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Discrete & Continuous Dynamical Systems - S

2010 , Volume 3 , Issue 4

Special issue
on KAM theory and its applications

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Preface
Henk W. Broer, Heinz Hanβmann and Mikhail B. Sevryuk
2010, 3(4): i-iii doi: 10.3934/dcdss.2010.3.4i +[Abstract](345) +[PDF](68.6KB)
Abstract:
This issue of Discrete and Continuous Dynamical Systems, Series S, is a collection of papers in the area of KAM theory and its applications.
   KAM theory, named after its founders A.N. Kolmogorov [1, 2], V.I. Arnol'd [3, 4, 5], and J.K. Moser [6, 7], is a major part of Dynamical Systems Theory and Qualitative Theory of Differential Equations, both ordinary and partial (the descriptive term "KAM theory'' was coined in a 1968 Russian preprint by F.M. Izrailev and B.V. Chirikov). It studies the typical occurrence of quasi-periodicity in non-integrable dynamical systems and is often regarded as one of the foremost branches of modern Nonlinear Dynamics.

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A geometric fractional monodromy theorem
Henk Broer, Konstantinos Efstathiou and Olga Lukina
2010, 3(4): 517-532 doi: 10.3934/dcdss.2010.3.517 +[Abstract](485) +[PDF](760.8KB)
Abstract:
We prove the existence of fractional monodromy for two degree of freedom integrable Hamiltonian systems with one-parameter families of curled tori under certain general conditions. We describe the action coordinates of such systems near curled tori and we show how to compute fractional monodromy using the notion of the rotation number.
Some KAM applications to Celestial Mechanics
Alessandra Celletti
2010, 3(4): 533-544 doi: 10.3934/dcdss.2010.3.533 +[Abstract](433) +[PDF](279.4KB)
Abstract:
The existence of invariant tori in Celestial Mechanics has been widely investigated through implementations of the Kolmogorov-Arnold-Moser (KAM) theory. We provide an introduction to some results on the existence of maximal and low-dimensional, rotational and librational tori for models of Celestial Mechanics: from the spin--orbit problem to the three-body and planetary models. We also briefly review a result on dissipative invariant attractors for the spin-orbit problem, whose existence is proven through a dissipative KAM theorem.
Properly-degenerate KAM theory (following V. I. Arnold)
Luigi Chierchia and Gabriella Pinzari
2010, 3(4): 545-578 doi: 10.3934/dcdss.2010.3.545 +[Abstract](453) +[PDF](526.6KB)
Abstract:
Arnold's "Fundamental Theorem'' on properly-degenerate systems [3, Chapter IV] is revisited and improved with particular attention to the relation between the perturbative parameters and to the measure of the Kolmogorov set. Relations with the planetary many-body problem are shortly discussed.
Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$
Hongzi Cong, Jianjun Liu and Xiaoping Yuan
2010, 3(4): 579-600 doi: 10.3934/dcdss.2010.3.579 +[Abstract](442) +[PDF](314.2KB)
Abstract:
In this paper we prove that there is a Cantorian branch of 2-dimensional KAM invariant tori for the complex Ginzburg-Landau equation with the nonlinearity $|u|^{2p}u,\ p\geq1$.
Convergence radius in the Poincaré-Siegel problem
Antonio Giorgilli and Stefano Marmi
2010, 3(4): 601-621 doi: 10.3934/dcdss.2010.3.601 +[Abstract](307) +[PDF](306.2KB)
Abstract:
We reconsider the Poincaré-Siegel center problem, namely the problem of conjugating an analytic system of differential equations in the neighbourhood of an equilibrium to its linear part $\Lambda=\diag(\lambda_1,\ldots,\lambda_n)$. If the linear part is non--resonant we show that the convergence radius $r$ of the conjugating transformation satisfies $\log r(\Lambda )\ge -CB+C'$ with $C=1$ and a constant $C'$ not depending on $\Lambda$. The convergence condition is the same as the Bruno condition since $B = -\sum_{r\ge 1}2^{-r}\log\alpha_{2^r-1}$, where $\alpha_r = \min_{0\le s\le r} \beta_r$ for $r\ge 0$ and $\beta_r = \min_{j=1,\ldots,n}\ \ \ \min_{k\inZ_+^n,|k|=r+1}$ | < k, $\lambda$ > $- \lambda_j$|. Our lower bound improves the previous results for $n\gt 1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for the discrete time version of the center problem when $n=1$, namely the linearization problem for germs of holomorphic maps when the eigenvalue of the fixed point is on the unit circle.
Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
Xuemei Li and Rafael de la Llave
2010, 3(4): 623-641 doi: 10.3934/dcdss.2010.3.623 +[Abstract](363) +[PDF](309.3KB)
Abstract:
In KAM theory and other areas of analysis, one is often led to consider sums of functions defined in decreasing domains. A question of interest is whether the limit function is differentiable or not.
   We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.
   We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.
Gevrey normal form and effective stability of Lagrangian tori
Todor Mitev and Georgi Popov
2010, 3(4): 643-666 doi: 10.3934/dcdss.2010.3.643 +[Abstract](463) +[PDF](348.2KB)
Abstract:
A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.
Finite smooth normal forms and integrability of local families of vector fields
Vincent Naudot and Jiazhong Yang
2010, 3(4): 667-682 doi: 10.3934/dcdss.2010.3.667 +[Abstract](399) +[PDF](273.0KB)
Abstract:
In this paper we study a class of smooth vector fields which depend on small parameters and their eigenvalues may admit certain resonances. We shall derive the polynomial normal forms for such systems under $C^k$ conjugacy, where $k$ can be arbitrarily large. When the smoothness of normalization is less required, we can even reduce these systems to their quasi-linearizable normal forms under $C^{k_0}$ conjugacy, where $k_0$ is good enough to preserve certain qualitative properties of the original systems while the normal forms are as convenient as the linearized ones in applications. Concerning the normalization procedure, we prove that the transformation can be expressed in terms of Logarithmic Mourtada Type (LMT) functions, which makes both qualitative and quantitative analysis possible.
KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character
Helmut Rüssmann
2010, 3(4): 683-718 doi: 10.3934/dcdss.2010.3.683 +[Abstract](516) +[PDF](455.0KB)
Abstract:
In this paper we present a new variant of the KAM theory, containing an artificial parameter $q$, $0 < q < 1$, which makes the steps of the KAM iteration infinitely small in the limit $q$↗$1$. This KAM procedure can be compared for $q<1$ with a Riemann sum which tends, for $q$↗$1$, to the corresponding Riemann integral. As a consequence this limit has all advantages of an integration process compared with its preliminary stages: Simplification of the conditions for the involved parameters and global linearization which therefore improves numerical results. But there is a difference from integrals: The KAM iteration itself works only for $q<1$, however, $q$ can be chosen as near to $1$ as we want and the limit $q$↗$1$ exists for all involved parameters. Hence, the mentioned advantages remain mainly preserved. The new technique of estimation differs completely from all what has appeared about KAM theory in the literature up to date. Only Kolmogorov's idea of local linearization and Moser's modifying terms are left. The basic idea is to use the polynomial structure in order to transfer, at least partially, the whole KAM procedure outside of the original domain of definition of the given dynamical system.
A parametrised version of Moser's modifying terms theorem
Florian Wagener
2010, 3(4): 719-768 doi: 10.3934/dcdss.2010.3.719 +[Abstract](429) +[PDF](727.9KB)
Abstract:
A sharpened version of Moser's 'modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is 'structural' and can be applied to dissipative, Hamiltonian, and symmetric vector fields; moreover, we give variants of the result for real analytic, Gevrey regular ultradifferentiable and finitely differentiable vector fields. In the first two cases, the conjugacy constructed in the theorem is shown to be Gevrey smooth in the sense of Whitney on the set of parameters that satisfy a "Diophantine'' non-resonance condition.

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