Invariant Tori Research Articles

Thermalization of Isolated Harmonic Networks Under Conservative Noise

We study a scalar harmonic network with pair interactions and a binary collision rule, exchanging the momenta of a randomly-chosen couple of sites. We consider the case of the isolated network where the total energy is conserved. In the first part, we recast the dynamics as a stochastic map in normal modes (or action-angle) coordinates and provide a geometric interpretation of it. We formulate the problem for generic networks but, for completeness, also reconsider the translation-invariant lattices. In the second part, we examine the kinetic limit and its range of validity. A general form of the linear collision operator in terms of eigenstates of the network is given. This defines an action network, whose connectivity gives information on the out-of-equilibrium dynamics. We present a few examples (ordered and disordered chains and elastic networks) where the topology of connections in action spaces can be determined in a neat way. As an application, we consider the classic problem of relaxation to equipartition from the point of view of the dynamics of linear actions. We compare the results based on the spectrum of the collision operator with numerical simulation, performed with a novel scheme based on direct solution of the equations of motion in normal modes coordinates.

Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structure.

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic [E. J. Heller and S. Tomsovic, Phys. Today 46, 38 (1993)0031-922810.1063/1.881358]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B=0.5) with strong stickiness effects [Č. Lozej et al., Phys. Rev. E 103, 012204 (2021)2470-004510.1103/PhysRevE.103.012204], as well as on the three billiards with a simple mixed-type phase space and no stickiness [Č. Lozej et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)1817-245810.33581/1561-4085-2021-24-1-1-18]. Here we study two classical and quantum lemon billiards, for the cases B=0.1953,0.083, which are mixed-type billiards with a complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently [ Č. Lozej, D. Lukman, and M. Robnik, Physics 3, 888 (2021)10.3390/physics3040055]. We calculate a very large number (10^{6}) of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We introduce an overlap index, which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions, the relative fraction of mixed-type states decreases as a power law with increasing energy, thus, in the strict semiclassical limit, leaving only purely regular and chaotic states. Our approach offers a general phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.

Escape dynamics of a particle from a purely nonlinear truncated quartic potential well under harmonic excitation

This paper focuses on the escape problem of a harmonically forced classical particle from a purely quartic truncated potential well. The latter corresponds to various engineering systems that involve purely cubic restoring force and absence of linear stiffness even under the assumption of small oscillations, such as pre-tensioned metal wires and springs, and compliant structural components made of polymer materials. This is in contrast to previous studies where the equivalent potential well could be treated as linear at first approximation under the assumption of small perturbations. Due to the strong nonlinearity of the current potential well, traditional analytical methods are inapplicable for describing the transient bounded and escape dynamics of the particle. The latter is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle variables and the corresponding reduced resonance manifold. The escape envelope is formulated analytically. Surprisingly, despite the essential nonlinearity of the well investigated, it exhibits a universal property of a sharp minimum due to the existence of multiple intersecting escape mechanisms. Unlike previous studies, three underlying mechanisms that govern the transient dynamics of the particle were identified: two maximum mechanisms and a saddle mechanism. The first two correspond to a gradual increase in the system’s response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system’s response and therefore more potentially hazardous. The response of the particle is described in terms of energy-based response curves. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. All theoretical predictions are in complete agreement with numerical results.

Invariant tori in dissipative hyperchaos.

