Three-dimensional Dynamical System Research Articles

Energy function of 2D and 3D dynamical systems

It is far well-known that energy function of a two-dimensional autonomous dynamical system can be simply obtained by multiplying its corresponding second-order ordinary differential equation, i.e., its equation of motion by the first time derivative of its state variable. In the nineties, one of us (J.C.S.) stated that a three-dimensional autonomous dynamical system can be also transformed into a third-order ordinary differential equation of motion todays known as jerk equation. Although a method has been developed during these last decades to provide the energy function of such three-dimensional autonomous dynamical systems, the question arose to determine by which type of term, i.e., by the first or second time derivative of their state variable, the corresponding jerk equation of these systems should be multiplied to deduce their energy function. We prove in this work that the jerk equation of such systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. We thus established that it is possible to obtain the energy function of a three-dimensional dynamical system directly from its corresponding jerk equation. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua’s models confirms their validity.

High paracrine activity of hADSCs cartilage microtissues inhibits extracellular matrix degradation and promotes cartilage regeneration

Due to its unique structure, articular cartilage has limited self-repair capacity. Microtissues are tiny tissue clusters that can mimic the function of target organs or tissues. Using cells alone for microtissue construction often results in the formation of necrotic cores. However, the extracellular matrix (ECM) of native cartilage can provide structural support and is an ideal source of microcarriers. Autologous adipose-derived mesenchymal stem cells (ADSCs) and bone marrow mesenchymal stem cells (BMSCs) are widely used in cartilage tissue engineering. In this study, we fabricated microcarriers and compared the behavior of two homologous cell types in the microcarrier environment. The microcarrier environment highlighted the advantages of ADSCs and promoted the proliferation and migration of these cells. Then, ADSCs microtissues (ADSCs-MT) and BMSCs microtissues (BMSCs-MT) were fabricated using a three-dimensional dynamic culture system. In vitro and in vivo experiments verified that the cartilage regeneration ability of ADSCs-MT was significantly superior to that of BMSCs-MT. Transcriptomics revealed that ADSCs-MT showed significantly lower expression levels of ECM degradation, osteogenesis, and fibrocartilage markers. Finally, the protective effect of microtissues on inflammatory chondrocytes was validated. Overall, the ADSCs-MT constructed in this study achieved excellent cartilage regeneration and could be promising for the autologous application of cartilage microtissues.

Kinematic gait analysis for patients with meniscus injury

To investigate application of the Codamotion 2CX1 three-dimensional dynamic joint motion capture system to analyze the kinematic characteristics of patients with different degrees of meniscus injury. From December 2020 to June 2022, 135 patients with meniscus injury and recruited normal people were collected, including 82 males and 53 females, aged 14 to 29 years old, with disease duration of 1 to 3 months. Combined with clinical symptoms and MRI examination, the diagnosis of meniscus injury was confirmed, and the patients were divided into stages, including 37 cases of grade 0(normal), 30 cases of gradeⅠ, 33 cases of gradeⅡ, 35 cases of grade Ⅲ according to Stoller grading standard. Subjects in each group were tested walking by using Codamotion 2CX1 three-dimensional dynamic joint motion capture system. Quantitative indexes of walking and kinematics were collected, including knee flexion and extension, internal and external rotation and internal and external turning, and their kinematics were analyzed. In the distribution of knee flexion/extension, there were significant differences in maximum knee flexion, minimum knee extension and knee flexion and extension range among 4 groups(P<0.05). In the distribution of internal/external rotation of knee joint, there were significant differences in the range of internal rotation and rotation of knee joint among 4 groups(P<0.05). In the distribution of internal/external turning of knee joint, there were significant differences in the range of internal and external turning of knee joint among 4 groups(P<0.05). The clinical stage progression was positively correlated with the range of motion of knee extension, external rotation, internal and external turning and turning range(P<0.05). It was negatively correlated with knee flexion, internal rotation, flexion extension and rotation range(P<0.05). The internal and external rotation angles of knee joint could be used as independent factors influencing the clinical stage of meniscus injury (P=0.006, 0.019<0.05). The knee movement of patients with meniscus injury has obvious changes, and the changes are different under different clinical stages. Gait analysis provides a reliable basis for the kinematic analysis of meniscus injury, helps to better understand the kinematic indexes of joints, and provides a reliable auxiliary diagnosis and treatment plan, which provides a new direction for the follow-up medical research.

