Analytic Dynamical Systems Research Articles

Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction. In this paper, we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component. In addition, we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.

About Linearization of Infinite-Dimensional Hamiltonian Systems

This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We first define the subset of elements which are formally symplectically conjugacted to a (formal) Birkhoff normal form. We prove that if the quadratic Hamiltonian satisfies a Diophantine-like condition and if such a perturbation is formally symplectically conjugated to the quadratic Hamiltonian, then it is also analytically symplectically conjugated to it. Of course what is an analytic symplectic change of variables depends strongly on the choice of the phase space. Here we work on periodic functions with Gevrey regularity.

Over- and Under-Approximations of Reachable Sets With Series Representations of Evolution Functions

In this article, we investigate both over- and under-approximations of reachable sets for analytic autonomous dynamical systems beyond polynomial dynamics. We start with the concept of evolution function, whose subzero-level set can be used to describe reachable set, and find a series representation of the evolution function with its Lie derivatives. Afterwards, based on the partial sums of this series, two different methodologies are introduced to compute over- and under-approximations of reachable sets, using numerical quantifier elimination for the semi-algebraic constraints and remainder estimation of the partial sum, respectively. Some benchmarks are given, including an eight-dimensional nonpolynomial quadrotor model, to show the advantages of our computational methods over some existing methods in the literature. Especially, our methods also work for nonconvex initial sets.

A remark on the intersections of subanalytic leaves

We discuss a new sufficient condition - weaker than the usual transversality condition - for the intersection of two subanalytic leaves to be smooth. It involves the tangent cone of the intersection and, as typically non-transversal, it is of interest in analytic geometry or dynamical systems. We also prove an identity principle for real analytic manifolds and subanalytic functions.

Theory of the norm-induced metric in atmospheric dynamics

Abstract. We suggest that some metrics for quantifying distances in phase space are based on linearized flows about unrealistic reference states and hence may not be applicable to atmospheric flows. A new approach of defining a norm-induced metric based on the total energy norm is proposed. The approach is based on the rigorous mathematics of normed vector spaces and the law of energy conservation in physics. It involves the innovative construction of the phase space so that energy (or a certain physical invariant) takes the form of a Euclidean norm. The metric can be applied to both linear and nonlinear flows and for small and large separations in phase space. The new metric is derived for models of various levels of sophistication: the 2-D barotropic model, the shallow-water model and the 3-D dry, compressible atmosphere in different vertical coordinates. Numerical calculations of the new metric are illustrated with analytic dynamical systems as well as with global reanalysis data. The differences from a commonly used metric and the potential for application in ensemble prediction, error growth analysis and predictability studies are discussed.

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.

On metric properties of limit sets of contractive analytic non-Archimedean dynamical systems

Let K be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean absolute value |⋅|. We study metric properties of the limit set Λ of a semigroup G generated by a finite set of contractive analytic functions on O={z∈K||z|⩽1}. We prove that the limit set Λ of G is uniformly perfect if the derivative of each generating function of G does not vanish on O. Furthermore, we show that if each coefficient of the generating functions is in the field Qp of p-adic numbers, or the limit set Λ satisfies the strong open set condition, then Λ has the doubling property. This yields that the limit set Λ is quasisymmetrically equivalent to the space Z2 of 2-adic integers. We also give a counterexample to show that not all limit sets have the doubling property. The Berkovich space is introduced to study the limit set Λ, and we prove that the limit set Λ has a positive capacity in the Berkovich space which yields that there exists an equilibrium measure μ whose support is contained in the limit set Λ. We also show that if the semigroup is generated by a countable set of contractive analytic functions, then its limit set Λ can be non-compact. However, if coefficients of the generating functions lie in Qp, then the limit set Λ is compact.

Analytic integrable systems: Analytic normalization and embedding flows

In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism F(x)=Bx+f(x) in (Cn,0) with B having eigenvalues not modulus 1 and f(x)=O(|x|2) is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system x˙=Ax+f(x) in (Cn,0) with A having nonzero eigenvalues and f(x)=O(|x|2) is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moserʼs one in J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956) 673–692 and of Poincaréʼs one in H. Poincaré, Sur lʼintégration des équations différentielles du premier order et du premier degré, II, Rend. Circ. Mat. Palermo 11 (1897) 193–239. These results also improve the ones in Xiang Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differential Equations 244 (2008) 1080–1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in N.T. Zung, Convergence versus integrability in Poincaré–Dulac normal form, Math. Res. Lett. 9 (2002) 217–228 in the way that our paper presents the concrete expression of the normal form in a restricted case.

