Regular Islands Research Articles

Chaos-Assisted Tunneling.

The ability of particles to "tunnel" through potential energy barriers is a purely quantum phenomenon. A classical particle in a symmetric double-well potential, with energy below the potential barrier, will be trapped on one side of the potential well. A quantum particle, however, can sit on both sides, in either a symmetric state or an antisymmetric state. An analogous phenomenon occurs in conservative classical systems with two degrees of freedom and no potential barriers. If only the energy is conserved, the phase space will be a mixture of regular "islands" embedded in a sea of chaos. Classically, a particle sitting in one regular island cannot reach another symmetrically located regular island when the islands are separated by chaos. However, a quantum particle can sit on both regular islands, in symmetric and antisymmetric states, due to chaos-assisted tunneling. Here, we give an overview of the theory and recent experimental observations of this phenomenon.

Bloch theorem dictated wave chaos in microcavity crystals

Universality class of wave chaos emerges in many areas of science, such as molecular dynamics, optics, and network theory. In this work, we generalize the wave chaos theory to cavity lattice systems by discovering the intrinsic coupling of the crystal momentum to the internal cavity dynamics. The cavity-momentum locking substitutes the role of the deformed boundary shape in the ordinary single microcavity problem, providing a new platform for the in situ study of microcavity light dynamics. The transmutation of wave chaos in periodic lattices leads to a phase space reconfiguration that induces a dynamical localization transition. The degenerate scar-mode spinors hybridize and non-trivially localize around regular islands in phase space. In addition, we find that the momentum coupling becomes maximal at the Brillouin zone boundary, so the intercavity chaotic modes coupling and wave confinement are significantly altered. Our work pioneers the study of intertwining wave chaos in periodic systems and provide useful applications in light dynamics control.

Invariant regions in piecewise linear area-preserving map

This paper studies regular islands (invariant regions) associated with elliptic periodic points of a piecewise linear area-preserving map. We reveal why regular island chains are interspersed in a chaotic sea, which is typical of the behavior exhibited by the area-preserving map. We find that regular islands may be ellipses or polygons and this depends on whether the rotation number of the rotation matrix that defines the linear map is irrational or rational. We also present a method for determining the boundary of these regular islands and chaotic seas (the largest invariant set). We show that critical lines play an important role in determining the boundaries of regular islands and chaotic seas.

Chaos and bipartite entanglement between Bose-Josephson junctions.

The entanglement between two weakly coupled bosonic Josephson junctions is studied in relation to the classical mixed phasespace structure of the system, containing symmetry-related regular islands separated by chaos. The symmetry-resolved entanglement spectrum and bipartite entanglement entropy of the system's energy eigenstates are calculated and compared to their expected structure for random states that exhibit complete or partial ergodicity. The entanglement spectra of chaos-supported eigenstates match the microcanonical structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic invariant that restricts ergodization on the energy shell. The symmetry-resolved entanglement entropy of these quasistochastic states consists of a mean-field maximum entanglement term and a fluctuation correction due to the finite size of the constituent subsystems. The total bipartite entanglement entropy of the eigenstates correlates with their chaoticity. Island-supported eigenstates are macroscopic Schrödinger cat states for particles and excitations with substantially lower entanglement.

Occupational risk assessment levels before and after improvement on fueling devices in ammoniation facilities from 2 nuclear plants

