Successive Bifurcations Research Articles

Analysis of Successive Doubly Nested Mixed-Mode Oscillations

In previous works [Inaba & Kousaka, 2020; Inaba & Tsubone, 2020; Inaba et al., 2023], significant bifurcation structures referred to as nested Mixed-Mode Oscillations (MMOs) were found to be present in forced Bonhoeffer–van der Pol (BVP) oscillators. It is well known that unnested Mixed-Mode Oscillation-Incrementing Bifurcations (MMOIBs) can generate [Formula: see text] oscillations (i.e. [Formula: see text] followed by [Formula: see text] repeated [Formula: see text] times) for successive values of [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent fundamental simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. Furthermore, it has been confirmed that MMOIBs can generate nested MMOs. Let two adjacent unnested MMOIB-generated MMOs be denoted [Formula: see text] and [Formula: see text]. Then, singly nested MMOIBs can generate [Formula: see text] for successive values of [Formula: see text], i.e. [Formula: see text] followed by [Formula: see text] repeated [Formula: see text] times, between the [Formula: see text]- and [Formula: see text]-generating regions. The sequential generation of singly nested MMOs has been investigated in detail in previous work [Ito et al., 2021]. Nested MMOs can, however, be at least doubly nested. In this study, we investigate doubly nested MMOs considering a constrained nonautonomous BVP oscillator containing an idealized diode. Based on the observed dynamics of this system, Poincaré return maps are rigorously constructed in one dimension. Therefore, we can solve the successive saddle-node bifurcations using a nested (i.e. double-loop) bisection method. We track 60 successive doubly nested MMOIBs and we do not rule out the possibility that the 58 scaling constants corresponding to the MMOIB intervals converge to unity. We note that because we solve the bifurcation equation avoiding the use of a method that requires the careful selection of the initial conditions (e.g. the Newton–Raphson), we can accurately track the saddle-node bifurcations without missing any doubly nested MMO sequences.

On the collision dynamics in a molecular model

We study the problem of the hydrogen atom submitted to a circularly polarized microwave field. This problem, analyzed from a classical mechanics approach, can be modeled by an autonomous Hamiltonian depending on one parameter K>0. The paper is focused on the so called n-ejection–collision orbits (n-EC orbits), that is orbits that the electron describes when it ejects from the nucleus and collides with it at the n relative minimum in the distance with respect to the nucleus. In this work, we analyze the evolution of the families of n-EC orbits. We conduct a comprehensive numerical analysis of the bifurcations, which involves multiple precision computations, to characterize the successive bifurcation families that emerge. Additionally, we examine the periodic and quasi-periodic motion of the n-EC orbits belonging to these bifurcation families.

Microfluidic devices for precise measurements of cell directionality reveal a role for glutamine during cell migration

Cancer cells that migrate from tumors into surrounding tissues are responsible for cancer dissemination through the body. Microfluidic devices have been instrumental in discovering unexpected features of cancer cell migration, including the migration in self-generated gradients and the contributions of cell–cell contact during collective migration. Here, we design microfluidic channels with five successive bifurcations to characterize the directionality of cancer cell migration with high precision. We uncover an unexpected role for glutamine in epithelial cancer cell orientation, which could be replaced by alfa-keto glutarate but not glucose.

Rapid-reaction kinetics of the bifurcating NAD+-dependent NADPH:ferredoxin oxidoreductase NfnI from Pyrococcus furiosus