Coupled Quintessence Inspired by Warm Inflation

We investigate a coupled quintessence cosmological model in which a dark-energy scalar field with an exponential potential interacts directly with a dark-matter fluid through a dissipative term inspired by warm inflation. The evolution equations of this model give rise to a three-dimensional dynamical system for which a thorough qualitative analysis is performed for all values of the relevant parameters. We find that the model is able to replicate the observed sequence of late-time cosmological eras, namely, a long enough matter-dominated era followed by a present era of accelerated expansion. In situations where there is a significant transfer of energy from dark energy to dark matter, temporary scaling-type solutions may arise, but, asymptotically, all solutions are dominated by dark energy.

A Comparative Evaluation of Real-Time Guided Dynamic Navigation and Conventional Techniques for Post Space Preparation During Post Endodontic Management: An In Vitro Study.

The three-dimensional (3D) dynamic navigation system (DNS; Navident, ClaroNav Technology, Toronto, ON) is a revolutionary technique in endodontics that offers superior precision and efficiency compared to existing techniques for post space preparation. The aim is to evaluate and contrast the efficacy and efficiency of the DNS with conventional post space preparation techniques. This assessment considers severalparameters, notably canal deviation (global coronal and apical deviation and angular deflection), duration of the procedure, and total volumetric loss of instrumented root canal and volumetric loss of instrumented root canal above 4 mm from the apex. Freshly extracted maxillary central incisors were chosen for this study. A total sample size of 60 (n) was included. The extracted teeth samples were divided into two groups: 3D DNS (group I; n = 30)and conventional techniques (group II; n = 30). The samples were taken, and 50% of the crown structure was reduced for post space preparation to ensure standardization between thetwo groups. The samples were root canal-treated and mounted in a 3D-printed maxillary cast. Preoperative micro-computed tomography (micro-CT) and cone-beam computed tomography (CBCT) were taken for both groups.For group I, post space preparation was conducted with the DNS, which provided comprehensive guidance. The procedure was stopped when post space preparation was 4 mm short of the apex, as indicated in the system display. For group II, post space preparation was done without the guidance of DNS. Time taken for the procedure was assessed using a timer; canal deviation was evaluated using CBCT analysis, and volumetric loss was estimated using micro-CT analysis. The dynamic navigation group achieves significantly more precise outcomes in post space preparation than the conventional technique. The DNS group has significantly lesser global coronal and apical deviation and angular deflection compared to the conventional group (p < 0.05). The DNS group has reduced the volumetric loss of instrumented root canals compared to the conventional group (p < 0.05). Furthermore, the DNS group requires significantlyless time than the conventional method, with a mean difference of about 10.567 minutes (p < 0.05). Implementing dynamic navigation improves precision in post space preparation, with a notable reduction in canal deviation and volumetric loss and a decrease in procedure time compared to the conventional method.

Research on three-dimensional dynamic design and decision system of airport

Abstract Due to the lack of elevation information, the traditional airport design system based on the AutoCAD 2D platform is often unable to make a comprehensive and objective analysis and judgment when the designer evaluates the advantages and disadvantages of the design scheme. Therefore, this paper focuses on the development of airport 3D dynamic design and decision systems based on the Cesium framework. Firstly, based on the 3D geographic information platform, a multi-type 3D digital map loading and display module will be developed to realize the combination of 3D geographic information and the airfield model. Then, according to the 3D model data and the characteristics of the platform development, the method of calculating the earth volume of clearance processing, the evaluation algorithm of airport clearance obstacle limit surface, the optimization analysis method of tower position, and the analysis method of navigation station and flight track visibility are constructed.

