Parameter Plane Research Articles

Nonlinear dynamics of continuous steady-state tunable mechanical metamaterials based on planetary gears

Mechanical metamaterials with tunable mechanical properties have received extensive attention. In this work, a continuous steady-state metamaterial with tunable mechanical based on planetary gear cells is designed. Firstly, the excellent tunable properties of metamaterials are studied, which shows the designed metamaterials has a wide range of programmable stiffness and a significantly tunning band gap. The configuration relationship between tunability and structural parameters is analyzed. Then, a rigid-flexible coupling nonlinear dynamic model of planetary gear metamaterials is established and the dynamic characteristics of planetary gear metamaterials in the process of dynamic tunning are explored by numerical simulation. Moreover, the distribution characteristics of the system vibration response with the coupling parameter plane are disclosed, and the influence of the system dynamic parameters on its bifurcation characteristics and stability under coupling excitation is explored. Furthermore, an improved non-iterative cell mapping method is used to analyze the dependence of the vibration response of the system on the initial conditions. This work shows that the planetary gear metamaterials have a wide range of stiffness tunability and significantly variable band gap interval. Under the influences of internal and external excitations, the system exhibits various dynamic characteristics. The dynamic characteristics of flexible construction are mainly affected by structural parameters, and the dynamic characteristics of rigid construction are affected by multiple sets of coupling parameters and initial conditions.

Unraveling the dynamics of firing patterns for neurons with impairment of sodium channels.

Various factors such as mechanical trauma, chemical trauma, local ischemia, and inflammation can impair voltage-gated sodium channels (Nav) in neurons. These impairments lead to a distinctive leftward shift in the activation and inactivation curves of voltage-gated sodium channels. The resulting sodium channel impairments in neurons are known to affect firing patterns, which play a significant role in neuronal activities within the nervous system. However, the underlying dynamic mechanism for the emergence of these firing patterns remains unclear. In this study, we systematically investigated the effects of sodium channel dysfunction on individual neuronal dynamics and firing patterns. By employing codimension-1 bifurcation analysis, we revealed the underlying dynamical mechanism responsible for the generation of different firing patterns. Additionally, through codimension-2 bifurcation analysis, we theoretically determined the distribution of firing patterns on different parameter planes. Our results indicate that the firing patterns of impaired neurons are regulated by multiple parameters, with firing pattern transitions caused by the degree of sodium channel impairment being more diverse than those caused by the ratio of impaired sodium channel and current. Furthermore, we observed that the firing pattern of tonic firing is more likely to be the norm in impaired sodium channel neurons, providing valuable insights into the signaling of impaired neurons. Overall, our findings highlight the intricate relationships among sodium channel impairments, neuronal dynamics, and firing patterns, shedding light on the impact of disruptions in ion concentration gradients on neuronal function.

Shrimp-shaped structure and period-bubbling route to chaos in a one-dimensional economic model.

A beautiful feature of nature is its complexity. The chaos theory has proved useful in a variety of fields, including physics, chemistry, biology, and economics. In the present article, we explore the complex dynamics of a rather simple one-dimensional economic model in a parameter plane. We find several organized zones of "chaos and non-chaos" and different routes to chaos in this model. The study reveals that even this one-dimensional model can generate intriguing shrimp-shaped structures immersed within the chaotic regime of the parameter plane. We also observe shrimp-induced period-bubbling phenomenon, three times self-similarity of shrimp-shaped structures, and a variety of bistable behaviors. The emergence of shrimp-shaped structures in chaotic regimes can enable us to achieve favorable economic scenarios (periodic) from unfavorable ones (chaotic) by adjusting either one or both of the control parameters over broad regions of these structures. Moreover, our results suggest that depending on the parameters and initial conditions, a company may go bankrupt, or its capital may rise or fall in a regular or irregular manner.