Invariant manifolds for analytic dynamical systems over ultrametric fields

We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric Banach space over K, an analytic diffeomorphism f:M→M, and a fixed point p of f. Under suitable assumptions on the tangent map Tp(f), we construct a centre–stable manifold, a centre manifold, respectively, an a-stable manifold around p, for a given real number a∈]0,1].

A Euclidean Skolem–Mahler–Lech–Chabauty Method

Using the theory of o-minimality we show that the p-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if f1, . . . , fn is a finite sequence of real analytic functions fi : (−1, 1)→ (−1, 1) for which fi(0) = 0 and | f ′ i (0)| ≤ 1 (possibly zero), a = (a1, . . . , an) is an n-tuple of real numbers close enough to the origin and H(x1, . . . , xn) is a real analytic function of n variables, then the set {m ∈N : H( f ◦m 1 (a1), . . . , f ◦m n (an)) = 0} is either all of N, all of the odd numbers, all of the even numbers, or is finite.

Small quotients in Euclidean algorithms

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M→∞ were provided by Hensley (J. Number Theory 40:336–358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166–172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371–385, 1987) and Vallée (J. Number Theory 72:183–235, 1998) considered the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237–255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M 0; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336–358, 1992) about the Hausdorff dimension σ M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531–570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331–386, 2005).

Poincaré theorems for random dynamical systems

In this paper, we establish analytic conjugacy theorems of Poincare type and Poincare–Dulac type for analytic random dynamical systems based on their Lyapunov exponents.

Truncated Lie groups and almost Klein models

Abstract We consider a real analytic dynamical system G×M→M with nonempty fixed point subset M G. Using symmetries of G×M→M, we give some conditions which imply the existence of transitive Lie transformation group with G as isotropy subgroup.

Attractor and Repeller Points for a Several-Variable Analytic Dynamical System in a Non-Archimedean Setting

We study discrete several-variable analytic dynamical systems over a complete non-Archimedean field with a nontrivial valuation and give sufficient conditions for a fixed point of the system to be an attractor, a repeller, or an indifferent point.

Actions toriques et groupes d'automorphismes de singularités des systèmes dynamiques intégrables

We show that in the neighborhood of each “finite type” singular orbit of a real analytic integrable dynamical system (Hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this orbit. We also show that the local automorphism group of the system near such an orbit is essentially Abelian. To cite this article: N.T. Zung, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

An example of a minimal torus of weak mixing for nearly integrable Hamiltonian systems

In this paper, we give an analytic dynamical system having a minimal torus of weak mixing, so that it leads to a partial answer to the Kolmogorov problem on the existence of invariant tori (mixing).

On invariant manifolds of nonlinear discrete-time input-driven dynamical systems

The present research work proposes a new approach to the problem of finding invariant manifolds for nonlinear discrete-time input-driven real analytic dynamical systems. The formulation of the problem is conveniently realized through a system of nonlinear first-order functional equations (NFEs) and a rather general set of necessary and sufficient conditions for solvability is derived. The solution of the aforementioned system of NFEs is proven to be a locally analytic invariant manifold that under certain conditions coincides with the stable or unstable manifold of the system, and which can be easily computed with the aid of a symbolic software package such as MAPLE. Finally, the proposed method is applied to an enzymatic bioreactor dynamics example.

Small denominators in complex p-adic dynamics

We study the problem of small denominators in the field of complex p-adic numbers C p . We prove that, in fact, we can always obtain the estimate from below D n ( x) ≥ CA n , C( x), A( x) > 0, for the denominator D n ( x) which is used in the construction of a congugate map for a dynamical system f having the derivative x = f′( a) in the fixed point a. Moreover, we find (via a long chain of technical computations) forms of coefficients C( x) and A( x). These are complicated functions of x which depend on the relative position of x and p sth roots of unity. Our result can be used to find radius of convergence for conjugate maps for C p -analytic dynamical systems at neutral fixed points (or cycles).

Limit at resonances of linearizations of some complex analytic dynamical systems

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ and of the semi-standard map.We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map.

Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems

The present research work proposes a new approach to the problem of finding invariant manifolds for nonlinear real analytic dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear partial differential equations (PDEs) and a rather general set of conditions for solvability is derived using Lyapunov's auxiliary theorem. The solution of the aforementioned system of PDEs is proven to be a locally analytic invariant manifold that under certain conditions coincides with the stable or unstable manifold, and which can be easily computed with the aid of a symbolic software package.