Two-parameter Space Research Articles

Arnold tongues, shrimp structures, multistability, and ecological paradoxes in a discrete-time predator–prey system

This paper explores a discrete-time system derived from the well-known continuous-time Rosenzweig–MacArthur model using the piecewise constant argument. Examining the impact of increasing carrying capacity and harvesting efforts, we uncover intricate phenomena, such as periodicity, quasiperiodicity, period-doubling, period-bubbling, and chaos. Our analysis reveals that increasing the carrying capacity of prey species can lead to both system stabilization and destabilization. We delve into normal forms associated with different bifurcation types, accompanied by numerical examples, observing multistabilities with intricate basin structures. Bistable, tristable, and quadruple attractors characterize the model’s multistable states. Additionally, we find that enriching prey species negatively affects predator abundance, and increasing carrying capacity can lead to a sudden jump in predator population to the brink of extinction. Examining the two-parameter space of predator and prey harvesting efforts, we identify organized periodic structures: Arnold tongues and shrimp-like structures within quasiperiodic and chaotic regions. Arnold tongues exhibit a sequence of periodic adding. The shrimp structures indicate the existence of period-doubling and period-bubbling phenomena. Discussions on ecological interpretations of predator harvesting, including the paradoxical hydra effect, are provided.

Dispersal induced catastrophic bifurcations, Arnold tongues, shrimp structures, and stock patterns in an ecological system.

This paper presents a comprehensive analysis of a discrete-time predator-prey model within a homogeneous two-patch environment, incorporating both prey and predator dispersal. We consider a logistic growth for both prey and predator species, and the predation process is based on the Holling type-II functional response in the isolated patches. We explore the existence of multiple coexisting equilibria and establish their stability conditions. By independently varying the prey and predator dispersal rates, we discover a sequence of phenomena including bifurcations, quasiperiodicity, and chaos. In addition, we observe a 10-period orbit, each point of the periodic orbit gives birth to a closed invariant curve. Such large number of closed invariant curves are generally not reported in spatially coupled population models. The system exhibits both catastrophic (non-smooth) jumps and smooth transitions in the dynamics whenever a bifurcation occurs. Commonly, dispersal can only destabilize the coexisting equilibrium. However, we found the stabilization of the coexisting equilibrium, which is a rare occurrence. Furthermore, a two-parameter space analysis reveals intricate dynamics when both dispersal rates are varied simultaneously, showcasing complex phenomena and the emergence of organized periodic regimes such as Arnold tongues and shrimp structures. We also investigate the stock pattern of both species with respect to the dispersal. This study enhances the understanding of predator-prey interactions in spatially homogeneous environments, illuminating their intricate and dynamic nature.

Generalized linear model study of the T90–T50 relation of gamma-ray bursts

ABSTRACT Gamma-ray bursts (GRBs) can be studied with their linearly dependent parameters alongside the standard $T_{90}$ distribution. The generalized linear model (GLM) identifies the number of linear dependences in a two-parameter space. Classically, GRBs are classified into two classes by the presence of bimodality in the histogram of $T_{90}$. However, additional classes and different features of GRBs are fascinating topics to explore. In this work, we investigate GRB features in the $T_{90} {-}T_{50}$ plane using the GLM for three major catalogues: Swift, the Fermi Gamma-ray Burst Monitor (GBM), and the Burst And Transient Source Experiment (BATSE). This study shows five linear features for the Fermi GBM catalogue and four linear features for the BATSE catalogue, directing us towards the possibility of non-Gaussianity in the light curves of GRBs.

Spatiotemporal Dynamics of an Intraguild Predation Model with Intraspecies Competition

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

Study of Bifurcation and Delay-Driven Chaos in a Prey–Predator Model with Fear in Prey Reproduction and Two Forms of Harvesting

In this paper, we delve into a predator–prey model incorporating a fear effect in prey reproduction, influenced by both delay and harvesting. The model accounts for delayed fear dynamics to capture more realistic dynamics. Initially, our focus lays on the nondelayed model, examining each biologically plausible equilibrium points and assessing their stability concerning the parameters of the model. Next, detailed mathematical results are provided, encompassing the asymptotic stability of all equilibria, Hopf bifurcation, and the direction and stability of bifurcated periodic solutions. Also, the stability analysis of the Hopf-bifurcating periodic solution is confirmed through the computation of first Lyapunov coefficient. Furthermore, we observed that the nondelayed system experiences Bogdanov–Takens bifurcation in a two-parameter space. Subsequently, we analyzed the corresponding delayed system, establishing the existence of a stable limit cycle through Hopf bifurcation concerning the delay parameter. Additionally, the inclusion of delay can prompt critical dynamics within the system, resulting in period-doubling routes toward chaotic oscillations. To validate our analytical findings, we conducted comprehensive and meticulous numerical simulations. The findings of the numerical simulations suggest that the impact of fear can be used as a measure of chaos control.

