Labyrinth Patterns Research Articles

Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response

In this work, a tritrophic food chain model has been proposed by incorporating Ivlev-like nonmonotonic functional response, where prey is equipped with defense ability. We have performed a detailed dynamical study and pattern formation analysis to obtain complex dynamics of the proposed system. Stability and bifurcation analysis have been performed in the model system. Persistence and permanence are discussed. Bifurcations of codimension-1, in particular, saddle-node, transcritical and Hopf bifurcation are observed. The model system also exhibits bifurcations of codimension-2 such as cusp, Bogdanov-Takens and generalized Hopf bifurcation. Interestingly, it is observed that the middle and top predator population become extinct due to defense ability of prey. Chaotic dynamics is observed via a period-doubling route to chaos with the change in the value of parameter β. The quantification of chaotic dynamics is done, using Lyapunov spectrum and sensitivity analysis. Diffusion induced chaos is studied in the spatiotemporal model system. Hopf bifurcation is seen in the case of a spatially extended system. Further, conditions for Turing instability have been obtained. Pattern formation study is done. In the two-dimensional spatial domain, various non-Turing patterns such as hot-spot, cold-spot, labyrinth patterns are obtained. Ripple and stripe Turing patterns are obtained in case of one-dimensional spatial domain. Also, labyrinth and patchy Turing patterns are obtained in the two-dimensional spatial domain. The spatial distribution of the species shows Turing patterns at the low cost of β, while the increased cost of β changes Turing patterns to non-Turing patterns. Throughout the study, we observe that the parameter β plays an important role in group defense mechanism and is the most sensitive parameter leading to vital change in system dynamics. A wide range of Turing and non-Turing patterns obtained in this work has not been reported so far in literature in any model with group defense.

Inkjet-Printed Quantum Dot Fluorescent Security Labels with Triple-Level Optical Encryption.

Optical security labels play a significant role in protecting both our wealth and health. However, simultaneously meeting the requirements including low-cost fabrication, easy detection, and high-level security is still challenging for security labels. Here, we design an unclonable anti-counterfeiting system with triple-level security by using the inkjet printing technique, which can be authenticated by naked eyes, a portable microscope, and a fluorescence microscope. These labels are achieved by printing microscale quantum dot (QD) ink droplets on premodified substrates with random-distributed glass microspheres. Due to the unique capillary action induced by the glass microspheres, QDs in the ink droplets are deposited around the microspheres, forming microscale multicircular patterns. Multiple pinning of QDs at the three-phase contact lines appears during the evaporation of the droplet, resulting in the formation of a nanoscale labyrinthine pattern around the microspheres. The nanoscale labyrinth pattern and the microscale multicircular microsphere array, together with the printed macroscopic image, constitute a triple-level progressive anti-counterfeiting system. Moreover, the system is compatible with an artificial intelligence-based identification strategy that allows rapid identification and verification of the unclonable security labels.

Transition from regular to labyrinth pattern in a neuronal network with fast inhibitory synapses

Pattern selection among spiral wave or target wave and irregular patterns plays a key role in cortical information processing in mammalian neocortex. In this paper, we study transition between spiral wave and labyrinth pattern in neural network in which neurons are coupled with fast inhibitory synapse. Transition between regular patterns when neurons are connected by fast inhibitory synapse is quite different from that of network with fast excitatory synapse. It is a complex scenario of transition from regular wave to labyrinth pattern. Four types of spatiotemporal patterns are discovered in the network, namely, rectangular spiral wave, chimera state, circular spiral wave and labyrinth pattern. When the labyrinth pattern occupies the whole domain of network, firing of every neuron is periodic motion. But transition from the regular wave to labyrinth pattern is accompanied with onset of intermittency of neuron firing at the corners of network. Occurrence of intermittency of individual neuron perfectly matches the appearance of circular spiral wave, which is explained by nonlinear analysis. Indeed, dynamics of single neuron electrical activities can also be considered as an index of pattern transition.