The rapid-reaction kinetic behavior of the NAD+-dependent NADPH:ferredoxin oxidoreductase (NfnI) has been investigated. NfnI is a heterodimeric transhydrogenase of the flavin-based electron bifurcating class of flavoproteins and in the course of catalysis takes two electron pairs from NADPH to reduce two ferredoxins and one NAD+ through two successive bifurcation events. NADPH reduction takes place at the bifurcating FAD of NfnI’s large subunit, with high-potential electrons transferred to the [2Fe-2S] cluster and S-FADH of the small subunit, ultimately on to NAD+, and low-potential electrons transferred to two [4Fe-4S] clusters of the large subunit, ultimately on to ferredoxin. Reduction of NfnI by NADPH goes to completion only at higher pH, with a limiting kred of 36±1.6 s-1 and apparent KdNADPH of 5±1.2 μM. Reduction of one of the [4Fe-4S] clusters of NfnI occurs within a second, as monitored by electron paramagnetic resonance (EPR) spectroscopy, indicating that in the absence of NAD+ the system can bifurcate with electrons entering the low potential pathway. When enzyme is reduced by NADPH in the absence of NAD+ but the presence of ferredoxin, up to three equivalents of ferredoxin become reduced in the course of the reaction, although the reaction is considerably slower than steady-state turnover, likely due to a previously identified conformational change in NfnI. Bifurcation thus appears to be rate-limited by transfer of the first, high-potential electron into the high-potential pathway. The observation of ferredoxin reduction in the absence of NAD+ demonstrates that the process of electron bifurcation is an intrinsic property of the bifurcating FAD and not strictly dependent on the simultaneous presence of NAD+ and ferredoxin. The tight coupling between NAD+ and ferredoxin reduction observed in steady-state assays with NfnI is instead simply due to the need to remove reducing equivalents from the high-potential electron pathway under multiple-turnover conditions. Reoxidation of pre-reduced enzyme with NAD+ is well-behaved, with a limiting kox of 50±0.7 s-1 and apparent KdNAD+ of 8±0.6 μM.

Prediction and suppression of chaos following flutter in wind turbines

Chaotic motion in a fluttering wind turbine blade is investigated by the development of an efficient analytical predictive model that is then used to suppress the phenomenon. Flutter is a dynamic instability of an elastic structure in a fluid, such as an airfoil section of a wind turbine blade. It is presently modelled using generalised two degree of freedom coupled modes of a blade airfoil section (pitch and plunge) combined with local unsteady aerodynamics, based on flutter derivatives and a continuous bilinear lift curve under damping. The mode coupling causes instability and limit cycle flutter due to a Hopf bifurcation. Following the critical flutter speed, the response can transition to chaos through successive other bifurcations like period doubling. New closed-form conservative analytical conditions for chaos following blade flutter are identified and discussed for the wind turbine section taking into account the blade geometry and optimal design of the wind turbine. These predictions are numerically verified for a range of conditions including stall slope and damping. The results confirm that chaos following blade flutter can occur due to nonlinearities in the aerodynamics, i.e. due to a bilinear lift law. This phenomenon is then suppressed to unrealistically high wind speeds and/or eliminated by quantified variation of system parameters using the predictive model. The results show that small changes in tip speed ratio (−15%), and stall slope factor (−17%) can eliminate or suppress chaos following flutter, while, in general, larger magnitude changes in dynamic parameters (i.e. mass, inertia > 81%, stiffness > 97%, damping > 100%) are required to achieve the same, by detuning the coupled plunge and pitch natural frequencies or damping out overlapping parametric resonances. These results also highlight that the analytical predictions can remarkably be generalized to any parameter set and provide almost instantaneous calculations representing many thousands of numerical simulations from many bifurcation diagrams (computational acceleration factor of 107 times). General insight is also provided into the occurrence and suppression of airfoil chaos following flutter in aeroelastic structures like wind turbines.

Numerical study on wrinkling of gradient shell on rigid cylindrical substrate for patterned functional fabric

Inspired by wrinkled surfaces such as Earthworms in nature, the construction of wrinkled patterns on fiber surfaces provides a novel idea for high-performance functional fibers and fabrics. Recent theoretical and experimental studies focused on wrinkling patterns on compliant substrates, however, most industrial fibers are rigid. Based on previous experiments of wrinkling on rigid PET fiber from gradient shell, herein the wrinkling formation and evolution based on a novel gradient shell-rigid core (GSRC) system is studied via the nonlinear finite element method (FEM). The results demonstrate that wrinkling is formed due to the stress-induced mutagenesis generated from the gradient shell modulus. The higher gradient factor of shell modulus leads to smaller critical mismatched strain, i.e., easier formation of surface wrinkling. Then a post-buckling phenomenon in this system is studied, involving a successive bifurcation: smooth-wrinkle-ridge transition. Meanwhile, the evolution of topological patterns is significantly influenced by shell-induced curvature. This study not only benefits the understanding of wrinkling gradient shell on rigid cylindrical substrate but also provides useful guides for the fabrication of surface patterned functional fiber or fabrics.