To evaluate and compare possible changes of individual health risk levels for ammonia exposed positions from ammoniation facilities in 2 nuclear power plants(X and Y) through occupational risk assessment models(OHRA) before and after devices improvement. This study tried to understand the improvement of fueling devices in ammoniation facilities, to identify the ammonia exposed positions and their working content, to collect information on protective measures and personal protective equipment situation through on-sites survey and detailed records of work days. Next, detection on C_(STE) of ammonia at different worksites and C_(TWA) of 10 involved positions were conducted, fresh air requirement and ventilation air changing rates were calculated through wind speed detection or collected directly. Finally, a semi-quantitative comprehensive index model and a quantitative non-carcinogenic model from standard GBZ/T 298-2017 were utilized to assess risk levels before and after improvement. After devices improvement, a series of innovations were observed, followed by:(1) ammonia steal tanks in stacking storage were replaced by metal pressure sealing tanks for reduction of exposure frequency and long time storage. (2) manual filling pattern was totally substituted by automatic filling devices with new installed ventilation systems and alarming apparatus. (3) increasing ventilation effect by doors opening when ammonia was filling. In this regard, conentration levels(C_(STE)) referred to positions of chemical analysis engineers, chemical sampling engineers, nuclear maintenance workers at 3# unit from X, and regular island operators at 1#unit, patrolling operators at 2# unit from Y were declined to qualified levels after improvement from unqualified levels before that, and significant difference of C_(STE )detected before and after improvement were found statistically(Z=10.856, P<0.001). C_(TWA )were all within the qualified ranges before and after improvement for relatively short cumulative exposure time. Moreover, the ventilation air changing rates from related indoor sectors were increased to 13.0 to 30.9 times/h after improvement which indicated a statistical difference as compared to quantitative ranges before started(Z=11.670, P<0.001). Further, a relative negative correlation was observed between C_(STE )and ventilation air changing rates(r=-0.39, P<0.05) while no correlation between them was spotted after improvement(r=-0.051, P>0.05). In addition, most positions like chemical analysis engineers and others changed their risk levels to a lower one from that they used to be after improvement with an observed significant difference(Z=1.345, P<0.05), by contrast, risk levels of positions like nuclear maintenance workers at 3#unit or patrolling operators at 4#unit increased a level or remained the same before and after improvement might be result ed from enhancement of cumulative exposure time. No significant difference among position risk levels under the quantitative non-carcinogenic model was observed(Z=0.447, P>0.05). The measure of devices improvement indeed decreased the ambient ammonia concentrations at workplaces of ammoniation facilities in X and Y and changed individual risk levels for most involved positions. The semi-quantitative comprehensive index model was more appropriated for self-contrast risk assessment application than quantitative non-carcinogenic model did, especially when improvement occurred.

Static Elastoplastic Analysis of Turbine House under Different Lateral Loading MoDES in Conventional Island of Nuclear Power Plant

Nuclear island in nuclear power plant requires that under the extreme safety shutdown earthquake action, the collapse of the turbine house of conventional island shall not affect the safety of nuclear island, so as to ensure the functional integrity of nuclear island. At present, the way to meet this requirement is to carry out elastoplastic analysis on the turbine house of the regular island under the action of the ultimate safe earthquake to ensure that it does not collapse and its deformation meets the deformation requirements of the standard rare earthquake. Structure elastic-plastic analysis method with static analysis and dynamic elastic-plastic time history analysis, in this paper, the ci turbine room nappe analyzed, and compared the five kinds of lateral load loading mode of static elastoplastic analysis results, the seismic performance of the structure, puts forward the layer shear lateral loading mode suitable for ci turbine room structure static elastoplastic analysis.

Spontaneous shape transition of Mn x Ge1- x islands to long nanowires.

We report experimental evidence for a spontaneous shape transition, from regular islands to elongated nanowires, upon high-temperature annealing of a thin Mn wetting layer evaporated on Ge(111). We demonstrate that 4.5 monolayers is the critical thickness of the Mn layer, governing the shape transition to wires. A small change around this value modulates the geometry of the nanostructures. The Mn–Ge alloy nanowires are single-crystalline structures with homogeneous composition and uniform width along their length. The shape evolution towards nanowires occurs for islands with a mean size of ≃170 nm. The wires, up to ≃1.1 μm long, asymptotically tend to ≃80 nm of width. We found that tuning the annealing process allows one to extend the wire length up to ≃1.5 μm with a minor rise of the lateral size to ≃100 nm. The elongation process of the nanostructures is in agreement with a strain-driven shape transition mechanism proposed in the literature for other heteroepitaxial systems. Our study gives experimental evidence for the spontaneous formation of spatially uniform and compositionally homogeneous Mn-rich GeMn nanowires on Ge(111). The reliable and simple synthesis approach allows one to exploit the room-temperature ferromagnetic properties of the Mn–Ge alloy to design and fabricate novel nanodevices.

Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

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 in Phys. Today 46 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard (B = 0:5) with strong stickiness effects published in Phys. Rev. E 103 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases B = 0:42; 0:55; 0:6, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following universal properties of mixed-type systems without stickiness in the chaotic regions: (i) Using the Poincare-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent β, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a bimodal universal distribution, where the narrow peak at small A encompasses the regular eigenstates, theoretically well understood, while the peak at larger A comprises the chaotic eigenstates, and is well described by the beta distribution. (iii) Thus the BRB energy level spacing distribution captures two effects: the divided phase space dictated by the classical Berry-Robnik parameter ρ2 measuring the relative size of the largest chaotic region, in agreement with the Berry-Robnik picture, and the localization of chaotic PH functions characterized by the level repulsion (Brody) parameter β. (iv) Examination of the PH functions shows that they are supported either on the classical invariant tori in the regular islands or on the chaotic sea, where they are only weakly localized. With increasing energy the localization of chaotic states decreases, as the PH functions tend towards uniform spreading over the classical chaotic region, and correspondingly β tends to 1.

Transient Chaos, Hyperchaotic Dynamics, and Transport Properties in a Bailout Embedding Web Map

In this work, we show that the bailout embedding method is responsible for creating different dynamical behaviors and for destroying intrinsic features present in mixed phase spaces of the area-preserving Hamiltonian maps, where the sticking to regular (or resonant) islands degrades chaotic properties. In particular, the base map chosen for the study is the two-dimensional (2D) Web Map (WM). The four-dimensional (4D) embedded Web Map dynamics is governed by four-parameters: ([Formula: see text]) in the WM control the nonlinearity and the type of symmetry structures (crystalline or quasi-crystalline) in phase space, respectively; ([Formula: see text]) in the embedding equations determine the mass density ratio and dissipation, respectively. For specific parameter combinations we explore the existence of transient chaos phenomenon, hyperchaotic dynamics and control the degradation of the underlying diffusive behaviors observed in phase space of the WM. If the WM is subjected to large enough dissipation through the embedding equations, stable periodic points (inside resonance islands) become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase space into chaotic and regular domains. As area-preserving maps obtained from Hamiltonian flows usually share the crucial property that resonance islands can be found immersed in chaotic sea (characterizing the mixed phase space) for appropriated parameter combinations, the results obtained here for the 4D embedded WM should be considered generic for such whole class of nonlinear systems.

Guiding Smooth Li Plating and Stripping by a Spherical Island Model for Lithium Metal Anodes.

Lithium metal anodes are considered as promising candidates for next-generation high-energy-density batteries. However, the dendrite formation of Li metal anodes during charge-discharge results in some serious issues. Herein, we show a simple way to flatten the Li metal deposition surface on Ag-modified Cu foil using a spherical island model. In this model, Ag nuclei induce the deposition of Li atoms with low nucleation potentials at the initial heterogeneous nucleation stage. Then, Li homogeneously grows around the spherical islands and these regular islands overlap each other and form a flat Li surface. On the bare Cu foil surface, the Li growth behavior is random, and the deposition surface is porous and covered with dendrites. Stable long-term plating/stripping of a symmetric battery over 800 h at 1 mA cm-2 was achieved. Moreover, the super flat Li structure can be achieved by constructing islands into a three-dimensional (3D) current collector using the spherical island model. Benefiting from the spherical island model, Li||LiFePO4 and Li||O2 batteries with this 3D anode structure can obtain a stable performance.

Slow Quantum Thermalization and Many-Body Revivals from Mixed Phase Space

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

An Iterative Two-Head Two-Track Detection Method for Staggered Bit-Patterned Magnetic Recording Systems

One of the major problems in bit-patterned magnetic recording (BPMR) systems is an inter-track interference (ITI) that can significantly degenerate the system performance, especially at high areal densities (ADs). Generally, the BPMR system with staggered islands experiences less ITI effect than that with regular islands. Therefore, we propose an iterative two-head two-track detection method for a staggered BPMR system to further reduce the ITI effect, which employs two read heads to read two adjacent tracks and decodes them simultaneously. To lessen the ITI, the weighted readback signal of one track, serving as an ITI signal, is subtracted from the readback signal of the other track, and vice versa, before passing the refined readback signal to an iterative decoding process. Hence, the decoded data will be utilized to estimate the ITI signal for the next global iteration, $N_{\mathrm{ G}}$ . Supposing that the refined readback signal has only one-side ITI effect. We also present the design of the $2{\times }3$ target and its compatible equalizer that will be employed in the proposed system. Results demonstrate that the proposed system not only is superior to the conventional system but also has low complexity. Specifically, the proposed system can perform close to the conventional system whose readback signal encounters only one-side ITI effect when $N_{\mathrm{ G}} = 2$ , 3 and 5 iterations for $\text {AD} = 2$ , 2.5, and 3 Tb/in2, respectively.

Collapse of hierarchical phase space and mixing rates in Hamiltonian systems

We employ statistical properties of Poincaré recurrences to investigate dynamical behaviors of coupled Hamiltonian maps with a mixed phase space, where sticking to regular (or resonant) islands degrades chaotic properties of the system. In particular we investigate two and three coupled Chirikov–Taylor standard mappings (SM) and choose nonlinearity parameters corresponding to a mixed phase space. Our first set of findings are: (i) the collapse of hierarchical phase space depends of the coupling intensity, (ii) the penetration of resonance islands generates a trapping regime responsible for a long intermediate algebraic decay which is suppressed as coupling increases; and (iii) the projection of recurrence-time statistics (RTS) onto two-dimensional planes of the high-dimensional phase space of coupled maps is useful to understand different dynamical features, when coupling strengths are varied. We also estimate the asymptotic polynomial decay exponent of RTS γ for both cases, and elaborate an indirect way to estimate the decay exponent of correlations χ by using large deviations theory applied to the probability distributions of finite-time largest Lyapunov exponents (FTLLEs). In the asymptotic regime both methods yield γ∼1.20 (2 SM) and γ∼1.10 (3 SM): the corresponding power-law exponents for time correlation decay χ is given by the known relationship χ=γ−1. As higher-dimensional Hamiltonian systems (with mixed phase space) share the crucial property that resonance islands are no more forbidden domains in phase space, the results obtained here for two and three coupled maps should be extended to even higher-dimensional systems.

Few Islands Approximation of Hamiltonian System with divided Phase Space

It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. Hamiltonian systems with sharply divided phase space between regions of regular and chaotic dynamics are much easier to analyze, but there are only few cases or families of such systems known to date. In this article, we outline a new approach for a systematic construction, starting from a generic KAM Hamiltonian system, of a system with a sharply divided phase space with an arbitrary number of regular islands which are in one-to-one correspondence with islands of the initial KAM system. In this procedure a typical Hamiltonian system, for example a KAM billiard, is replaced by a sequence of Hamiltonian systems having an increasing (but finite) number of islands of regular motion. The islands in the substituting systems are sub-islands of the KAM islands in the initial system. We apply this idea to two-dimensional lemon-shaped billiards, where the substituting systems are obtained by replacing parts of the curved boundaries by chords, so that in the limit of infinite number of islands the boundary of the substituting system becomes arbitrary close to the original billiard’s boundary.

Ray dynamics and wave chaos in circular-side polygonal microcavities

We systematically study mode characteristics in circular-side polygonal microcavities (CSPMs), particularly in these cavities with chaotic ray dynamics, in order to gain insights into the wave chaos in the CSPMs. The circular sides could improve the light confinement of the CSPMs as concave mirrors, in that regular islands are formed around the stable fixed points in the Poincar\'e surface of sections (SOS). However, the fixed points become unstable under some specific deformations, and global chaos with quasistable ``star islands'' appears around these fixed points in the Poincar\'e SOS accordingly. The phenomenon can be well explained by the ray dynamic analysis under the second-order approximation, and the results show that the high-order terms play an important role in the motions of light rays and destroy the regular islands in the phase space leading to chaotic ray dynamics. The destruction of regular islands results in degradation of mode quality factors and dispersed mode field distributions according to the finite-element method simulation of the confined modes. Furthermore, an unusual variation of mode quality factor is observed by varying the refractive index of the outside media for the CSPM with chaotic ray dynamics.

Characterization of magnetic field lines in connection with their transport in Beltrami magnetic fields

Single and double curl Beltrami magnetic fields are studied numerically in connection with their transport properties, and the results are compared and analyzed by characterizing the magnetic field lines. In the phase space of the single curl Beltrami field, islands of regular field lines are embedded into the chaotic sea, whereas the field lines for certain solutions of the double curl Beltrami field are chaotic over the entire space. Due to the presence of regular islands in phase space, the chaotic trajectories show stickiness phenomena which are characterized by the distribution of a chaotic field line. The dynamical traps in chaotic orbits due to stickiness phenomena are also characterized by the distribution of finite distance Lyapunov exponents. Finally, the recurrence length distribution of chaotic trajectories is plotted to understand their global behaviour.