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

Testing the conservative character of particle simulations: II. Spurious heating of guiding centers and full orbits subject to fluctuations expressed in terms of E and B

For an axisymmetric tokamak plasma, Hamiltonian theory predicts that the orbits of charged particles must stay on invariant tori of conserved energy in the moving frame of reference of a wave that propagates along the torus with fixed angular phase velocity, amplitude, and shape. The mode structure in the poloidal plane is arbitrary if the fluctuations are expressed in terms of potentials Φ and A, which satisfy Faraday's law and the solenoidal condition by definition. Consequently, smoothing operations (such as gyroaveraging and noise suppression) do not violate the conservative laws. However, this is not guaranteed for models expressed in terms of the physical fields E and B. Here, we demonstrate that manipulations of E and B in the poloidal (R, z) plane can cause spurious heating that is independent of time steps or numerical methods, but can be sensitive to geometry. In particular, we show that secular acceleration is enhanced when one imposes nonnormal modes that possess strong up–down asymmetry instead of the usual in–out asymmetry of normal toroidal (eigen)modes. We compare full gyro-orbit and guiding center models and find similar behavior. We also examine the effect of ad hoc N-point gyroaveraging in a guiding center model, as is done in some simulation codes. If one uses Faraday's law to (re)compute B(t) after gyroaveraging E, the guiding center motion remains conservative. Otherwise, spurious heating should be expected and monitored, but it may be tolerable when normal modes dominate.

Steady-state solutions of split beams in electron storage rings

Recently, a novel operation method for synchrotron light sources with transversely split beams has been explored to fulfill the rising demand for flexible and high-throughput X-ray sources required in such diverse fields as time-resolved X-ray spectroscopy, molecular chemistry in organic cells, high-resolution medical imaging, quantum materials science or sustainable energy research. Within that novel operation mode, additional stable regions are produced in the horizontal phase space by operating an electron storage ring on a resonance that is driven by the nonlinear sextupole or octupole magnets. In the longitudinal phase space, a similar split can be produced by introducing an oscillation of the synchrotron phase via a modulation of the phase of the radiofrequency resonator. Strong radiation damping in electron storage rings, however, has to be overcome before additional regions in phase space can become populated by particles and form stable islands. This damping mechanism changes the dynamics of the system and causes diffusion between the different islands in phase space, raising the question what kind of equilibrium state exists in the asymptotic temporal limit. In this paper, a finite-differences approximation in rotating action-angle coordinates is used to solve the Vlasov–Fokker–Planck equation and to study the obtained equilibrium states for the longitudinal as well as the transverse case. The number of solution vectors and the magnitude of the corresponding singular values of the matrix of the underlying finite-differences equation are used as abstract indicators to define the required parameter set that provides stable additional beamlets. As a consequence, the beamlets have a stability that is close to that of the main beam in terms of diffusion caused by the radiation damping and quantum excitation.

Dynamics of a hybrid cubic vibro-impact oscillator and nonlinear energy sink

In the current work, we model the HCVI oscillator by a forced particle in a hybrid quartic-square potential well with infinite depth under periodic external forcing. The slow-flow dynamics of the system is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined and described analytically. The former is associated with the SNO-regime and the HCVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. Unlike previous studies, three underlying dynamical mechanisms that govern the occurrence of bifurcations are identified: two maximum mechanisms and one saddle mechanism. The first two correspond to a gradual increase in the system’s response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system’s response and therefore more potentially hazardous when takes place in engineering systems and more effective for energy pumping when takes place in the HCVI-NES. Both mechanism types are universal for systems that undergo escape from a potential well. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. The response curves of the system are obtained analytically, allowing a full perspective on the HCVI-oscillator dynamical response, and the HCVI-NES performances and design optimization. All theoretical results are supported by numerical simulations.

Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u t t + Δ 2 u + ε f u = 0 , x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f u is a real analytic odd function of the form f u = f 5 u 5 + ∑ i ^ ≥ 3 f 2 i ^ + 1 u 2 i ∧ + 1 , f 5 ≠ 0 . It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ † 2 = r = r 1 , r 2 , r 1 ∈ 4 ℤ − 1 , r 2 ∈ 4 ℤ . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.

