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Nguyen Tien Zung

Publications and source records attributed to Nguyen Tien Zung.

At least 19 recordsLinked to original sources
arXiv2025

Normalization of rationally integrable systems

In two previous papers we showed that any analytically integrable vector field admits a local analytic Poincaré-Birkhoff normalization in the neighborhood of a singular point. The aim of this paper is to extend this analytic normalization result to the case of rationally integrable systems, where the first integrals and commuting vector fields are not required to be analytic, but just rational (i.e., quotients of analytic functions or vector fields by analytic functions).

math.DS↗
arXiv2020

Epidemic Dynamics via Wavelet Theory and Machine Learning, with Applications to Covid-19

We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number $I(t)$ of infectious individuals at time $t$ in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the John Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida.

math.OC↗
arXiv2019

Local normal forms of dynamical systems with a singular underlying geometric structure

In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $\mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $\mathcal{G}$ is singular but $X$ does not vanish at the origin.

math.DS↗
arXiv2019

Integrable systems in planar robotics

The main purpose of this paper is to investigate commuting flows and integrable systems on the configuration spaces of planar linkages. Our study leads to the definition of a natural volume form on each configuration space of planar linkages, the notion of cross products of integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals of our systems are functions of Bott-Morse type, which may be used to study the topology of configuration spaces. Dedicated to Anatoly T. Fomenko on the occasion of his 75th birthday

math.DS↗
arXiv2019

Normalization of singular contact forms and primitive 1-forms

A differential 1-form $α$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $α\wedge (dα)^n \neq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $α(O) = 0$, is called a \textit{singular contact form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $ω$ in dimension $2n$, i.e. differential 1-forms $γ$ which vanish at a point and such that $dγ= ω$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin \cite{Lychagin-1stOrder1975}, Webster \cite{Webster-1stOrder1987} and Zhitomirskii \cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems \cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.

math.DG↗
arXiv2018

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently big then the local-global convexity principle breaks down, and the base spaces can be globally non-convex even for compact manifolds. As one of surprising examples, we construct a 2-dimensional "integral affine black hole", which is locally convex but for which a straight ray from the center can never escape.

math.SG↗
arXiv2018

Singular fibers of the Gelfand--Cetlin system on $\mathfrak{u}(n)^*$

In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibers can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibers, and give a detailed description of these singular fibers in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibers are degenerate for the Gelfand--Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids, and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.

math.SG↗
arXiv2018

A conceptual approach to the problem of action-angle variables

In this paper we develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions: anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: a) the shortest and most conceptual easy to understand proof of the classical Arnold--Liouville--Mineur theorem; b) basically all known results in the literature about the existence of action-angle variables in various contexts can be recovered in a unifying way, with simple proofs, using our approach; c) new results on action-angle variables in many different contexts, including systems on contact manifolds, systems on presymplectic and Dirac manifolds, action-angle variables near singularities, stochastic systems, and so on. Even when there are no natural action variables, our approach still leads to useful normal forms for dynamical systems, which are not necessarily integrable.

math.DS↗
arXiv2017

Orbital Linearization of Smooth Completely Integrable Vector Fields

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.

math.DS↗
arXiv2017

Second order stochastic differential models for financial markets

Using agent-based modelling, empirical evidence and physical ideas, such as the energy function and the fact that the phase space must have twice the dimension of the configuration space, we argue that the stochastic differential equations which describe the motion of financial prices with respect to real world probability measures should be of second order (and non-Markovian), instead of first order models à la Bachelier--Samuelson. Our theoretical result in stochastic dynamical systems shows that one cannot correctly reduce second order models to first order models by simply forgetting about momenta. We propose some simple second order models, including a stochastic constrained n-oscillator, which can explain many market phenomena, such as boom-bust cycles, stochastic quasi-periodic behavior, and "hot money" going from one market sector to another.

q-fin.MF↗
arXiv2017

Presymplectic convexity and (ir)rational polytopes

In this paper, we extend the Atiyah--Guillemin--Sternberg convexity theorem and Delzant's classification of symplectic toric manifolds to presymplectic manifolds. We also define and study the Morita equivalence of presymplectic toric manifolds and of their corresponding framed momentum polytopes, which may be rational or non-rational. Toric orbifolds, quasifolds and non-commutative toric varieties may be viewed as the quotient of our presymplectic toric manifolds by the kernel isotropy foliation of the presymplectic form.

math.SG↗
arXiv2014

Reduction and integrability of stochastic dynamical systems

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction and integrability. In particular, we show that an SDS which is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS's, and extend the results about the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS's. We also show how integrable SDS's are related to compatible families of integrable Riemannian metrics on manifolds.

math.DS↗
arXiv2014

Geometry of integrable non-Hamiltonian systems

This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal forms and associated torus actions for integrable systems, and the geometry of integrable systems of type $(n,0)$. Most of the results presented in this text are very recent, and some theorems in this text are even original in the sense that they have not been written down explicitly elsewhere.

math.DS↗
arXiv2013

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu structures.

math.DG↗
arXiv2013

Nondegenerate singularities of integrable dynamical systems

We give a natural notion of nondegeneracy for singular points of integrable non-Hamiltonian systems, and show that such nondegenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case, and prove this conjecture for systems of type $(n,0)$, i.e. $n$ commuting smooth vector fields on a $n$-manifold.

math.DS↗
arXiv2013

Geometry of nondegenerate $\mathbb{R}^n$-actions on $n$-manifolds

This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

math.DS↗
arXiv2012

Action-Angle variables on Dirac manifolds

The main purpose of this paper is to show the existence of action-angle variables for integrable Hamiltonian systems on Dirac manifolds under some natural regularity and compactness conditions, using the torus action approach. We show that the Liouville torus actions of general integrable dynamical systems have the structure-preserving property with respect to any underlying geometric structure of the system, and deduce the existence of action-angle variables from this property. We also discover co-affine structures on manifolds as a by-product of our study of action-angle variables. 22 pages.

math.SG↗