Impact of multiplexing noise on multilayer networks of bistable maps

We explore numerically the complex dynamics of multilayer networks (consisting of three and one hundred layers) of cubic maps in the presence of noise-modulated interlayer coupling (multiplexing noise). The coupling strength is defined by independent discrete-time sources of color Gaussian noise. Uncoupled layers can demonstrate different complex structures, such as double-well chimeras, coherent and spatially incoherent regimes. Regions of partial synchronization of these structures are identified in the presence of multiplexing noise. We elucidate how synchronization of a three-layer network depends on the initially observed structures in the layers and construct synchronization regions in the plane of multiplexing noise parameters “noise spectrum width – noise intensity”. It is shown that in a hundred-layer network, clusters of synchronized layers can be formed at certain optimal values of multiplexing noise parameters. The performed numerical studies confidently indicate that the spatial dynamics of multilayer networks can be controlled by varying multiplexing noise parameters.

Dependence of wear of friction pairs of the mechanism for loading solid household waste into a garbage truck on the characteristics of antifriction materials

The article is dedicated to the establishment a regression dependence of wear of friction units of the mechanism of loading a garbage truck on the properties of antifriction materials. By using a first-order planning of experiment with first-order interaction effects using the Box-Wilson method, an appropriate regularity of wear of friction units of the garbage truck loading mechanism on the properties of antifriction materials has been determined. It is established that, according to the Student's criterion, among the studied factors of influence, the wear of friction units of the garbage truck loading mechanism is most influenced by the pressure in the friction zone, and the least – by the Brinell hardness of the antifriction material. The response surfaces of the objective function – wear of friction units of the garbage truck loading mechanism and their two-dimensional cross-sections in the planes of the influence parameters are shown, which allow to clearly illustrate the specified dependence of this objective function on individual influence parameters. It has been established that an increase in the pressure in the friction zone and sliding speed by 60% leads to an increase in the wear of friction units of the garbage truck loading mechanism during reverse motion by 6.3...34%, depending on the properties of antifriction materials. The expediency of conducting further research to determine ways to further improve the wear resistance of friction units of the mechanism for loading solid household waste into a garbage truck has been shown.

Analyzing top-down visual attention in the context of gamma oscillations: a layer- dependent network-of- networks approach.

Top-down visual attention is a fundamental cognitive process that allows individuals to selectively attend to salient visual stimuli in the environment. Recent empirical findings have revealed that gamma oscillations participate in the modulation of visual attention. However, computational studies face challenges when analyzing the attentional process in the context of gamma oscillation due to the unstable nature of gamma oscillations and the complexity induced by the layered fashion in the visual cortex. In this study, we propose a layer-dependent network-of-networks approach to analyze such attention with gamma oscillations. The model is validated by reproducing empirical findings on orientation preference and the enhancement of neuronal response due to top-down attention. We perform parameter plane analysis to classify neuronal responses into several patterns and find that the neuronal response to sensory and attention signals was modulated by the heterogeneity of the neuronal population. Furthermore, we revealed a counter-intuitive scenario that the excitatory populations in layer 2/3 and layer 5 exhibit opposite responses to the attentional input. By modification of the original model, we confirmed layer 6 plays an indispensable role in such cases. Our findings uncover the layer-dependent dynamics in the cortical processing of visual attention and open up new possibilities for further research on layer-dependent properties in the cerebral cortex.

A 5D super-extreme-multistability hyperchaotic map based on parallel-cascaded memristors

A new 5-dimensional discrete memristors-based hyperchaotic map (5DMHM) is constructed by introducing four charge-controlled discrete memristors (DMs, coupled through parallel-cascade mixing operation) into the sine map. This 5DMHM has 4-dimensional space fixed points and its stability distributions are partitioned in coupling strength, memristor initial condition, and parameter planes. The dynamics of the map are investigated by employing 2D dynamical behavior and the first Lyapunov exponent (LE) distribution. The super extreme multistability is discovered through the basin of attraction, the initial-value-dependent LE spectra, bifurcation diagrams, the mean value of state variables, and trajectory plots. Moreover, a high level of complexity can be observed by calculating the spectral entropy complexity for various parameter planes. Subsequently, a microcontroller-based digital circuit is designed to implement the 5DMHM. The security analysis of the pseudorandom number generator (PRNG) designed by the 5DMHM is presented and the PRNG passes the NIST SP800-22 test. The comparison of the 5DMHM with the recent maps indicates a higher performance in secure communication and image encryption.