An extended Goodwin model with endogenous technical change and labor supply

This paper extends the Goodwin model of distributive cycles by incorporating the simultaneous endogeneity of technical change and labor supply within a classical-Marxian framework. It reinterprets induced innovation, suggesting that firms optimize mechanization to maximize cost reduction, obtaining a micro-founded relationship between mechanization and the wage share. Additionally, it assumes a positive relationship between labor supply and the employment rate. The resulting three-dimensional dynamical system includes wage share, employment rate, and capital-output ratio as state variables. The Hopf bifurcation theorem reveals the emergence of limit cycles as the employment rate's effect on labor productivity (reserve-army-creation effect) approximates a critical value from below. Numerical simulations for 10 OECD countries illustrate the cyclical nature of the model and its consistency with empirical patterns. Furthermore, a sensitivity analysis explores the effect of parameters variations, emphasizing the social dimensions of productivity and labor supply as critical factors defining the stability of distributive cycles.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

Real-Time Three-dimensional Dynamic Navigation for Post Space Preparation in Root Canal-Treated Teeth: An In vitro Study

IntroductionThe aim of this study was to investigate the feasibility of a real-time three-dimensional dynamic navigation system (3D-DNS) for post space preparation (PSP) in root canal-treated teeth and to compare the accuracy and efficiency of 3D-DNS to freehand (FH) for PSP. MethodsFifty-four maxillary molars were divided into two groups: 3D-DNS (n = 27) and FH group (n = 27). Cone beam computed tomography (CBCT) scans were taken preoperatively and postoperatively. The drilling path for the PSP was virtually planned in the preoperative CBCT scan in the X-guide software (X-Nav Technologies, Lansdale, PA). For the 3D-DNS group, the PSP drilling was conducted under dynamic navigation. The 3D deviations and angular deflections were calculated. The residual dentin thickness (RDT) was determined after PSP. The operation time and the total number of mishaps were recorded. Shapiro-Wilk, t-test or Mann-Whitney rank sum, weighted Cohen's kappa, and Fisher exact tests were used (P < .05). ResultsThe PSP was completed in all samples (54/54). The 3D-DNS was more accurate than the FH, with significantly fewer 3D deviations and angular deflections (all, P < .05). The 3D-DNS required less operating time than the FH (P < .05). For the 3D-DNS, no teeth had RDT < 1 mm, whereas 6/27 in the FH showed RDT < 1 mm after the PSP. There was no difference in the total number of mishaps (P > .05). ConclusionWithin the limitations of this in vitro study, the 3D-DNS is feasible for PSP. The 3D-DNS improved the accuracy and efficiency of PSP. The dynamic navigation system can potentially become a safe and reliable technology for PSP.

Investigation of the stability of trajectories for three-dimensional nonlinear dynamic system using an accompanying coordinate reference

The development of methods of stability analysis and the study of various types of dynamic systems stability belongs to an urgent scientific direction. Among the problems solved in the framework of this direction, an important place is occupied by the problems of finding conditions for stability and stabilization in the sense of N.E. Zhukovsky of trajectories of dynamic systems modeled by systems of three nonlinear differential equations of the first order. The aim of the work is to study the stability in the Zhukovsky sense of the trajectories of a dynamic system described by three autonomous nonlinear first-order differential equations using a differential geometric method called the accompanying coordinate reference method. The formulation of the stability problem in the Zhukovsky sense of the trajectories of dynamic systems described by systems of three nonlinear differential equations of the first order is considered. The definitions of stability of trajectories of a three-dimensional nonlinear system are specified. Using the accompanying coordinate reference, the conditions of exponential stability and instability according to Zhukovsky are obtained. The results can be used in studying the stability of processes in systems of natural science and technology, as well as in solving theoretical and applied problems of mathematical modeling and qualitative research of trajectories of nonlinear dynamic systems.

Conformists and contrarians on spheres

We investigate the conformists–contrarians model of identical Kuramoto oscillators evolving on a sphere. Using group-theoretic and geometric approach, we reduce the model to the dynamical system on extended Ott–Antonsen manifold. Further reduction yields the system of three scalar ODE’s for global variables. This three-dimensional dynamical system is studied analytically in order to investigate an interplay between conformists and contrarians on spheres. Our study demonstrates that conformists–contrarians models on spheres display the same types of equilibria and dynamical phenomena in all dimensions. However, critical combination of parameters, for which particular equilibrium states arise, does depend on the dimension. In particular, models on spheres exhibit traveling waves consisting of contrarians. We derive an exact formula for the relation between parameter values for which such waves arise in different dimensions. Finally, we take a closer look at trajectories of traveling waves on spheres, demonstrating subtleties of this dynamical phenomenon.

Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion

The behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and a complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion.