Bifurcation and deformation during the evolution of periodic patterns on a gel film bonded to a soft substrate

In this study, we investigate the bifurcation and deformation during the evolution of periodic patterns on a gel film bonded to a soft substrate. 3D finite element analysis is performed using an inhomogeneous field theory for polymeric gels. Step-by-step eigenvalue buckling analysis is conducted to explore not only the first bifurcation, but also sequential bifurcations on bifurcated paths. When the hexagonal dimple mode occurs as the first bifurcation, the second bifurcation consists of rectangular checkerboard modes in three symmetric directions. The resulting deformation patterns are in good agreement with experiments and, surprisingly, are analogous to the in-plane buckling behavior of hexagonal honeycombs. Uniaxial, biaxial, and equibiaxial (flower-like) patterns are produced by the periodic arrangements of distorted dimples. The third and fourth bifurcations cause the coalescence of the selected dimples. This reveals the occurrence of the rectangular checkerboard modes at the second bifurcation to be the missing link in the pattern evolution from hexagonal dimples to herringbone and labyrinth patterns.

Formation of zero-field skyrmion arrays in asymmetric superlattices

We demonstrate the formation of metastable Néel-type skyrmion arrays in Pt/Co/Ni/Ir multi-layers at zero-field following the ex situ application of an in-plane magnetic field using Lorentz transmission electron microscopy. The resultant skyrmion texture is found to depend on both the strength and misorientation of the applied field as well as the interfacial Dzyaloshinskii–Moriya interaction. To demonstrate the importance of the applied field angle, we leverage bend contours in the specimens, which coincide with transition regions between skyrmion and labyrinth patterns. The subsequent application of a perpendicular magnetic field near these regions reveals the unusual situation where skyrmions with opposite magnetic polarities are stabilized in close proximity.

Controlled wrinkling patterns and surface friction by thickness gradient

Gradient changing patterns often endow the surfaces with intriguing functionalities. Spontaneous wrinkling of film-substrate bilayer opens up a new possibility to create heterogeneous surfaces. In this work, we prepared the gradient changing wrinkle surfaces of Ag films and Mo films with identical thickness gradient on polydimethylsiloxane substrates by magnetron sputtering. It is revealed that Ag films show only labyrinth patterns, regardless of thickness gradient, while Mo films exhibit the pattern transition from labyrinth to herringbone along the thickness gradient direction. This can be attributed to the combined effect of the compressive stress, the thickness gradient and material property. The wavelength of Ag films and Mo films increases with increasing distance. Ag films show isotropic friction, which increases with increasing distance due to the increased surface roughness. In contrast, the friction properties of Mo films exhibit two-regime behavior, namely, isotropic friction and anisotropic friction, depending on the wrinkling patterns. The isotropic/anisotropic friction is closely related to the orientation-dependent topography. These results provide better understanding of the combined effect of thickness gradient and film constituent on the gradient changing wrinkle patterns and pave a new way for designing gradient surfaces with controlled friction behavior by harnessing wrinkling instability.

Thermal wrinkling of liquid crystal polymer shell/core spheres

Smart soft materials that can flexibly respond to external multi-physics stimuli, have shown intriguing applications in shape-morphing and morphology control. Here, we present tunable wrinkling patterns in core–shell spheres under thermal load via controlling the orientation of director in nematic liquid crystal polymer (LCP). To analyze nonlinear instability and morphological evolution of LCP shell/core spheres, we develop a shallow core–shell model that accounts for director-induced anisotropic spontaneous strains. By tuning the alignment of liquid crystal director, we explore effects of anisotropy of spontaneous strains on wrinkling pattern formation and evolution. When the director is along the out-of-plane direction, the surface shell undergoes equi-biaxial expansion, resulting in hexagonal, checkerboard, herringbone or labyrinth pattern, determined by a single dimensionless parameter Cs which characterizes the stiffness ratio and curvature of the system. Moreover, stability analysis yields an analytical solution of the critical wrinkling strain and wavelength for equi-biaxially stressed shell/core spheres. When the director is in-plane aligned, leading to primary uniaxial expansion upon heating, the core–shell usually buckles into stripes. For general spatial director alignment, the ultimate wrinkling patterns depending on the anisotropy state of spontaneous strains, can be hexagonal, parallel bead-chain, stripe, herringbone, labyrinth or hybrid. Lastly, we provide director-affected phase diagrams on pattern selection, which could be used to quantitatively guide the effective design of morphology-related smart surfaces such as anticounterfeiting system.

Supermixed labyrinth fractals

Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.

Joule self-heating effects and controlling oxygen vacancy in La0.8Ba0.2MnO3 ultrathin films with nano-sized labyrinth morphology

Electric current induced Joule heating effects have been investigated in La0.8Ba0.2MnO3 ultrathin films deposited on a LaAlO3(001) single crystal substrate with a smaller lattice constant by using the sol–gel method. By applying moderate bias currents (∼10 mA), it is found that Joule self-heating simply gives rise to a temperature deviation between the thermostat and the test sample, but the intrinsic ρ(T) relationship measured at a low current (0.1 mA) changes a little. However, it is noteworthy that the low-temperature transport behavior degrades from the metallic to the insulating state after applying higher bias currents (>31 mA) in vacuum. Furthermore, the metallic transport can be recovered by placing the degraded film in air. The results clearly suggest that the oxygen vacancy in the La0.8Ba0.2MnO3 films is controllable in different atmospheres, particularly with the aid of the Joule self-heating. According to the SEM images, we attribute the controlled oxygen vacancy to the nano-sized labyrinth pattern of the films, where the large surface-to-volume ratio plays a crucial role.

TRIANGULAR LABYRINTH FRACTALS

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.

Wrinkling pattern evolution on curved surfaces

We investigate the effects of surface curvature on the wrinkling pattern evolution in soft materials. Theoretical analysis and the Fourier spectral method are combined to understand the occurrence of surface wrinkling and pattern transitions. We reveal that surface curvature and its anisotropy play a key role in wrinkling pattern transitions, for example, from the sinusoidal to the hexagonal mode. Based on the nonlinear equilibrium equations, a Fourier spectral method is developed to track the surface wrinkling pattern evolution in a curved bilayer system. This method is validated by comparison between the simulation results and relevant experiments. The simulations show that a hexagonal phase may evolve into either a bistable or labyrinth phase, depending on the curvature anisotropy, the excess stress, a dimensionless curvature parameter, and the curvature gradient. This work can not only help understand the pattern formation in some natural systems, but the methods presented here can also find a broad range of technical applications, for examples, design of anticounterfeiting systems with high-security levels using labyrinth patterns on curved surfaces.

Predator\u2013prey pattern formation driven by population diffusion based on Moore neighborhood structure

Diffusion-driven instability is a basic nonlinear mechanism for pattern formation. Therefore, the way of population diffusion may play a determinative role in the spatiotemporal dynamics of biological systems. In this research, we launch an investigation on the pattern formation of a discrete predator–prey system where the population diffusion is based on the Moore neighborhood structure instead of the von Neumann neighborhood structure widely applied previously. Under pattern formation conditions which are determined by Turing instability analysis, numerical simulations are performed to reveal the spatiotemporal complexity of the system. A pure Turing instability can induce the self-organization of many basic types of patterns as described in the literature, as well as new spiral-spot and labyrinth patterns which show the temporally oscillating and chaotic property. Neimark–Sacker–Turing and flip–Turing instability can lead to the formation of circle, spiral and much more complex patterns, which are self-organized via spatial symmetry breaking on the states that are homogeneous in space and non-periodic in time. Especially, the emergence of spiral pattern suggests that spatial order can generate from temporal disorder, implying that even when the predator–prey dynamics in one site is chaotic, the spatially global dynamics may still be predictable. The results obtained in this research suggest that when the way of population diffusion changes, the pattern formation in the predator–prey systems demonstrates great differences. This may provide realistic significance to explain more general predator–prey coexistence.

Confined 1D Propulsion of Metallodielectric Janus Micromotors on Microelectrodes under Alternating Current Electric Fields.

There is mounting interest in synthetic microswimmers ("micromotors") as microrobots as well as a model system for the study of active matters, and spatial navigation is critical for their success. Current navigational technologies mostly rely on magnetic steering or guiding with physical boundaries, yet limitations with these strategies are plenty. Inspired by an earlier work with magnetic domains on a garnet film as predefined tracks, we present an interdigitated microelectrodes (IDE) system where, upon the application of AC electric fields, metallodielectric (e.g., SiO2-Ti) Janus particles are hydrodynamically confined and electrokinetically propelled in one dimension along the electrode center lines with tunable speeds. In addition, comoving micromotors moved in single files, while those moving in opposite directions primarily reoriented and moved past each other. At high particle densities, turbulence-like aggregates formed as many-body interactions became complicated. Furthermore, a micromotor made U-turns when approaching an electrode closure, while it gradually slowed down at the electrode opening and was collected in large piles. Labyrinth patterns made of serpentine chains of Janus particles emerged by modifying the electrode configuration. Most of these observations can be qualitatively understood by a combination of electroosmotic flows pointing inward to the electrodes, and asymmetric electrical polarization of the Janus particles under an AC electric field. Emerging from these observations is a strategy that not only powers and confines micromotors on prefabricated tracks in a contactless, on-demand manner, but is also capable of concentrating active particles at predefined locations. These features could prove useful for designing tunable tracks that steer synthetic microrobots, as well as to enable the study of single file diffusion, active turbulence, and other collective behaviors of active matters.

Curvature-Controlled Wrinkling Surfaces for Friction.

Topographical patterns endow material surfaces with unique and intriguing physical and chemical properties. Spontaneously formed wrinkling has been harnessed to generate surface topography for various functionalities. Despite promising applications in biomedical devices and robot engineering, the friction behavior of wrinkling on curved surfaces remains unclear. Herein, wrinkled surfaces are induced by sputtering metals on soft polymer microspheres. The wrinkle morphologies and length scales can be controlled precisely by tailoring the microsphere radius (substrate curvature) and film thickness. The wrinkled surfaces exhibit controlled friction property, depending on the wrinkling patterns and length scales. An increase in friction force with increasing surface roughness is generally found for dimple patterns and labyrinth patterns. The dimple patterns show the lowest friction due to strong curvature constraint. The herringbone patterns exhibit apparent friction anisotropy with respect to topographic orientation. These results will guide future design of wrinkled surfaces for friction by harnessing substrate curvature.

Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response

Diffusion has long been known to induce pattern formation in predator prey systems. For certain prey-predator interaction systems, self diffusion conditions ceases to induce patterns, i.e., a non-constant positive solution does not exist, as seen from the literature. We investigate the effect of cross diffusion on the pattern formation in a tritrophic food chain model. In the formulated model, the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley-Martin functional response. We prove that the stationary uniform solution of the system is stable in the presence of diffusion when cross diffusion is absent. However, this solution is unstable in the presence of both self diffusion and cross diffusion. Using a priori analysis, we show the existence of a inhomogeneous steady state. We prove that no non-constant positive solution exists in the presence of diffusion under certain conditions, i.e., no pattern formation occurs. However, pattern formation is induced by cross diffusion because of the existence of non-constant positive solution, which is proven analytically as well as numerically. We performed extensive numerical simulations to understand Turing pattern formation for different values of self and cross diffusivity coefficients of the top level predator to validate our results. We obtained a wide range of Turing patterns induced by cross diffusion in the top population, including floral, labyrinth, hot spots, pentagonal and hexagonal Turing patterns.

Extending the Mathematical Palette for Developmental Pattern Formation: Piebaldism.

Here, we present a theoretical investigation with potential insights on developmental mechanisms. Three biological factors, consisting of two diffusing factors and a cell-autonomous immobile transcription factor are combined with different feedback mechanisms. This results in four different situations or fur patterns. Two of them reproduce classical Turing patterns: (1) regularly spaced spots, (2) labyrinth patterns or straight lines with an initial slope in the activation of the transcription factor. The third situation does not lead to patterns, but results in different homogeneous color tones. Finally, the fourth one sheds new light on the possible mechanisms leading to the formation of piebald patterns exemplified by the random patterns on the fur of some cows' strains and Dalmatian dogs. Piebaldism is usually manifested as white areas of fur, hair, or skin due to the absence of pigment-producing cells in those regions. The distribution of the white and colored zones does not reflect the classical Turing patterns. We demonstrate that these piebald patterns are of transient nature, developing from random initial conditions and relying on a system's bistability. We show numerically that the presence of a cell-autonomous factor not only expands the range of reaction diffusion parameters in which a pattern may arise, but also extends the pattern-forming abilities of the reaction-diffusion equations.

Effects of Spreading Conditions on the Aggregation Behavior of a Symmetric Diblock Copolymer Polystyrene- block-poly(methyl methacrylate) at the Air/Water Interface.