Red blood cell lingering modulates hematocrit distribution in the microcirculation

The distribution of red blood cells (RBCs) in the microcirculation determines the oxygen delivery and solute transport to tissues. This process relies on the partitioning of RBCs at successive bifurcations throughout the microvascular network, and it has been known since the last century that RBCs partition disproportionately to the fractional blood flow rate, therefore leading to heterogeneity of the hematocrit (i.e., volume fraction of RBCs in blood) in microvessels. Usually, downstream of a microvascular bifurcation, the vessel branch with a higher fraction of blood flow receives an even higher fraction of RBC flux. However, both temporal and time-average deviations from this phase-separation law have been observed in recent studies. Here, we quantify how the microscopic behavior of RBC lingering (i.e., RBCs temporarily residing near the bifurcation apex with diminished velocity) influences their partitioning, through combined in vivo experiments and in silico simulations. We developed an approach to quantify the cell lingering at highly confined capillary-level bifurcations and demonstrate that it correlates with deviations of the phase-separation process from established empirical predictions by Pries et al. Furthermore, we shed light on how the bifurcation geometry and cell membrane rigidity can affect the lingering behavior of RBCs; e.g., rigid cells tend to linger less than softer ones. Taken together, RBC lingering is an important mechanism that should be considered when studying how abnormal RBC rigidity in diseases such as malaria and sickle-cell disease could hinder the microcirculatory blood flow or how the vascular networks are altered under pathological conditions (e.g., thrombosis, tumors, aneurysm).

Bifurcations of Grid-Following Rectifiers and Routes to Voltage Instability in Weak AC Grids

As indispensable components in modern power systems, grid-following rectifiers would interact with the grid. Such interaction, however, may lead to unexpected collapse of the ac bus voltage. In this paper, three routes to voltage instability are identified, namely, (i) supercritical Hopf bifurcation and successive saddle-node bifurcation of periodic orbits (SNPO); (ii) subcritical Hopf bifurcation; and (iii) departure from a stable basin of attraction. The SNPO and subcritical Hopf bifurcation are reported for the first time in rectifier-based power systems. It is also found that the rectifier-based system may undergo a Bautin bifurcation, which changes a Hopf bifurcation from supercritical to subcritical, or vice versa. The current loop of the grid-following rectifier has been found to play a crucial role in these bifurcation behaviors. Considering the inevitable and immediate voltage breakdown after the onset of a subcritical Hopf bifurcation, it is recommended to increase the bandwidth of the current loop to trigger the Bautin bifurcation, so that only supercritical Hopf bifurcation may occur. Furthermore, the original high-order and nonlinear model of the rectifier-based system can be simplified to a reduced order system by focusing on the center manifold. Based on the reduced system, the occurrence mode of Hopf bifurcation can be confirmed by inspecting the first Lyapunov coefficient. Finally, relevant bifurcation diagrams are derived, providing practical design guidelines to avoid an eventual system collapse.

Temporal-spatial heterogeneity of hematocrit in microvascular networks

Hematocrit, defined as the volume percentage of red blood cells in blood, is an important indicator of human health status, which demonstrates the capability of blood to deliver oxygen. It has been studied over many decades using in vivo, in vitro, and in silico experiments, and recent studies have shown that its major feature in microvascular networks is the temporal-spatial heterogeneity. The present work is a numerical study of such temporal-spatial heterogeneity, based on direct simulations of cellular-scale blood flow in complex microvascular networks. The simulations take into account the cell deformation and aggregation and thus are able to capture both the three-dimensional dynamics of each individual cell and the temporal-spatial distribution of cell population. The results showed that the temporal-spatial heterogeneity is more pronounced in the network that has the vessels with smaller diameters or with more complex geometry. Such heterogeneity is largely attributed to the existence of bifurcations, where the positively correlated hypotactic (feeding-branch) and paratactic (branch-branch) relations are generally observed in both the time-averaged hematocrit and temporal hematocrit ranges. This suggests that the successive bifurcations have a substantial impact on the temporal-spatial heterogeneity of hematocrit. However, these positive correlations may be broken up if the diameter of the feeding vessel is small enough or the bifurcation is asymmetric extremely, due to the vessel blockage. The present study is of great clinical significance to help doctors make more accurate diagnosis and treatment, by providing more information about the temporal-spatial distribution of the hematocrit in microvascular networks.