Influence of sinusoidal forcing on the master FitzHugh–Nagumo neuron model and global dynamics of a unidirectionally coupled two-neuron system

In this paper, we investigate a seven-parameter, five-dimensional dynamical system, specifically a unidirectional coupling of two FitzHugh–Nagumo neuron models, with one neuron being sinusoidally driven. This master–slave configuration features neuron N1 as the master, subjected to an external sinusoidal electrical current, and neuron N2 as the slave, interacting with N1 through an electrical force. We report numerical results for three distinct scenarios where N1 operates in (i) periodic, (ii) quasiperiodic, and (iii) chaotic regimes. The primary objective is to explore how the dynamics of the master neuron N1 influence the coupled system’s behavior. To achieve this, we generated cross sections of the seven-dimensional parameter space, known as parameter planes. Our findings reveal that in the periodic regime of N1, the coupled system exhibits period-adding sequences of Arnold tongue-like structures in the parameter planes. Furthermore, regions of multistability can also be identified in these parameter planes of the coupled system. In the quasiperiodic regime, regions of periodic motion are absent, with only regions of quasiperiodic and chaotic dynamics present. In the chaotic regime of N1, the parameter planes display regions of chaos, hyperchaos, and transient hyperchaos.

Further study on the crossing curves in two-delay differential equations with delay-dependent coefficients

The crossing curves method, which allows delays to vary simultaneously, is generalized to the scenario that four terms exist in the characteristic equations with delay-dependent coefficients. The crossing curves on the two-delay parameter plane are first plotted by our generalized algorithms. The criteria to determine the crossing direction are also given. Finally, an example is provided to support our method and illustrate its fortes.

Strongly Anisotropic Vortices in Dipolar Quantum Droplets.

We construct strongly anisotropic quantum droplets with embedded vorticity in the 3D space, with mutually perpendicular vortex axis and polarization of atomic magnetic moments. Stability of these anisotropic vortex quantum droplets (AVQDs) is verified by means of systematic simulations. Their stability area is identified in the parametric plane of the total atom number and scattering length of the contact interactions. We also construct vortex-antivortex-vortex bound states and find their stability region in the parameter space. The application of a torque perpendicular to the vorticity axis gives rise to robust intrinsic oscillations or rotation of the AVQDs. The effect of three-body losses on the AVQD stability is considered too. The results show that the AVQDs can retain the topological structure (vorticity) for a sufficiently long time if the scattering length exceeds a critical value.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus.

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

SemanticMesh: parameterized fusion of semantic components for photogrammetric meshes

ABSTRACT The façades of photogrammetric building mesh models frequently lack detailed semantics, such as windows, doors, and balconies. To address this, we introduce SemanticMesh, a methodology designed to enrich façades with detailed semantics by integrating discrete components seamlessly. The process begins by transforming a localized area of the building surface into a 2D planar mesh using geodesic vectors, based on the component poses. This is followed by the application of 2D constrained triangulation to map the component boundaries onto the parameterized plane, after which the mesh is reconverted into 3D space. To achieve a smooth and aesthetically pleasing integration, we employ a Laplacian mesh deformation technique along the 3D embedding boundary. Our experiments across three distinct datasets – featuring near-planar, non-planar, and noisy surfaces – demonstrate that SemanticMesh provides superior modeling outcomes compared to conventional approaches.

Classification of unimodal parametric plane curve singularities in positive characteristic

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating “very short” parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest.

Complex dynamical behaviors of a honeybee-mite model in parameter plane

The honeybee has a significant impact on ecosystem stability, biodiversity conservation, and pollinating crops. However, during the past few years, there has been a sharp reduction in both the honeybee population and their colonies. It has been found that the parasitic mite Varroa destructor is responsible for the decline in honeybee colonies around the world, which in turn immensely affects the economic growth of a country. To investigate how the dynamics of a system is influenced by the growth of honeybees and the parasitism of mites, we study a nonlinear honeybee-mite population model in a discrete-time setup. We observe that an increase in the value of the queen’s egg-laying rate drives the system towards chaos. However, chaos can be controlled as well if the parasite attachment effect is increased. The intrinsic dynamical properties of the proposed system are investigated with the simultaneous variation of the queen’s egg-laying rate and the mite’s parasite attachment effect by constructing several largest Lyapunov exponent and isoperiodic diagrams. The investigation reveals the existence of several periodic structures in the quasiperiodic and chaotic regimes of the parameter plane, including Arnold tongues, saddle area, spring area, connected shrimp-shaped structure, and connected saddle area. We also find the appearance of a novel ‘jellyfish’-shaped periodic structure. One of the most fascinating findings of this study is the appearance of Arnold tongues along the inner boundary region of another Arnold tongue. In addition, this work also reveals different types of multistability, e.g., the coexistence of two, three, and even four attractors. What is more interesting is that the current analysis unveils the coexistence of five attractors as well, more specifically, four different periodic attractors coexist with the trivial fixed point attractor, which is quite rare in ecological systems. The structures of the basins of these coexisting attractors are either smooth or very complex in nature. Furthermore, the present study also discloses the fact that variation in the initial condition of the system can significantly change the appearance of the periodic structures in the parameter plane.

Propagation dynamics of multipole solitons generated in dissipative systems

The propagation dynamics of multipole solitons generated in dissipative systems are investigated numerically based on the fractional complex cubic-quintic Ginzburg–Landau equation using the Airy beam as the input beam. The effect of different parameter values on the generation of stable solitons is explored. In addition, we observe different resultant domains of the input beam evolving in the linear loss coefficient or cubic gain coefficient and Lévy index parameter planes. The results show that the evolution can lead to the formation of stable multipole solitons. It is also demonstrated that two solitons merge to form a single soliton. And, the relation between the merger distance and the initial amplitude is given.

Homoclinic behavior around a degenerate heteroclinic cycle in a Lorenz-like system

In this work, we analyze a degenerate heteroclinic cycle that appears in a Lorenz-like system when one of the involved equilibria changes from real saddle to saddle-focus. First, from a theoretical model based on the construction of a Poincaré return map, we demonstrate that an infinite number of homoclinic connections arise from the point of the parameter plane where the degenerate heteroclinic cycle appears. The subsequent numerical study not only illustrates the presence of the first homoclinic orbits in the infinite succession but also allows to find other important local and global organizing centers of codimension two (Bogdanov–Takens bifurcations, degenerate homoclinic and heteroclinic connections, T-points) and three (triple-zero bifurcation, doubly-degenerate heteroclinic cycles, degenerate T-points).

Bistable click mechanism for dipteran flight robot

The flight mechanism of the dipteran insects is commonly used to design the flapping robot. However, the nonlinear dynamics of the buckling click mechanism of the flapping robot with bistability still unclear. In this paper, a novel flapping robot model with the click mechanism proposed based on the bistability of snap-through buckling. The nonlinear governing equations of the flight robot are obtained by using the Euler–Lagrange equation. The nonlinear potential energy, moment of the elastic force, equilibrium bifurcation, as well as equilibrium stability are investigated to show the bistable characteristics. The transitional and persistent sets of bifurcation regions in the parameter plane and the corresponding phase portraits are obtained with single and double well behaviors. Then, the periodic response of the free flight are defined by the analytical solution of three kinds of elliptical functions, as well as the amplitude frequency responses are investigated by numerical integration. Based on the topological equivalent, the chaotic thresholds of the homo-clinic and heteroclinic orbits for the chaotic vibration of the perturbed robot system are derived. Finally, the prototype of nonlinear flapping robot is manufactured and the experimental system is setup. The nonlinear static moment of force curves, periodic response and dynamic flight vibration of dipteran robot system are carried out. It is shown that the test results are agree well with the theoretical analysis and numerical simulation. Those results have the potential application for the structure design of the efficient flight robot.