Langmuir monolayers and Langmuir-Blodgett (LB) films of a symmetric diblock copolymer polystyrene- block-poly(methyl methacrylate) (PS- b-PMMA) were characterized by the film balance technique and tapping mode atomic force microscopy, respectively. Effects of both the spreading solution concentration and the surface concentration on the aggregation behavior of PS- b-PMMA at the air/water interface and the morphologies of its LB films were studied in detail. When the monolayers spread in different concentrations (≤0.50 mg/mL), all their initial morphologies exhibit tiny circular micelles because of the long hydrophilic PMMA block in the copolymer. The initial tiny circular micelles form spontaneously and then aggregate into small ones upon compression, which can further coalesce into rodlike aggregates or large micelles depending on the spreading concentrations. The LB films of PS- b-PMMA usually exhibit various mixed structures of rodlike aggregates and circular micelles, which can further transform into labyrinth patterns under some special spreading conditions. Besides spreading concentration and volume, we discover that the detailed spreading process should also be responsible for the initial and final morphologies of the LB films. Furthermore, the LB films prepared under different spreading conditions can be regarded as in the equilibrium or nonequilibrium structures because of the kinetic effect difference resulting from the different PS chain entanglement degrees.

Tunable wrinkle structure formed on surface of polydimethylsiloxane microspheres

Soft materials with controlled surface patterns are useful for a range of applications, including microdevices, sensors and adhesives. In this work, the spontaneously wrinkled morphologies on the surfaces of crosslinked polydimethylsiloxane (PDMS) microspheres were observed. These wrinkled microspheres may have great potentials for construction of high-performance functional materials. They were fabricated in curable PDMS/water emulsions stabilized by surfactants. After curing and drying, the PDMS droplets in the emulsions formed microspheres with different surface wrinkling morphologies, including dimples and labyrinth patterns. The buckling patterns could be well tuned by the surfactant concentration, curing temperature, evaporation and compositions of PDMS phase. A power-law dependence of the wrinkling wavelength on microsphere radius was also observed. Furthermore, the wrinkling morphologies showed good reversibility by simply drying after wetting with water. It is speculated that the evaporation of dissolved water in the PDMS phase triggers interfacial instabilities and result in the wrinkled morphologies. This work gives new insights into the surface wrinkling behavior of soft microspheres, and provides a simple, efficient and environmentally friendly route to fabricate microspheres with complex surface patterns.

Effects of feedback regulation on vegetation patterns in semi-arid environments

• We systematically reveal a mechanism of positive feedback on vegetation pattern. • Regular spot, stripe and the mixed patterns appear in the range of appropriate feedback intensities. • Positive feedback may induce the emergence of desertification. • Our findings highlight the relationship among feedback intensity, rainfall and pattern dynamics of the vegetation. It is well known that vegetation patterns characterize the distribution of the vegetation and provide some signs for vegetation protection. The positive feedbacks regulation between the water and plant biomass play an important role in the vegetation patterns in semi-arid environments, yet its influence on vegetation patterns is far from being well understood. In order to reveal a mechanism of positive feedbacks on pattern formation, a water-biomass model in semi-arid environments with soil-water diffusion feedback is presented. Our results reveal that, as the soil-water diffusion intensity decreases, the pattern transitions: gap patterns → stripe (labyrinth) patterns → spot patterns emerge. More importantly, when the soil-water diffusion feedback intensity is smaller, the feedback will promote the growth of the vegetation; when the feedback intensity is much larger, the vegetation biomass will decrease and the feedback may induce the emergence of desertification. Additionally, the rainfall can also induce the pattern transition. As the rainfall capacity decreases, the vegetation disappears and becomes the bare soil state. Our findings highlight the relationship among feedback intensity, rainfall and pattern dynamics of the vegetation.

Surface wrinkling of anisotropic films bonded on a compliant substrate

Material anisotropy regulates the instabilities of film–substrate systems at different length scale. In this paper, we investigate the surface wrinkling and morphological evolution of an orthotropic thin film resting on a compliant substrate. Under different loading conditions, the system may buckle into various surface patterns, e.g., stripe, checkerboard, and herringbone, which are analyzed by using the Föppl–von Kármán plate theory. The Fourier spectral method is employed to simulate the morphological evolutions of surface patterns under different loading biaxialities. We find that both loading biaxiality and material anisotropy affect the characteristics of surface wrinkling patterns and their evolutions. Stripe and checkerboard modes often emerge at the critical buckling and they tend to transform into herringbone and labyrinth patterns during postbuckling. Phase diagrams are established to reveal the dependence of surface patterns on material anisotropy, Poisson's effect, and loading biaxiality. This study may help design diverse functional surfaces and deepen our understanding of the morphogenesis of some soft biological tissues and organs.