing equations of periodic coefficients undergoing suitable impulses are bounded for all time and that there are many quasi-periodic solutions clustering at infinity.

KAM theory for the reversible perturbations of 2D linear beam equations

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.

Stability of Weyl semimetals with quasiperiodic disorder

Weyl semimetals are phases of matter with excitations effectively described by massless Dirac fermions. Their critical nature makes unclear the persistence of such phase in presence of disorder. We present a theorem ensuring the stability of the semimetallic phase in presence of weak quasiperiodic disorder. The proof relies on the subtle interplay of the relativistic Quantum Field Theory description combined with number theoretical properties used in KAM theory.

Whiskered KAM tori of conformally symplectic systems

We investigate the existence of whiskered tori in some dissipative systems, called conformally symplectic systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family f μ of conformally symplectic maps which depends on a drift parameter μ.

Reducible KAM Tori for the Degasperis\u2013Procesi Equation

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation.

Cauchy distributions for the integrable standard map

We consider the integrable (zero perturbation) two–dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for KAM tori.