Exceptional Nexus in Bose-Einstein Condensates with Collective Dissipation.

In multistate non-Hermitian systems, higher-order exceptional points and exotic phenomena with no analogues in two-level systems arise. A paradigm is the exceptional nexus (EX), a third-order EP as the cusp singularity of exceptional arcs (EAs), that has a hybrid topological nature. Using atomic Bose-Einstein condensates to implement a dissipative three-state system, we experimentally realize an EX within a two-parameter space, despite the absence of symmetry. The engineered dissipation exhibits density dependence due to the collective atomic response to resonant light. Based on extensive analysis of the system's decay dynamics, we demonstrate the formation of an EX from the coalescence of two EAs with distinct geometries. These structures arise from the different roles played by dissipation in the strong coupling limit and quantum Zeno regime. Our Letter paves the way for exploring higher-order exceptional physics in the many-body setting of ultracold atoms.

Multipliers on bi-parameter Haar system Hardy spaces

Let (hI)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(h_I)$$\\end{document} denote the standard Haar system on [0, 1], indexed by I∈D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I\\in \\mathcal {D}$$\\end{document}, the set of dyadic intervals and hI⊗hJ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_I\\otimes h_J$$\\end{document} denote the tensor product (s,t)↦hI(s)hJ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(s,t)\\mapsto h_I(s) h_J(t)$$\\end{document}, I,J∈D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I,J\\in \\mathcal {D}$$\\end{document}. We consider a class of two-parameter function spaces which are completions of the linear span V(δ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {V}(\\delta ^2)$$\\end{document} of hI⊗hJ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_I\\otimes h_J$$\\end{document}, I,J∈D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I,J\\in \\mathcal {D}$$\\end{document}. This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces Lp[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p[0,1]$$\\end{document} or the Hardy spaces Hp[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p[0,1]$$\\end{document}, 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p < \\infty $$\\end{document}. We say that D:X(Y)→X(Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D:X(Y)\\rightarrow X(Y)$$\\end{document} is a Haar multiplier if D(hI⊗hJ)=dI,JhI⊗hJ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D(h_I\\otimes h_J) = d_{I,J} h_I\\otimes h_J$$\\end{document}, where dI,J∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{I,J}\\in \\mathbb {R}$$\\end{document}, and ask which more elementary operators factor through D. A decisive role is played by the Capon projectionC:V(δ2)→V(δ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}:\\mathcal {V}(\\delta ^2)\\rightarrow \\mathcal {V}(\\delta ^2)$$\\end{document} given by ChI⊗hJ=hI⊗hJ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C} h_I\\otimes h_J = h_I\\otimes h_J$$\\end{document} if |I|≤|J|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|I|\\le |J|$$\\end{document}, and ChI⊗hJ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C} h_I\\otimes h_J = 0$$\\end{document} if |I|>|J|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|I| > |J|$$\\end{document}, as our main result highlights: Given any bounded Haar multiplier D:X(Y)→X(Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D:X(Y)\\rightarrow X(Y)$$\\end{document}, there exist λ,μ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ,\\mu \\in \\mathbb {R}$$\\end{document} such that λC+μ(Id-C)approximately 1-projectionally factors throughD,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lambda \\mathcal {C} + \\mu ({{\\,\ extrm{Id}\\,}}-\\mathcal {C})\ ext { approximately 1-projectionally factors through }D, \\end{aligned}$$\\end{document}i.e., for all η>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta > 0$$\\end{document}, there exist bounded operators A, B so that AB is the identity operator Id\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Id}\\,}}$$\\end{document}, ‖A‖·‖B‖=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert A\\Vert \\cdot \\Vert B\\Vert = 1$$\\end{document} and ‖λC+μ(Id-C)-ADB‖<η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\lambda \\mathcal {C} + \\mu ({{\\,\ extrm{Id}\\,}}-\\mathcal {C}) - ADB\\Vert < \\eta $$\\end{document}. Additionally, if C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}$$\\end{document} is unbounded on X(Y), then λ=μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda = \\mu $$\\end{document} and then Id\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Id}\\,}}$$\\end{document} either factors through D or Id-D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Id}\\,}}-D$$\\end{document}.