Exploring the dynamics of a four-dimensional Chua-like system: Numerical simulations and hardware investigations

In this work, we employ simulations to investigate deterministic and stochastic dynamics of a system governed by four coupled nonlinear ordinary differential equations (ODEs) derived from the canonical three-dimensional Chua model, replacing the piecewise nonlinearity with a cubic function. We employed two simulation approaches: (i) using software — where we performed numerical simulations (digital computation) in the ODEs model, and (ii) using hardware — where we performed experimental studies (analog computation) in the ODEs model. In the initial software-based approach for simulations, we performed stability and bifurcation analyses of the trivial equilibrium point, which were analytically conducted, along with the numerical continuation method. We solved the model using digital computation and explored its intricate dynamical behavior through nonlinear dynamics techniques applied to the parameter planes. In the second approach, employing hardware for simulations through analog computation, we designed an electronic circuit using analog electronics to solve the model in a continuous-time integration. The experimental setup mirrored the numerical simulations, yielding comparable results in terms of dynamical behaviors. This study underscores significant agreement between numerical and experimental findings, highlighting the robustness of the model across digital and analog contexts and offering practical insights into the various nonlinear dynamics presented in this system.

Dynamical analysis of a periodically forced chaotic chemical oscillator.

We present a comprehensive dynamical analysis of a chaotic chemical model referred to as the autocatalator, when subject to a periodic administration of one substrate. Our investigation encompasses the dynamical characterization of both unforced and forced systems utilizing isospikes and largest Lyapunov exponents-based parameter planes, bifurcation diagrams, and analysis of complex oscillations. Additionally, we present a phase diagram showing the effect of the period and amplitude of the forcing signal on the system's behavior. Furthermore, we show how the landscapes of parameter planes are altered in response to forcing application. This analysis contributes to a deeper understanding of the intricate dynamics induced by the periodic forcing of a chaotic system.

The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System

The complex service environment of railway vehicles leads to changes in the wheel–rail adhesion coefficient, and the decrease in critical speed may lead to hunting instability. This paper aims to reveal the diversity of periodic hunting motion patterns and the internal correlation relationship with wheel–rail impact velocities after the hunting instability of a bogie system. A nonlinear, non-smooth lateral dynamic model of a bogie system with 7 degrees of freedom is constructed. The wheel–rail contact relations and the piecewise smooth flange forces are the main nonlinear, non-smooth factors in the system. Based on Poincaré mapping and the two-parameter co-simulation theory, hunting motion modes and existence regions are obtained in the parameter plane consisting of running speed v and the wheel–rail adhesion coefficient μ. Three-dimensional cloud maps of the maximum lateral wheel–rail impact velocity are obtained, and the correlation with the hunting motion pattern is analyzed. The coexistence of periodic hunting motions is further revealed based on combined bifurcation diagrams and multi-initial value phase diagrams. The results show that grazing bifurcation causes the number of wheel–rail impacts to increase at a low-speed range. Periodic hunting motion with period number n = 1 has smaller lateral wheel–rail impact velocities, whereas chaotic motion induces more severe wheel–rail impacts. Subharmonic periodic hunting motion windows within the speed range of chaotic motion, pitchfork bifurcation, and jump bifurcation are the primary forms that induce the coexistence of periodic motion.