相关论文: Beltrami fields exhibit knots and chaos almost sur…

We characterize the thermodynamical equilibrium states of axisymmetric Euler-Beltrami flows. They have the form of coherent structures presenting one or several cells. We find the relevant control parameters and derive the corresponding…

In present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with…

Bohr chaoticity is a topological notion of dynamical complexity defined through non-orthogonality to all non-trivial weights. It is strictly stronger than positivity of topological entropy and also has strong consequences for the…

Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$ on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued trigonometric polynomial of degree $L$ whose coefficients are independent standard…

It has recently been speculated that statistical properties of chaos may be captured by weighted sums over unstable invariant tori embedded in the chaotic attractor of hyperchaotic dissipative systems; analogous to sums over periodic orbits…

Deformations of the heterotic superpotential give rise to a topological holomorphic theory with similarities to both Kodaira-Spencer gravity and holomorphic Chern-Simons theory. Although the action is cubic, it is only quadratic in the…

In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove several results of instability when the frequency of the…

In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the…

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant…

The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the…

We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action-angle-angle maps with degeneracy, satisfying the intersection property. Such degenerate action-angle-angle maps arise…

We consider the class of Beltrami fields (eigenfields of the curl operator) on three-dimensional Riemannian solid tori: such vector fields arise as steady incompressible inviscid fluids and plasmas. Using techniques from contact geometry,…

This paper investigates the effect of random perturbations, in particular multiplicative noise, on the integrable structure of Hamiltonian systems, with a particular focus on KAM theory for stochastic Hamiltonian dynamics. We prove that,…

In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows…

We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct steady nonsingular solutions to the Euler equations on a Riemannian $S^3$ whose flowlines trace out closed curves of all…

The core of this paper is a novel group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin, which allows for a more systematic classification and algorithmic construction of Beltrami…

Spatio-temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio-temporally periodic solutions. The periodic orbit theory yields the global averages characterizing the chaotic…

This thesis focuses on two research areas: non-relativistic field theories and complexity. In the first part we review the general classification of the trace anomaly for 2+1 dimensional field theories coupled to a Newton-Cartan background…

We establish a criterion for the existence of a topological horseshoe in a class of planar systems generated by periodic switching between two subsystems, each admitting a family of closed orbits, where the mechanism for chaos arises from…

The perturbative expansion of Chern-Simons gauge theory leads to invariants of knots and links, the finite type invariants or Vassiliev invariants. It has been proven that at any order in perturbation theory the resulting expression is an…