Commonality and difference in the eigenfunctions of various types of acoustic trapped modes

We investigate and compare various types of acoustic trapped modes (TMs) in resonator–waveguide systems. The goal is to understand the commonality and difference between the mechanisms of common (symmetry protected, invisibility protected and symmetry–periodicity protected) and accidental TMs, occurring continuously and discretely in the resonator length–frequency two-parameter space. The latter type cannot yet be explained via an operator decomposition. Here, all TMs are explained in the same way by analysing why and how the propagating-wave loops in the eigenfunctions can satisfy the eigenmode condition (loop magnitude and phase constraints for closure) and the wave-trapping condition (loop zero-radiation condition) simultaneously. Firstly, the unified analysis reveals the commonality that one or multiple coupled propagating-wave loops satisfy TM conditions, and the difference. In common TMs, the loop zero radiation is independent of the single loop phase constraint that selects the TM frequency as a continuous function of resonator length. On the other hand, loop zero radiation in accidental TMs depends on the loop phase constraints and there are two phase constraints. Only at the crossing points of the two phase constraints can zero-radiation loops be ensured. Secondly, in contrast to previous studies, it suggests that modal degeneracy, avoided crossing and resonance-width bifurcation are not the mechanisms of accidental TMs.

On two-parameter bifurcation and analog circuit implementation of a Chameleon chaotic system

In this paper, the two-parameter space bifurcation of a three-dimensional Chameleon system is investigated. It is called Chameleon since the type and the number of the system equilibrium are adjustable for different parameter configurations. Aided by the computation analysis, the graphic structures of two-parameter bifurcation of the Chameleon system are characterized for the first time. With different two-parameter configurations, the bifurcation evolution shows that various self-excited and hidden attractors exist. In addition, numerical demonstration of the two-dimensional slice through the attraction basin space is presented. The results show that the basin of attraction of the typical hidden chaotic attractor does not associated with the origin, which makes the attractor difficult to be numerically localized and experimentally observed. To solve the problem, offset boost scheme is adopted to control the basin of attraction and make it touch the origin, which allows to coin the hidden attractor via configuring zero initial value and making it feasible in experimental observation. Finally, the analog circuit-assisted experiment validated the feasibility of the scheme.

Exploring static bifurcations in a controlled dynamical system with cubic and quadratic nonlinearities: 2D and 3D visualization

This work delves into the investigation of static bifurcation control and vibration reduction of a two-degree-of-freedom dynamical system. The system under study simulates the lateral oscillations of rotating machinery and encompasses both cubic and quadratic nonlinearities. The nonlinear system is augmented with a magnetic bearing actuator, incorporating a novel control strategy that combines two first-order filters. The system model is derived based on classical mechanics and electromagnetic theories. Then, an analytical solution is extracted for the obtained dynamical model. The solutions obtained have been utilized to visualize the static bifurcations of the system in both two and three-dimensional spaces, using various system parameters as bifurcation variables. The mono-stable and multiple-stable solution regions have been distinguished in two-parameter space. Subsequently, an investigation has been conducted to evaluate the effectiveness of the introduced control technique in eliminating the catastrophic bifurcation of the rotor and suppressing undesirable vibrations. Furthermore, as a precautionary measure, the impact of the controller’s sudden malfunction on the stability of the system was explored. The main findings revealed that the implemented control approach effectively eliminates dangerous bifurcation characteristics and induces the nonlinear rotor to exhibit a response like a linear system with minimal vibration amplitudes. Furthermore, it was observed that the abrupt failure of the controller does not affect the stability of the system; however, the nonlinearities regain dominance in the system’s response

Hydrodynamic Anisotropy of Depletion in Nonequilibrium.

Active colloids in a bath of inert particles of smaller size cause anisotropic depletion. The active hydrodynamics of this nonequilibrium phenomenon, which is fundamentally different from its equilibrium counterpart and passive particles in an active bath, remains scarcely understood. Here we combine mesoscale hydrodynamic simulation as well as theoretical analysis to examine the physical origin for the active depletion around a self-propelled noninteractive colloid. Our results elucidate that the variable hydrodynamic effect critically governs the microstructure of the depletion zone. Three characteristic states of anisotropic depletion are identified, depending on the strength and stress of activity. This yields a state diagram of depletion in the two-parameter space, captured by developing a theoretical model with the continuum kinetic theory and leading to a mechanistic interpretation of the hydrodynamic anisotropy of depletion. Furthermore, we demonstrate that such depletion in nonequilibrium results in various clusters with ordered organization of squirmers, which follows a distinct principle contrary to that of the entropy scenario of depletion in equilibrium. The findings might be of immediate interest to tune the hydrodynamics-mediated anisotropic interactions and active nonequilibrium organizations in the self-propulsion systems.