The Theory of Probabilistic Hierarchical Learning for Classification

Providing the ability of classification to computers has remained at the core of the faculty of artificial intelligence. Its application has now made inroads towards nearly every walk of life, spreading over healthcare, education, defence, economics, linguistics, sociology, literature, transportation, agriculture, and industry etc. To our understanding most of the problems faced by us can be formulated as classification problems. Therefore, any novel contribution in this area has a great potential of applications in the real world. This paper proposes a novel way of learning from classification datasets i.e., hierarchical learning through set partitioning. The theory of probabilistic hierarchical learning for classification has been evolved through several works while widening its scope with each instance. The theory demonstrates that the classification of any dataset can be learnt by generating a hierarchy of learnt models each capable of classifying a disjoint subset of the training set. The basic assertion behind the theory is that an accurate classification of complex datasets can be achieved through hierarchical application of low complexity models. In this paper, the theory is redefined and revised based on four mathematical principles namely, principle of successive bifurcation, principle of two-tier discrimination, principle of class membership and the principle of selective data normalization. The algorithmic implementation of each principle is also discussed. The scope of the approach is now further widened to include ten popular real-world datasets in its test base. This approach does not only produce their accurate models but also produced above 95% accuracy on average with regard to the generalising ability, which is competitive with the contemporary literature.

Surface tension force on a partially submerged horizontal concave cylinder

A horizontal cylinder with a concave cross-section partially submerged in a liquid at a given position may permit multiple menisci around itself. The number and stabilities of the menisci are analysed, and how the menisci change during the processes of gradually hoisting and lowering the cylinder is explained by bifurcation theory. The restoring force on the concave cylinder and the rebounding potential energy (defined as the work done by the restoring force during the whole hoisting process to represent the potential rebounding capacity of a cylinder on water) are also investigated. The results show that, when the radius of the concave arc is smaller than the critical value, the concave cylinder at a given position permits multiple possible menisci. The equilibria form fold bifurcations with the position of the cylinder as the bifurcation parameter, and two successive fold bifurcations can form a one-fold hysteresis loop. The force–distance curve representing the relation between the restoring force and the position of the cylinder also has corresponding hysteresis loops, where the restoring force will jump (i.e. change discontinuously) at the bifurcation points. In contrast to a convex cylinder, a concave cylinder can have different values of the restoring force at the same height because of multiple menisci, and the values depend on whether it is hoisted or lowered. Under the condition of a fixed cross-sectional area, the optimal cross-sectional shape is determined when the maximum rebounding potential energy is reached, and it is close to the shape with the critical concave arc angle for the existence of multiple possible menisci. The cross-sections with concave parts are preferable to circular, laterally planed and corner-concave cross-sections. This paper provides an effective method of enhancing the restoring force and potential rebounding height of a robotic water strider insect or particles on the surface of water.

Navigating Knowledge Frameworks at the Intercultural Interface

As we emerge not altogether unscathed in 2022 into what optimistically might be called a postpandemic world, we are confronted by the pressing need to address global and climate instabilities against a general backdrop of complexity. Potential solutions must be balanced against environmental and societal concerns that cannot take for granted that any system is somehow isolated. Here then is the crux of new materialist and post-humanist approaches – a shift “away from Kant”1 and away it seems, from humancentric understandings of who, or what, has agency in the world. Despite acknowledging the agencies of non-human others, such as electrical grids2 and quantum entanglement,3 or proposing new speculative realist frameworks by which to engage with such agentic capacities,4 finding workable solutions within such dynamics remains stubbornly difficult. What does become clear, at least, is that these Eurocentric traditions, arising from the European Enlightenment project, have not served the environment particularly well. Newtonian physics can no longer claim mastery over the tangible world through recourse to universal laws acting in isolation, and liberal humanism is revealed to be underpinned by Eurocentric cultural traditions of human exceptionalism and the rights of the individual exceeding the rights of the collective. As I have argued elsewhere,5 such traditions within the European imaginary arise from Judeo-Christian notions of dominion over the nonhuman and are reinforced by successive bifurcations between nature and culture through Plato/Aristotle-Descartes-Kant metaphysical trajectories.