Noise induced escape from stable invariant tori

In the present paper, the noise-induced escape from stable invariant tori with the frame of general Langevin dynamics is investigated under weak random perturbations. Based on the large deviation theory, the quasi-potential, a quantity exponentially dominating the mean first escape time and the stationary or quasi-stationary probability density, is explored both analytically and numerically. The results show that whether the stochastic trajectories are ergodic on a torus has a crucial influence on the behavior of the quasi-potential. Specifically, there are two sources of ergodicity. One is the ergodicity of the deterministic flow, and the other requires the nondegeneracy of the noise in the tangential directions of a torus. It is found that if the ergodicity holds, the quasi-potential will be independent of the initial position on a torus, but not when the ergodicity is broken. In particular, it indicates that, for nonlinear systems driven by combined Gaussian white noise and multiple harmonic excitations, the mean first exit time varies discontinuously with respect to the frequency vector of these harmonic excitations as the noise intensity approaches zero. Adding noise to the phase of these harmonic excitations will eliminate the dependence on the initial position and thus, make the systems more robust. It reminds us that great care must be taken when dealing with noise-induced problems involving systems that possess unstable geometric structures.

Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets

We construct quasimodes for some non-selfadjoint semiclassical operators at the boundary of the pseudo-spectrum using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a subelliptic finite-type condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on non-damped periodic orbits or invariant tori satisfying a weakgeometric- control condition.

Four-body cislunar quasi-periodic orbits and their application to ballistic lunar transfer design

Examining the properties of quasi-periodic orbits provides insight into the Sun-perturbed environment in cislunar space. In this investigation, quasi-periodic trajectories and their properties are explored in the Sun-Earth-Moon four-body problem. Computation and the stability characteristics of families of invariant tori are detailed. Furthermore, this investigation offers a framework for construction of ballistic lunar transfer trajectories in the four-body problem. The framework leverages manifold trajectories to supply a set of initial conditions for construction of periapsis Poincaré maps. Periapsis maps reduce the dimensionality of the space and illuminate solutions of interest as a basis to produce feasible families of transfers. Through a continuation process, families of ballistic transfers and families of transfers that include powered lunar flybys are constructed. Ultimately, these solutions supply an initial guess for transition to the Sun-Earth-Moon ephemeris model.

Physics aware analytics for accurate state prediction of dynamical systems

It has been successfully demonstrated that synchronisation of physical prior, like conservation laws with a conventional neural network significantly decreases the amount of training necessary to learn the dynamics of non-linear physical systems. Recent research shows how parameterisation of Lagrangian and Hamiltonian using conventional Neural Network’s weights and biases are achieved, and then executing the Euler-Lagrangian and Hamilton’s equation of motion for prediction of future state vectors proved to be more efficient than conventional Neural Networks in predicting non-linear dynamical systems. In classical mechanics, the Lagrangian formalism predicts the trajectory by solving the second-order Euler-Lagrangian equation following all the symmetries of the system while, the Hamiltonian mechanics explicitly uses the Poisson bracket of canonical position and canonical momentum to arrive at two independent first-order equations of motion. For a physical system that follows energy conservation, we obtain the momentum taking the first derivative of the Lagrangian which supplements the energy conservation via Hamiltonian Mechanics. The dynamics of the physical system converges when the generalised coordinates transform to suitable action angle variables using Poisson bracket formalism conserving the energy. In this work, we have demonstrated the prediction by integrating Hamiltonian and Lagrangian neural network leading to faster convergence with the actual dynamics from less number of training epochs. Hence, integration of HNN(Hamiltonian Neural Network) and LNN(Lagrangian Neural Network) outperforms their autonomous implementation. We have tested our proposed model in three conservative non-linear dynamical systems and found promising results.

A simple three-dimensional quadratic flow with an attracting torus

Attracting torus is a rare phenomenon in the dynamics of low-dimensional autonomous systems. Adding an anti-damping term to the well-known Nosé-Hoover oscillator, this paper introduces a new system exhibiting attracting torus in a wide range of parameter values. This system has a variety of dynamical solutions like limit cycles, strange attractors, attracting tori, invariant tori, and chaotic sea. It is also demonstrated that the system is multistable in some regions of parameter space wherein different types of attractors coexist. However, the attracting torus is the leading bounded solution in a considerable area of parameter space. Moreover, the coexistence of four limit cycles is found in the time-reversed system. The study of the system's basin of attraction shows that the system owns a solid basin of attraction with rounded boundaries for the attracting torus, which is an exciting property.

