Invariant Tori Research Articles

Action–angle coordinates for motion in a straight magnetic field with constant gradient

The motion of a charged particle in a straight magnetic field B=B(y)ẑ with a constant perpendicular gradient is solved exactly in terms of elliptic functions and integrals. The motion can be decomposed in terms of a periodic motion along the y-axis and a drift motion along the x-axis. The periodic motion can be described as a particle trapped in a symmetric quartic potential in y. The canonical transformation from the canonical coordinates (y,Py) to the action–angle coordinates (θ,J) is solved explicitly in terms of a generating function S(θ,J) that is expressed in terms of Jacobi elliptic functions. The presence of a weak constant electric field E=E0ŷ introduces an asymmetric component to the quartic potential, and the associated periodic motion is solved perturbatively up to second order.

Resonance and energy transfer in forced vibro-impact systems with linear compliance

We address a non-stationary dynamic of a vibro-impact oscillator with linear compliance under harmonic excitation. The analytic treatment is based on the well-known approximation of isolated resonance. To this end, an appropriate set of action–angle variables is formulated for the single degree of freedom conservative vibro-impact oscillator with linear compliance. At the next stage, the dynamics of the forced system is considered in conditions of a primary 1:1 resonance, common in most applications. The approximation of the isolated resonance reduces the dynamics to a planar resonance manifold with foliation induced by a global invariant derived for the 1:1 isolated resonance flow. This foliation allows for an appropriate description of the steady-state, modulated and transient responses, and their bifurcations. The results are verified with direct numeric simulations.

Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.

On integrability of the Nosé–Hoover oscillator and generalized Nosé–Hoover oscillator

The Nosé–Hoover system is a basic primitive model for the molecular dynamics simulations, which describes the equilibrium characterized by canonical distributions at a constant temperature. Its simplest form, called the Nosé–Hoover oscillator (NH), is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories. Recently, a similar system, called the generalized Nosé–Hoover oscillator (GNH), was constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in [Formula: see text]. This paper aims to study the integrability of both NH and GNH models. We show that (i) in the case of [Formula: see text], both NH and GNH models are integrable by quadratures and the general solutions are given; (ii) in the case of [Formula: see text] the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a complete characterization of Darboux polynomials and exponential factors are given; (iii) both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by an extended Morales–Ramis theory; (iv) the reduced systems of NH and GNH models at the Poincaré compactification balls are integrable, which yields complete descriptions of dynamics at infinity for NH and GNH models. Interestingly, the topological structure of NH and GNH models at infinity strongly depends on the sign of [Formula: see text]. These results may help us better understand the complex and rich dynamics of nonlinear time-reversible systems.

On the graph theorem for Lagrangian invariant tori with totally irrational invariant sets

On the graph theorem for Lagrangian invariant tori with totally irrational invariant sets

On the divergence of Birkhoff Normal Forms

It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (dgeq 1) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d=1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.

Bifurcations of Invariant Tori in Second-Order Quasilinear Evolution Equations in Hilbert Spaces and Scenarios of Transition to Turbulence

In this paper, we consider second-order quasilinear differential equations in a separable Hilbert space for which the well-known Landau–Hopf scenario of transition to turbulence can be realized. We prove increasing of the control parameter leads to the consecutive appearance of invariant tori of increasing dimensions. In this case, the invariant torus of the largest possible dimension appears to be attractive. The results are obtained by using methods of the qualitative theory of dynamical systems with an infinite-dimensional space of initial conditions: the method of integral manifolds, the theory of normal forms, and also asymptotic methods of analysis of dynamical systems.

IWAVE-An adaptive filter approach to phase lock and the dynamic characterization of pseudo-harmonic waves.

We present a novel adaptive filtering approach to the dynamic characterization of waves of varying frequencies and amplitudes embedded in arbitrary noise backgrounds. This method, known as IWAVE (Iterative Wave Action angle Variable Estimator), possesses critical advantages over conventional techniques, making it a useful new tool in the dynamic characterization of a wide range of data containing embedded oscillating signals. After a review of existing techniques, we present the IWAVE algorithm, derive its key characteristics, and provide tests of its performance using simulated and real world data.

Strange attractors in a dynamical system inspired by a seasonally forced SIR model

We analyze a multiparameter periodically-forced dynamical system inspired in the SIR endemic model. We show that the condition on the basic reproduction numberR0<1 is not sufficient to guarantee the elimination of Infectious individuals due to a backward bifurcation. Using the theory of rank-one attractors, for an open subset in the space of parameters where R0<1, the flow exhibits persistent strange attractors. These sets are not confined to a tubular neighborhood in the phase space, shadow the ghost of a two-dimensional invariant torus and are numerically observable. Although numerical experiments have already suggested that periodically-forced biological models may exhibit observable chaos, a rigorous proof was not given before. Our results agree well with the empirical belief that intense seasonality induces chaos.This work provides a preliminary investigation of the interplay between seasonality, deterministic dynamics and the prevalence of strange attractors in a nonlinear forced system inspired by biology.

Regions without invariant tori of given class for the planar circular restricted three-body problem

A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for planets around a binary star are deduced. It is expected that lessons learnt from this problem will be useful for applications of the method to other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics.

Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.

Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite gap tori for the 2D cubic NLS equation

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the twodimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long time scales, they exhibit a strong form of transverse instability in Sobolev spaces H^s(\mathbb{T}^2)(0 &lt; s &lt; 1) . More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H^s topology and whose H^s norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.

Topological power pumping in quantum circuits

In this article, we develop a description of topological pumps as slow classical dynamical variables coupled by a quantum system. We discuss the cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps based frequency converters. This last case corresponds to a quantum topological coupling between classical modes described by action-angle variables on which we focus. We propose a realization of such a topological coupler based on a superconducting qutrit suitably driven by three modulated drives. A detailed experimental protocol allowing to measure the quantized topological power transfer between the different modes of a superconducting circuit is discussed.

An introduction to classical monodromy: Applications to molecules in external fields

An integrable Hamiltonian system presents monodromy if the action-angle variables cannot be defined globally. As a prototype of classical monodromy with azimuthal symmetry, we consider a linear molecule interacting with external fields and explore the topology structure of its phase space. Based on the behavior of closed orbits around singular points or regions of the energy–momentum plane, a semi-theoretical method is derived to detect classical monodromy. The validity of the monodromy test is numerically illustrated for several systems with azimuthal symmetry.

Instabilities of invariant quasi-periodic tori

We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic motion:

Linking bifurcation analysis of Holling–Tanner model with generalist predator to a changing environment

AbstractBifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.

Dynamical Organization of Recollisions by a Family of Invariant Tori

We consider the motion of an electron in an atom subjected to a strong linearly polarized laser field. We identify the invariant structures organizing a very specific subset of trajectories, namely recollisions. Recollisions are trajectories which first escape the ionic core (i.e., ionize) and later return to this ionic core, for instance, to transfer the energy gained during the large excursion away from the core to bound electrons. We consider the role played by the directions transverse to the polarization direction in the recollision process. We compute the family of two-dimensional invariant tori associated with a specific hyperbolic-elliptic periodic orbit and their stable and unstable manifolds. We show that these manifolds organize recollisions in phase space.

Approximation of potential function in the problem of forced escape

The paper revisits a famous problem of escape of a classical particle from a potential well under a harmonic forcing. In particular, the most prominent escape dynamics appears at 1:1 resonance and can be described by means of an isolated resonant (IR) approximation. In this case, the problem has to be reformulated in terms of action–angle variables (AA), which are available for only a few of the model potentials. It is proposed in the paper that realistic generic potentials can be approximated by low-order polynomial functions, admissible to the AA transformation, with possible truncation. In order to illustrate the idea, we first formulate the AA transformation and then solve the escape problem in the IR approximation for a generic quartic potential. Then, we apply this approach to the study of pull-in instability in microelectromechanical systems (MEMS). In particular, we consider a SDOF approximate model of a capacitive micromachined ultrasonic transducer (CMUT) device. The resonant external excitation corresponds to the acoustical wave incidental on the diaphragm of the CMUT. In general, pull-in effect is observed in MEMS due the electrostatic force growing faster than the restoring mechanical force. We propose to study the pull-in (escape) in the model electrostatic potential by approximating it with quartic polynomials (globally and locally). The quality of predicting the escape threshold is assessed numerically. Most accurate predictions are delivered by global L2-optimal heuristic approximation.

Small amplitude quasi-periodic solutions for the forced radial vibrations of cylindrical shells with incompressible materials

In this paper, we proved the existence of small amplitude quasi-periodic solutions for the forced radial vibrations of cylindrical shells with hyperelastic, homogeneous, isotropic, and incompressible materials. The proof mainly relies on action angle variables and a series of approximately identical transformations, which transform the Hamiltonian into the near integrable one. Owing to Moser’s twist theorem, there exist infinitely many invariant curves at any sufficiently small neighborhood of the equilibrium point of free radial oscillations, some of which are corresponding to quasi-periodic solutions. Moreover, this procedure also provides a new method to calculate the approximate period of the small amplitude free radial periodic oscillation. Some numerical examples demonstrate and support our results.