Further Analysis of the Passive Dynamics of the Compass Biped Walker and Control of Chaos via Two Trajectory Tracking Approaches

This work consists in analyzing and controlling the walk of the compass-type bipedal walker in order to stabilize its passive dynamic gait. The dynamic walking of the compass-gait walker is modeled by an impulsive hybrid nonlinear system. This impulsive hybrid nature is considered very complex as it can generate unwanted phenomena such as chaos and bifurcations. We show first by means of bifurcation diagrams and by varying the slope angle of the walking surface and also the length of the lower leg segment that the passive dynamic walking exhibits successive period-doubling bifurcations leading to chaos. Furthermore, in order to control chaos and hence obtain one-periodic walking behavior, we propose two control approaches based on tracking a desired trajectory. The first method consists in tracking the one-periodic passive dynamic walking generated by the compass model itself. The second control method lies in following a planned trajectory using the 4th-order Spline function. An optimization method is also achieved to design the parameters of the desired trajectory. Some features of the period-1 passive gait are used in the design of such Spline trajectory. Finally, we show some simulation results revealing the efficiency of the two proposed control methods in the control of the chaotic passive gait of the compass-gait walker. Moreover, we demonstrate the stabilization of the bipedal locomotion of the compass biped walker on different slopes: descending and ascending inclined planes and walking on a level ground. A comparison with the OGY-based control method is also performed to further show the superiority of these two control approaches.

Modulated rotating waves and triadic resonances in spherical fluid systems: The case of magnetized spherical Couette flow

The existence of triadic resonances in the magnetized spherical Couette system is related to the development of modulated rotating waves, which are quasiperiodic flows understood in terms of bifurcation theory in systems with symmetry. In contrast to previous studies in spherical geometry, the resonant modes are not inertial waves but related to the radial jet instability, which is strongly equatorially antisymmetric. We propose a general framework in which triadic resonances are generated through successive Hopf bifurcations from the base state. The study relies on an accurate frequency analysis of different modes of the flow, for solutions belonging to two different bifurcation scenarios. The azimuthal and latitudinal nonlinear coupling among the resonant modes is analyzed and interpreted using spherical harmonics, and the results are compared with previous studies in spherical geometry.

Nanosleeves: Morphology transitions of infilled carbon nanotubes

Morphology instability of substrate-supported carbon atomic layers can be harnessed to modulate physical properties and functions, which has drawn interesting attention. Curvature would be a critical factor affecting surface morphology and its stability characteristics. Infilled carbon nanotubes, that is to say carbon monolayers with curved geometry and infilled substrates, namely nanosleeves, widely exist in the literature and have many potential applications. Here, we reveal an unprecedented rich post-buckling phenomenon of nanosleeves, which involves multiple successive bifurcations: smooth-wrinkle-ridge-sagging transitions. The nanosleeve initially buckles into periodic axisymmetric wrinkles at the threshold and then evolves into a localized mode with one single ridge growing upon further axial compression. Finally, the amplitude of the ridge reaches a limit and the symmetry is broken with the ridge sagging into a recumbent fold that can be axisymmetric or non-axisymmetric. Such rich morphology transition is first revealed by molecular dynamics simulation, and then theoretically predicted by a finite strain core–shell continuum model. An analytical solution of the critical threshold of primary smooth-wrinkle transition is further provided, which explicitly indicates the crucial role played by curvature and van der Waals interaction between the carbon monolayers and infilled metals. Such multiple-bifurcation scenario and curvature-determined mechanism are found to be inherently universal.

Emergent cell-free layer asymmetry and biased haematocrit partition in a biomimetic vascular network of successive bifurcations.

Blood is a vital soft matter, and its normal circulation in the human body relies on the distribution of red blood cells (RBCs) at successive bifurcations. Understanding how RBCs are partitioned at bifurcations is key for the optimisation of microfluidic devices as well as for devising novel strategies for diagnosis and treatment of blood-related diseases. We report the dynamics of RBC suspensions flowing through a biomimetic vascular network incorporating three generations of microchannels and two classical types of bifurcations at the arteriole level. Our microfluidic experiments with dilute and semidilute RBC suspensions demonstrate the emergence of excessive heterogeneity of RBC concentration in downstream generations upon altering the network's outflow rates. Through parallel simulations using the immersed-boundary-lattice-Boltzmann method, we reveal that the heterogeneity is attributed to upstream perturbations in the cell-free layer (CFL) and lack of its recovery between consecutive bifurcations owing to suppressed hydrodynamic lift under reduced flow conditions. In the dilute/semidilute regime, this perturbation dominates over the effect of local fractional flow at the bifurcation and can lead to inherently unfavourable child branches that are deprived of RBCs even for equal flow split. Our work highlights the importance of CFL asymmetry cascading down a vascular network, which leads to biased phase separation that deviates from established empirical predictions.

Review article: Hilbert problems for the climate sciences in the 21st century – 20 years later

Abstract. The scientific problems posed by the Earth's atmosphere, oceans, cryosphere – along with the land surface and biota that interact with them – are central to major socioeconomic and political concerns in the 21st century. It is natural, therefore, that a certain impatience should prevail in attempting to solve these problems. The point of a review paper published in this journal in 2001 was that one should proceed with all diligence but not excessive haste, namely “festina lente”, i.e., “to hurry in a measured way”. The earlier paper traced the necessary progress through the solutions of 10 problems, starting with “What can we predict beyond 1 week, for how long, and by what methods?” and ending with “Can we achieve enlightened climate control of our planet by the end of the century?” A unified framework was proposed to deal with these problems in succession, from the shortest to the longest timescale, i.e., from weeks to centuries and millennia. The framework is that of dynamical systems theory, with an emphasis on successive bifurcations and the ergodic theory of nonlinear systems, on the one hand, and on pursuing this approach across a hierarchy of climate models, from the simplest, highly idealized ones to the most detailed ones. Here, we revisit some of these problems, 20 years later,1 and extend the framework to coupled climate–economy modeling.

A widespread interaction between generalist and specialist enemies: The role of intraguild predation and Allee effect

A natural food web system involves a diverse community of natural enemies and one of the widespread and common phenomena of such systems is intraguild predation (IGP) (i.e., presence of eating and killing among potential competitors). Here, we investigate the dynamics of a food web system with Allee effect and intraguild predation. In the present study, generalist and specialist natural enemies competing for a shared resource (prey) (intraguild predation) have been introduced. The Allee effect has also been incorporated in the shared prey growth rate. We investigate how parameters defining Allee effect and intraguild predation affect the long-term persistence, extinction and co-existence regions of such species. We outline the conditions under which different types of interior and non-interior equilibria exist and are locally stable. Bi-stable dynamics has also been investigated for the proposed model system for a suitable range of parametric values. A threshold condition on the strength of Allee effect has been obtained assuring the absence of IG predator population along with extinction region of shared prey. To understand the dynamics of the system, a comprehensive study of bifurcation analysis has also been provided taking Allee effect and fertility rate of intraguild predator as bifurcation parameters. These two parameters generate various interesting bifurcations like saddle-node bifurcation and Hopf-bifurcation. We have obtained different parametric regions of Allee parameter for the existence of different boundary and interior equilibria. All the analytical results related with local stability of equilibrium points and all possible successive bifurcations have been supported by different numerical examples, one and two parameter bifurcation diagrams, bi-stability diagram and stability regions of all possible equilibrium points. The impacts of Allee effect on co-existence, stability, extinction of species, their persistence, bistability and bifurcations have been explicitly discussed and the whole dynamics has also been successfully compared with the dynamics of food web without Allee effect. It is observed that the introduction of Allee effect and IG predator induce more rich dynamics and compel the system to be more sensitive to initial population densities.

Using chaotic advection for facile high-throughput fabrication of ordered multilayer micro- and nanostructures: continuous chaotic printing

This paper introduces the concept of continuous chaotic printing, i.e. the use of chaotic flows for deterministic and continuous extrusion of fibers with internal multilayered micro- or nanostructures. Two free-flowing materials are coextruded through a printhead containing a miniaturized Kenics static mixer (KSM) composed of multiple helicoidal elements. This produces a fiber with a well-defined internal multilayer microarchitecture at high-throughput (>1.0 m min−1). The number of mixing elements and the printhead diameter determine the number and thickness of the internal lamellae, which are generated according to successive bifurcations that yield a vast amount of inter-material surface area (∼102 cm2 cm−3) at high resolution (∼10 µm). This creates structures with extremely high surface area to volume ratio (SAV). Comparison of experimental and computational results demonstrates that continuous chaotic 3D printing is a robust process with predictable output. In an exciting new development, we demonstrate a method for scaling down these microstructures by 3 orders of magnitude, to the nanoscale level (∼150 nm), by feeding the output of a continuous chaotic 3D printhead into an electrospinner. The simplicity and high resolution of continuous chaotic printing strongly supports its potential use in novel applications, including—but not limited to—bioprinting of multi-scale layered biological structures such as bacterial communities, living tissues composed of organized multiple mammalian cell types, and fabrication of smart multi-material and multilayered constructs for biomedical applications.

On the Detuned $$2\!{:}\!4$$ Resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.