Application of the Slater criteria to localize invariant tori in Hamiltonian mappings.

We investigate the localization of invariant spanning curves for a family of two-dimensional area-preserving mappings described by the dynamical variables I and θ by using Slater's criterion. The Slater theorem says there are three different return times for an irrational translation over a circle in a given interval. The returning time, which measures the number of iterations a map needs to return to a given periodic or quasi periodic region, has three responses along an invariant spanning curve. They are related to a continued fraction expansion used in the translation and obey the Fibonacci sequence. The rotation numbers for such curves are related to a noble number, leading to a devil's staircase structure. The behavior of the rotation number as a function of invariant spanning curves located by Slater's criterion resulted in an expression of a power law in which the absolute value of the exponent is equal to the control parameter γ that controls the speed of the divergence of θ in the limit the action I is sufficiently small.

Extreme mass ratio inspirals into black holes surrounded by matter

Inspirals of stellar-mass compact objects into massive black holes, known as extreme mass ratio inspirals (EMRIs), are one of the key targets for upcoming space-based gravitational-wave detectors. In this paper we take the first steps needed to systematically incorporate the effect of external gravitating matter on EMRIs. We model the inspiral as taking place in the field of a Schwarzschild black hole perturbed by the gravitational field of a far axisymmetric distribution of mass enclosing the system. We take into account the redshift, frame-dragging, and quadrupolar tide caused by the enclosing matter, thus incorporating all effects to inverse third order of the characteristic distance of the enclosing mass. Then, we use canonical perturbation theory to obtain the action-angle coordinates and Hamiltonian for mildly eccentric precessing test-particle orbits in this background. Finally, we use this to efficiently compute mildly eccentric inspirals in this field and document their properties. This work shows the advantages of canonical perturbation theory for the modeling EMRIs, especially in the cases when the background deviates from the standard black hole fields.

Antaeus: A Retrograde Group of Tidal Debris in the Milky Way’s Disk Plane

We present the discovery of a wide retrograde moving group in the disk plane of the Milky Way using action-angle coordinates derived from the Gaia DR3 catalog. The structure is identified from a sample of its members that are currently almost at the pericenter of their orbit and are passing through the solar neighborhood. The motions of the stars in this group are highly correlated, indicating that the system is probably not phase mixed. With a width of at least 1.5 kpc and with a probable intrinsic spread in metallicity, this structure is most likely the wide remnant of a tidal stream of a disrupted ancient dwarf galaxy (age ∼12 Gyr, 〈[Fe/H]〉 ∼ −1.74). The structure presents many similarities (e.g., in energy, angular momentum, metallicity, and eccentricity) with the Sequoia merging event. However, it possesses extremely low vertical action J z , which makes it unique even among Sequoia dynamical groups. As the low J z may be attributable to dynamical friction, we speculate that these stars may be the remnants of the dense core of the Sequoia progenitor.

Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

We deal with the following system of coupled asymmetric oscillators x¨1+a1x1+-b1x1-+ϕ1(x2)=p1(t)x¨2+a2x2+-b2x2-+ϕ2(x1)=p2(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\ddot{x}_1+a_1x_1^+-b_1x^-_1+\\phi _1(x_2)=p_1(t) \\\\ \\ddot{x}_2+a_2\\,x_2^+-b_2\\,x^-_2+\\phi _2(x_1)=p_2(t), \\end{array} \\right. \\end{aligned}$$\\end{document}where ϕi:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi _i: \\mathbb {R} \\rightarrow \\mathbb {R}$$\\end{document} is locally Lipschitz continuous and bounded, pi:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_i: \\mathbb {R} \\rightarrow \\mathbb {R}$$\\end{document} is continuous and 2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\pi $$\\end{document}-periodic and the positive real numbers ai,bi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_i, b_i$$\\end{document} satisfy 1ai+1bi=2n,forsomen∈N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\dfrac{1}{\\sqrt{a_i}}+\\dfrac{1}{\\sqrt{b_i}}=\\dfrac{2}{n}, \\quad \ ext{ for } \ ext{ some } n \\in \\mathbb {N}. \\end{aligned}$$\\end{document}We define a suitable function L:T2→R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L: \\mathbb {T}^2 \\rightarrow \\mathbb {R}^2$$\\end{document}, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincaré map, in action-angle coordinates.

Conservative dynamics in a novel class of 3D generalized thermostatted systems.

This paper reports a method to derive a novel class of 3D generalized thermostatted oscillators from a simple damped harmonic oscillator. Its detailed procedure is obtained through a mathematical derivation. Then, we propose an example system to show the effectiveness of the method. Furthermore, the numerical analysis is performed to investigate its rich conservative dynamics, including chaotic sea, hierarchical invariant tori, and the coexistence of chaotic seas and invariant tori with islands-around-islands hierarchy, even if there are three dissipative terms in the example system. To verify the existence of conservative chaos at the physical level, an analog circuit is presented to observe the existing conservative chaotic flows in National Instruments multisim. We finally provide two systems, which can produce conservative chaotic flows with more complicated topologies, and draw our conclusions.

Maintenance management of large-size rolling bearings in heavy-duty machinery

Slewing bearings are one of the most important elements in the vast majority of large-size machines. They are widely used in the mining industry: tunnel cutters, bucket excavators and many other devices. In a Bucket-wheel excavator, continuous rotation of the body is most advantageous due to the technique of digging the input or coal. The rotational movement of the machine is then the basic cutting movement, and the delivery movement, carried out by driving the machine, is only an auxiliary movement. A similar kinematics occurs in tunnel cutters. Therefore, these bearings have played such a significant role and have been the subject of extensive research and continuous improvement over the years. High demands are placed on them in terms of load capacity, friction, accuracy, durability and reliability. It happens, however, that despite careful design and manufacture, the bearings do not achieve the required durability. Failures usually result in economic losses due to loss of production, damage to adjacent parts and repair costs. Premature bearing failure can occur for a variety of reasons. Each failure leaves its own special mark on the bearing. Consequently, by examining the damaged bearing, it is in most cases possible to find the root cause and define corrective actions, thus preventing further failures. This publication aims to provide basic knowledge about the factors determining the load capacity and durability of large-size slewing ring bearings and the analysis of their damage. The result of the considerations is finding the sources of errors in determining the load-bearing capacity characteristics of roller slewing bearings. For this purpose, the ISHIKAWA and FMEA methods were used and the risk level for errors was determined. Moreover, the article presents some forms of damage to raceways of slewing bearings and indicates their causes. Changes in the so-called angle of action of the rolling elements in the ball bearing due to the transferred loads. The influence of changes in this angle on the geometry of the contact zone of the rolling elements and raceways was investigated. It has been shown that the contact angles of some rolling elements increase significantly. This can damage the raceway by chipping or rolling the edge of the bearing ring. With the knowledge presented in this article, it is possible to evaluate various emergency situations and start their proper analysis.

Invariant tori for multi-dimensional integrable Hamiltonians coupled to a single thermostat

This paper demonstrates sufficient conditions for the existence of Kolmogorov-Arnol’d-Moser (KAM) tori in a singly thermostated, integrable Hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2—which include the Nosé thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends theorem 3.2 of Legoll et al (2009 Nonlinearity 22 1673–94) to prove that a ‘typical’ singly thermostated, integrable, real-analytic Hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable Hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion.