Lower-dimensional tori in reversible systems

On a (2n+d)-dimensional manifold M consider a vector field V reversible with respect to an involution G whose fixed point manifold is of dimension n+d. It is conjectured that generically for each 0相似文献   

Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems

For the Hamiltonian systems of KAM type, it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case.     

Dynamical systems on quantum tori Lie algebras

We use quantum tori Lie algebras (QTLA), which are a one-parameter family of sub-algebras ofgl , to describe local and non-local versions of the Toda systems. It turns out that the central charge of QTLA is responsible for the non-locality. There are two regimes in the local systems-conformal for irrational values of the parameter and non-conformal and integrable for its rational values. We also consider infinite-dimensional analogs of rigid tops. Some of these systems give rise to quantized (magneto-)hydrodynamic equations of an ideal fluid on a torus. We also consider infinite dimensional versions of the integrable Euler and Clebsch cases.     

Time-optimal torus theorem and control of spin systems

Given a compact, connected Lie group G with Lie algebra . We discuss time-optimal control of bilinear systems of the form

Harmonic balance/Galerkin method for non-smooth dynamic systems

Models of non-linear systems frequently introduce forces with bounded continuity resulting in non-smooth (even discontinuous) flow. Examples include systems with clearances, backlash, friction, and impulses. Asymptotic methods require smooth (differentiable) flow and are therefore ill-suited for analyzing non-smooth systems. In these cases, the traditional harmonic balance method may be used to obtain approximate periodic solutions, but the method suffers from extremely slow convergence in general. Generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence. These improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution. In particular, forming an infinite series of such functions results in an expansion in the same function space of the exact solution. This expansion converges pointwise to the exact solution and to all derivatives thereof. These improvements are illustrated by example upon re-evaluating a classical single degree-of-freedom model for friction-induced vibration.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Some new advance on the research of stochastic non-smooth systems

Due to the extensive applicability in real life, the non-smooth system with random factors attracted much attention in past two decades. A lot of methods and techniques have been proposed to research these systems by scholars. In this paper, we will summarize some new research advance on the stochastic non-smooth systems. The existing results about the stochastic vibro-impact system, the stochastic friction system, and the stochastic hysteretic system are introduced respectively. Some conclusions and outlook are given at the end.     

Generalized mechanical systems. A mathematical exposition of non-smooth classical Lagrangian mechanics

In this note we shall give a generalization of classical mechanical systems.     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.     

Polynomial integrals of mechanical systems on a torus with a singular potential

The problem on integrability of the equations of motion of a material point on an n-dimensional Euclidean torus under the action of a force field with the potential energy having singularities at a finite number of points is considered. It is assumed that these singularities contain logarithmic coefficients and, consequently, have a more general form in comparison with power features. The potentials having power-type singularities were considered previously by V.V. Kozlov and D.V. Treshchev. In this work, it is proved that the equations of motion in the problem under consideration admit no nontrivial momentum-polynomial first integral with integrable coefficients on this torus.     

Breaking and disappearance of tori

A mechanism is illustrated which can cause a torus to disappear in dissipative differential equations. Three different examples give evidence that a collision with a neighbouring unstable periodic orbit, possibly preceded by a transition into a weakly chaotic attractor, causes the sudden destruction of a torus.     

Multi-frequency tori in wide-aperture lasers

A distributed model of wide-aperture laser based on Maxwell?Bloch equations in onedimensional approximation is considered. It is shown that an increase in the pumping parameter in the system gives rise to a cascade of bifurcations of periodic and quasi-periodic dynamic modes, as a result of which attractors in the form of three-frequency tori can be observed.     

Coexisting structures in 119I

High-spin structures of ¹¹⁹I have been studied by using ¹³C and ¹⁵N induced reactions. In all, fifteen ΔI = 1 or 2 bands belonging to ¹¹⁹I were found. No evidence was found for bands with collective oblate shape, instead, all the observed rotational bands were interpreted to possess a collective prolate shape. A rich tapestry of noncollective states of both negative and positive parity was observed. Based on TRS calculations various configurations at β₂ ≈ 0.17 and γ = 60° were assigned to these states.     

Symmetric bursting behaviour in non-smooth Chua’s circuit

<正>The dynamics of a non-smooth electric circuit with an order gap between its parameters is investigated in this paper.Different types of symmetric bursting phenomena can be observed in numerical simulations.Their dynamical behaviours are discussed by means of slow-fast analysis.Furthermore,the generalized Jacobian matrix at the non-smooth boundaries is introduced to explore the bifurcation mechanism for the bursting solutions,which can also be used to account for the evolution of the complicated structures of the phase portraits.With the variation of the parameter,the periodic symmetric bursting can evolve into chaotic symmetric bursting via period-doubling bifurcation.     

Robust tori in a double-waved Hamiltonian model

A Hamiltonian system perturbed by two waves with particular wave numbers can present robust tori, which are barriers created by the vanishing of the perturbed Hamiltonian at some defined positions. When robust tori exist, any trajectory in phase space passing close to them is blocked by emergent invariant curves that prevent the chaotic transport. Our results indicate that the considered particular solution for the two waves Hamiltonian model shows plenty of robust tori blocking radial transport.