Versatile nanodot-patterned Gore-Tex fabric for multiple energy harvesting in wearable and aerodynamic nanogenerators

The ongoing expedition to harvest ambient renewable energies from the environment by wearable fabric-based nanogenerators is a promising route to sustainably drive the small electronics with unprecedented opportunities in next-generation self-powered devices. Here, we report a simple method to fabricate a washable, breathable and wearable triboelectric nanogenerator that harvests the energy of triboelectricity through an enhanced friction surface area made of the gold nanodot-pattern crafted by electron-beam sputtering on an inexpensive polyurethane surface. The gold deposition which crops-up as regular small islands, under oxygen plasma is subsequently, etched into nanodot-pattern on a polyurethane surface to convert mechanical energy into an electrical signal via in-plane sliding mode with a maximum output of ~2 μW. The nanodot engineering plays an important role to improve the active sliding frictional area, as well as the corresponding output-performance of the triboelectric nanogenerator. To demonstrate the potential applications of our approach, we designed a self-powered wearable device integrated with clothes to harvest different kinds of mechanical energies from the human motion. To elevate the power output-performance, we fabricated waterproof fiber with flutter membrane and quantified triboelectric charge against airflow speed. At mild wind speed, the fabricated triboelectric nanogenerator shows a maximum output of 70 µW. Besides, as an example of practical application, the nanogenerator constructed can produce an improved capacitor charge voltage to drive dozens of light-emitting diodes and apply them to low power consumption devices. This technology is produced in a simple and cost-effective manner and reports an easy way to produce an energy harvesting system based on triboelectric effects using a sustainable and renewable energy source of body motions and air flows. This system is expected to be one of the best green energy sources for portable and wearable electronic devices in the near future.

Exploring conservative islands using correlated and uncorrelated noise.

In this work, noise is used to analyze the penetration of regular islands in conservative dynamical systems. For this purpose we use the standard map choosing nonlinearity parameters for which a mixed phase space is present. The random variable which simulates noise assumes three distributions, namely equally distributed, normal or Gaussian, and power law (obtained from the same standard map but for other parameters). To investigate the penetration process and explore distinct dynamical behaviors which may occur, we use recurrence time statistics (RTS), Lyapunov exponents and the occupation rate of the phase space. Our main findings are as follows: (i) the standard deviations of the distributions are the most relevant quantity to induce the penetration; (ii) the penetration of islands induce power-law decays in the RTS as a consequence of enhanced trapping; (iii) for the power-law correlated noise an algebraic decay of the RTS is observed, even though sticky motion is absent; and (iv) although strong noise intensities induce an ergodic-like behavior with exponential decays of RTS, the largest Lyapunov exponent is reminiscent of the regular islands.

Hierarchical collapse of regular islands via dissipation

In this work we investigate how regular islands localized in a mixed phase-space of generic area-preserving Hamiltonian systems are affected by a small amount of dissipation. Mainly we search for a universality (hierarchy) in the convergence of higher-order resonances and their periods when dissipation increases. One very simple scenario is already known: when subjected to small dissipation, stable periodic points become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase-space in chaotic and regular domains. However, performing numerical experiments with the paradigmatic Chirikov–Taylor standard mapping we show that this presumably simple scenario can be rather complicated. The first, not trivial, scenario is what happens to chaotic trajectories, since they can be attracted by the sinks or by chaotic attractors, in cases when they exist. We show that this depends very much on how basins of attraction are formed as dissipation increases. In addition, we demonstrate that higher-order resonances are usually first affected by small dissipation when compared to lower-order resonances from the conservative case. Nevertheless, this is not a generic behaviour. We show that a local hierarchical collapse of resonances, as dissipation increases, is related to the area of the islands from the conservative case surrounding the periodic orbits. All observed resonance destructions occur via the bifurcation phenomena and are quantified here by determining the largest finite-time Lyapunov exponent.

Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.