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

Traveling waves in a quintic BBM equation under both distributed delay and weak backward diffusion

In this paper, we focus on the existence of periodic and solitary waves for a quintic Benjamin–Bona–Mahony (BBM) equation with distributed delay and diffused perturbation. The corresponding traveling wave equation is transformed into a three-dimensional dynamical system, which is regarded as a singularly perturbed system for small perturbation parameter. By geometric singular perturbation theory, the three-dimensional dynamical system can be reduced to a near-Hamiltonian planar system and a locally invariant manifold diffeomorphic to the critical manifold with normally hyperbolicity can be constructed. Further, the existence of periodic wave and solitary wave are established by proving the persistence of periodic and homoclinic orbits of the near-Hamiltonian planar system for sufficiently small perturbation, and their existence conditions for the delayed quintic BBM equation have also been obtained. The uniqueness of periodic wave is established by analyzing the monotonicity of ratios of Abelian integrals form a Chebyshev set. Moreover, the monotonicity of wave speed is proved, and the supremum and infimum of wave speed are gotten. Numerical simulations are in complete agreement with the theoretical predictions.

Алгоритм построения полиномиального решения задачи программного управления для динамической системы в частных производных

The completely controlled dynamic system in partial derivatives is considered. The problem of constructing state and control functions in an analytical form is solved. The basic method is the cascade decomposition method, which is algorithmically implemented in three stages: forward cascade decomposition, central stage and reverse. The method is based on the properties of the matrix coefficient at the derivative of the control function. Decomposition means a p-step transition from the original system to a reduced system that is quite similar in form to the original one, but with respect to functions from subspaces. The given conditions are reduced in the process of decomposition. When passing to the p-th step system, additional conditions appear on the partial derivatives of the components of the state function. The number of extra conditions at each point is equal to the number of decomposition steps. The matrix coefficient at the derivative of the control function of the reduced system of the last step is surjective. It is this property that determines the presence of the property of complete controllability of the system under consideration. The first stage of decomposition - the stage of the direct move ends with the detection of the number of decomposition steps and the identification of the property of complete controllability. The task of the central stage of decomposition is to construct the state function of the reduced system of the last step in an analytical form. The state function of the reduced system is the basis function that determines the form of the state function of the original system. Necessary and sufficient conditions for the existence of a basis function in polynomial form are established. The minimum degree of the polynomial is also set, which is determined by the number of decomposition steps. Formulas for constructing vector functions - coefficients of the basis function polynomial are given. Formulas for constructing the control function of the reduced system are given in polynomial form. During the last stage of decomposition, the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system in polynomial form as well. While the last stage of decomposition the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system also in polynomial form. A step-by-step algorithm for solving the program control problem for a dynamic system in partial derivatives has been developed. Formulas for constructing state and control functions in polynomial form are given. An example of a three-dimensional dynamical partial differential system with a surjective matrix coefficient in the first-step splitting system is given. The implementation of the proposed algorithm is demonstrated. The state and control functions are constructed in the form of a polynomial of minimum degree.

Placental scaffolds as a potential biological platform for embryonic stem cells differentiation into hepatic-like cells lineage: A pilot study

Hepatic microenvironment plays an essential role in liver regeneration, providing the necessary conditions for cell proliferation, differentiation and tissue rearrangement. One of the key factors for hepatic tissue reconstruction is the extracellular matrix (ECM), which through collagenous and non-collagenous proteins provide a three-dimensional structure that confers support for cell adhesion and assists on their survival and maintenance. In this scenario, placental ECM may be eligible for hepatic tissue reconstruction, once these scaffolds hold the major components required for cell support. Therefore, this preliminary study aimed to access the possibility of mouse embryonic stem cells differentiation into hepatocyte-like cells on placental scaffolds in a three-dimensional dynamic system using a Rotary Cell Culture System. Following a four-phase differentiation protocol that simulates liver embryonic development events, the preliminary results showed that a significant quantity of cells adhered and interacted with the scaffold through outer and inner surfaces. Positive immunolabelling for alpha fetus protein and CK7 suggest presence of hepatoblast phenotype cells, and CK18 and Albumin positive immunolabelling suggest the presence of hepatocyte-like phenotype cells, demonstrating the presence of a heterogeneous population into the recellularized scaffolds. Periodic Acid Schiff-Diastase staining confirmed the presence of glycogen storage, indicating that differentiate cells acquired a hepatic-like phenotype. In conclusion, these preliminary results suggested that mouse placental scaffolds might be used as a biological platform for stem cells differentiation into hepatic-like cells and their establishment, which may be a promissing biomaterial for hepatic tissue reconstruction

Mesenchymal stem cells paracrine proteins from three-dimensional dynamic culture system promoted wound healing in third-degree burn models.

Recovery of skin function remains a significant clinical challenge for deep burns owing to the severe scar formation and poor appendage regeneration, and stem cell therapy has shown great potential for injured tissue regeneration. Here, a cell-free therapy system for deep burn skin was explored using mesenchymal stem cell paracrine proteins (MSC-PP) and polyethylene glycol (PEG) temperature-sensitive hydrogels. A three-dimensional (3D) dynamic culture system for MSCs' large-scale expansion was established using a porous gelatin microcarrier crosslinked with hyaluronic acid (PGM-HA), and the purified MSC-PP from culture supernatant was characterized by mass spectrometric analysis. The results showed the 3D dynamic culture system regulated MSCs cell cycle, reduced apoptosis, and decreased lactic acid content, and the MSC-PP produced in 3D group can promote cell proliferation, migration, and adhesion. The MSC-PP + PEG system maintained stable release in 28 days of observation in vitro. The in vivo therapeutic efficacy was investigated in the rabbit's third-degree burn model, and saline, PEG, MSC-PP, and MSC-PP + PEG treatments groups were set. The in vivo results showed that the MSC-PP + PEG group significantly improved wound healing, inhibited scar formation, and facilitated skin appendage regeneration. In conclusion, the MSC-PP + PEG sustained-release system provides a potentially effective treatment for deep burn skin healing.

Robust guidance law for cooperative aerial target circumnavigation of UAVs based on composite system theory

Unmanned aerial vehicles (UAVs) are widely used in many fields and relevant research is becoming widespread. However, there are still many problems in the control design of multiple UAVs. This paper proposes a new robust guidance law for the cooperative circumnavigation of an aerial target by multiple UAVs. By incorporating an altitude channel, the controlled fixed-wing UAV was modeled as a simplified three-dimensional (3D) nonlinear dynamic system. Unlike existing studies, tracking and cooperation were considered as a composite system, and the coupling of the UAV linear velocity and angular velocity was considered in the design process. The proposed guidance law considers both the input and output constraints, which represent the system uncertainties and output limits, respectively, and are used to verify the robustness of the guidance law. The robust stability of the disturbed cooperative target circumnavigation system in 3D space is ensured through a strict mathematical proof. Finally, the effectiveness and superiority of the proposed robust guidance law are demonstrated by simulations using a UAV unicycle model and a six-degree-of-freedom (6-DOF) model.

Can Carbon Trading Promote Low-Carbon Transformation of High Energy Consumption Enterprises?—The Case of China

This paper explores the effect of carbon trading on low-carbon transformation of high energy consumption enterprises in China. Based on the mechanism of interaction and restriction among high energy consumption enterprises, carbon verification agencies and the government, a tripartite evolutionary game model is constructed. The three-dimensional dynamic system is built to analyze the behavior patterns of the three parties. The evolution path of the tripartite game is visualized, and the low-carbon transformation states of high energy consumption enterprises in different situations are described. The results show that the high energy consumption enterprises, verification organization and the government cannot reach the optimal game equilibrium (low-carbon transformation, verification and supervision) temporarily when seeking their own interests. The corresponding measures should be taken with different situations of the tripartite game. No matter what strategy the government chooses, the low-carbon transformation could be promoted by carbon trading through carbon verification mechanism.

Stability and Bifurcations in Banks and Small Enterprises—A Three-Dimensional Continuous-Time Dynamical System

Here, we discuss a three-dimensional continuous-time Lotka–Volterra dynamical system, which describes the role of government in interactions with banks and small enterprises. In Italy, during the COVID-19 emergency, the main objective of government economic intervention was to maintain the proper operation of the bank–enterprise system. We also review the effectiveness of measures introduced in response to the COVID-19 pandemic lockdowns to avoid a further credit crunch. By applying bifurcation theory to the system, we were able to produce evidence of the existence of Hopf and zero-Hopf bifurcating periodic solutions from a saddle focus in a special region of the parameter space, and we performed a numerical analysis.