Complex dynamical study of a delayed prey–predator model with fear in prey and square root harvesting of both species

In the current study, the dynamics of predator-prey systems under the influence of fear effect on the reproduction of prey population and harvesting on both species has been proposed. Assessing the dynamics of the system with the combined influence of fear and harvesting for various values of n is our central objective. We present comprehensive mathematical findings that cover fundamental dynamical features, the presence of positive equilibria, and the stability of all equilibria. Hopf-bifurcating periodic solutions have been demonstrated to emerge around the positive equilibrium point, and the direction of the Hopf-bifurcating limit cycle is determined using the first Lyapunov coefficient. Furthermore, in two-parameter space, we have seen that the system experiences the Bogdanov-Takens bifurcation. Moreover, we have included predator gestation delay and noticed some chaotic dynamics in the system. In addition, we run through numerical simulations to numerically validate our mathematical findings. The article is concluded with a conclusion at the end.

Study on the Influence of Microinjection Molding Processing Parameters on Replication Quality of Polylactic Acid Microneedle Array Product

Biodegradable microneedles with a drug delivery channel have enormous potential for consumers, including use in chronic disease, vaccines, and beauty applications, due to being painless and scarless. This study designed a microinjection mold to fabricate a biodegradable polylactic acid (PLA) in-plane microneedle array product. In order to ensure that the microcavities could be well filled before production, the influences of the processing parameters on the filling fraction were investigated. The results indicated that the PLA microneedle can be filled under fast filling, higher melt temperature, higher mold temperature, and higher packing pressure, although the dimensions of the microcavities were much smaller than the base portion. We also observed that the side microcavities filled better than the central ones under certain processing parameters. However, this does not mean that the side microcavities filled better than the central ones. The central microcavity was filled when the side microcavities were not, under certain conditions in this study. The final filling fraction was determined by the combination of all parameters, according to the analysis of a 16 orthogonal latin hypercube sampling analysis. This analysis also showed the distribution in any two-parameter space as to whether the product was filled entirely or not. Finally, the microneedle array product was fabricated according to the investigation in this study.

Nanopore-based immunoassay based on rolling circle amplification and DNA fragmentations.

In recent years, small-size portable nanopore-based sequencers have become a robust tool with unique advantages for genomics applications. However, a few challenges have impeded applying nanopore sequencers as sensitive, quantitative diagnostic tools. One of the major bottlenecks is the inefficient capture of molecules into a pore at pM or lower concentrations which are typical for disease biomarkers. To bridge this gap we have developed a signal amplification strategy utilizing immunocapture, isothermal rolling circle amplification, and sequence-specific fragmentation of the product to release multiple DNA reporter molecules for nanopore detection. This group of DNA reporter molecules that comprise different lengths and structures could produce a distinct nanopore cluster spectrum in the two-parameter space of dwell times and fractional current blockades. By identifying the nanopore cluster spectrum, we could identify the biomarker analytes and achieve quantification. As a proof of concept, we demonstrate the quantification of human epididymis protein 4 (HE4) to low pM levels using wild-type alpha-hemolysin. Future improvement that employ microfluidics and nanopore arrays can further reduce the limit of detection, allow multiplexed biomarker detection, and further, reduce the footprint, cost, and portability of existing laboratory and point-of-care devices.

Cubic fixed point in three dimensions: Monte Carlo simulations of the ϕ4 model on the simple cubic lattice

We study the cubic fixed point for $N=3$ and $4$ by using finite size scaling applied to data obtained from Monte Carlo simulations of the $N$-component $\phi^4$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the $O(N)$-invariance. For $N=4$, we determine the correction exponents $\omega_1=0.763(24)$ and $\omega_2=0.082(5)$. In the case of $N=3$, we obtain $Y_4=0.0142(6)$ for the RG-exponent of the cubic perturbation at the $O(3)$-invariant fixed point, while the correction exponent $\omega_2=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\nu=0.7202(7)$ and $\eta=0.0371(2)$ of the critical exponents of the cubic fixed point for $N=4$. For $N=3$, at the cubic fixed point, the $O(3)$-symmetry is only mildly broken and the critical exponents differ only by little from those of the $O(3)$-invariant fixed point. We find $-0.00001 \lessapprox \eta_{cubic}- \eta_{O(3)} \lessapprox 0.00007$ and $\nu_{cubic}-\nu_{O(3)} =-0.00061(10)$.

Change-Point Tests for the Tail Parameter of Long Memory Stochastic Volatility Time Series

We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.

Rank-based change-point analysis for long-range dependent time series

We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point height, the asymptotic distributions of corresponding test statistics are derived. For this, a uniform reduction principle for the sequential empirical process in a two-parameter Skorohod space equipped with a weighted supremum norm is proved. Moreover, we compare the efficiency of rank tests resulting from the consideration of different score functions. Under Gaussianity, the asymptotic relative efficiency of rank-based tests with respect to the CuSum test is 1, irrespective of the score function. Regarding the practical implementation of rank-based change-point tests, we suggest to combine self-normalized rank statistics with subsampling. The theoretical results are accompanied by simulation studies that, in particular, allow for a comparison of rank tests resulting from different score functions. With respect to the finite sample performance of rank-based change-point tests, the Van der Waerden rank test proves to be favorable in a broad range of situations. Finally, we analyze data sets from economy, hydrology, and network traffic monitoring in view of structural changes and compare our results to previous analysis of the data.

Sigmoid Function Model and Dynamic Characteristics Analysis for Inductive Wireless Power Transmission Control System

In this article, an inductive wireless power transmission (IWPT) system is a high-order nonlinear switching system with multiple operating modes. It is challenging to analyze the dynamic characteristics of such a system using the extended describing function, generalized state-space averaging, or other existing modeling methods. To address this challenge, this study proposes a modeling and dynamic characteristics analysis method for a closed-loop IWPT control system based on a sigmoid function model. A sigmoid function with a large steepness factor is adopted to approximate the switching process of the switch and a unified smooth continuous dynamic model is established for a phase-shift controlled IWPT system. Because this model is infinitely differentiable, a stability theory of continuous systems, such as the Floquet theory, can be directly applied to analyze the bifurcation type of the system and stability of periodic solutions. Analysis results reveal the transition of the phase-shift-controlled IWPT system from periodic behavior to chaotic behavior through saddle-node bifurcation and Neimark–Sacker bifurcation. A stable domain in the two-parameter space of the controller proportional coefficient and load is also obtained. Both simulation and experimental results validate the accuracy of the proposed model and theoretical analysis method, which can significantly reduce the difficulty of analyzing IWPT system dynamics.

Analysis and finite-time synchronization of a novel double-wing chaotic system with transient chaos

A chaotic system that can generate double-wing attractor is proposed. Through the analysis of Lyapunov exponents, bifurcation diagram, phase diagram and complexity, it is found that the system has rich dynamical phenomenon. The topological structure of the system will change with the change of parameters, which is analyzed by using two-parameter space. In addition, the novel system has phenomenon such as offset-boosting, transient chaos and coexisting attractors. The analog simulation circuit of the system is designed based on Multisim, and the actual digital circuit is realized with field programmable gate array (FPGA). The experimental phase diagram, numerical simulation results and simulation circuit results agree well, which prove the feasibility of the chaotic system. Finally, a controller is designed according to finite-time theory to achieve finite-time synchronization of systems with different structures. The simulation results are consistent with the theoretical analysis, which proves the feasibility of the controller and lays the foundation for the next application research.

A Two-Parameter Space to Tune Solid Electrolytes for Lithium Dendrite Constriction.

Li dendrite penetration, and associated microcrack propagation, at high current densities is one main challenge to the stable cycling of solid-state batteries. The interfacial decomposition reaction between Li dendrite and a solid electrolyte was recently used to suppress Li dendrite penetration through a novel effect of “dynamic stability”. Here we use a two-parameter space to classify electrolytes and propose that the effect may require the electrolyte to occupy a certain region in the space, with the principle of delicately balancing the two property metrics of a sufficient decomposition energy with the Li metal and a low critical mechanical modulus. Furthermore, in our computational prediction prepared using a combination of high-throughput computation and machine learning, we show that the positions of electrolytes in such a space can be controlled by the chemical composition of the electrolyte; the compositions can also be attained by experimental synthesis using core–shell microstructures. The designed electrolytes following this principle further demonstrate stable long cycling from 10 000 to 20 000 cycles at high current densities of 8.6–30 mA/cm2 in solid-state batteries, while in contrast the control electrolyte with a nonideal position in the two-parameter space showed a capacity decay that was faster by at least an order of magnitude due to Li dendrite penetration.