Three-dimensional Dynamical System Research Articles

Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.

Viewing tropical forest succession as a three-dimensional dynamical system

As tropical forests are complex systems, they tend to be modelled either roughly via scaling relationships or in a detailed manner as high-dimensional systems with many variables. We propose an approach which lies between the two whereby succession in a tropical forest is viewed as a trajectory in the configuration space of a dynamical system with just three dependent variables, namely, the mean leaf-area index (LAI) and its standard deviation (SD) or coefficient of variation along a transect, and the mean diameter at breast height (DBH) of trees above the 90th percentile of the distribution of tree DBHs near the transect. Four stages in this forest succession are identified: (I) naturally afforesting grassland: the initial stage with scattered trees in grassland; (II) very young forest: mostly covered by trees with a few remaining gaps; (III) young smooth forest: almost complete cover by trees of mostly similar age resulting in a low SD; and (IV) old growth or mature forest: the attracting region in configuration space characterized by fluctuating SD from tree deaths and regrowth. High-resolution LAI measurements and other field data from Khao Yai National Park, Thailand show how the system passes through these stages in configuration space, as do simple considerations and a crude cellular automaton model.

Arnold tongues and the Devil's Staircase in a discrete-time Hindmarsh–Rose neuron model

We investigate a three-dimensional discrete-time dynamical system, described by a three-dimensional map derived from a continuous-time Hindmarsh–Rose neuron model by the forward Euler method. For a fixed integration step size, we report a two-dimensional parameter-space for this system, where periodic structures, the so-called Arnold tongues, can be seen with periods organized in a Farey tree sequence. We also report possible modifications in this parameter-space, as a function of the integration step size.

Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras

The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L 3 . The investigation is based on Bianchi’s classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L 3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem.

Three-dimensional forced-damped dynamical systems with rich dynamics: Bifurcations, chaos and unbounded solutions

We consider certain autonomous three-dimensional dynamical systems that can arise in mechanical and fluid-dynamical contexts. Extending a previous study in Craik and Okamoto (2002), to include linear forcing and damping, we find that the four-leaf structure discovered in that paper, and unbounded orbits, persist, but may now be accompanied by three distinct period-doubling cascades to chaos, and by orbits that approach stable equilibrium points. This rich structure is investigated both analytically and numerically, distinguishing three main cases determined by the damping and forcing parameter values.

A three-dimensional dynamic coculture system enabling facile cell separation for chondrogenesis of mesenchymal stem cells

Coculture has emerged as a strategy to improve chondrogenesisof mesenchymal stem cells (MSCs) for cartilage regeneration. In the present study, a new system integrating cell encapsulation, dynamic stimulation and coculture with articular chondrocytes (ACs), termed as three-dimensional (3D) dynamic coculture, was developed for chondrogenic differentiation of MSCs. In addition, a novel mechanism was exploited to enable facile separation of cocultured cells. Rabbit bone-marrow derived MSCs (rMSCs) and rabbit ACs (rACs) were encapsualted in alginate gel beads, respectively. In rACs-laden beads, magnetic Fe3O4 nanoparticles were incorporated. Dynamic coculture was performed in a spinner flask. Fe3O4 nanoparticles had no negative effects on growth of rACs, although a slight inhibition on production of glycosaminoglycans (GAG) by rACs was noticed. An alginate concentration of above 1.8w/v% was favorable for dynamic culture of rMSCs and rACs. Coculturing rMSCs with rACs in chondrogenic medium in this system demonstrated an improved chondrogenesis than monocultured rMSCs according to GAG quantification, histological staining, and gene expression analysis of collagen I, II and X. However, supplementation of 2% FBS did not show posive effects on chondrogenesis of cocultured rMSCs. This proof-of-concept study demonstrates that such a separable 3D dynamic coculture system holds great potential in cartilage tissue regeneration.

Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann–Lemaître–Robertson–Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ‘attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation.

A Four-Leaf Chaotic Attractor of a Three-Dimensional Dynamical System

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

Complexity Uncertainty Analysis of Dynamic in a Dual-Channel Energy Supply Chain Model with Heterogeneous Retailers

This paper analyses the dynamics of dual-channel energy supply chain model with heterogeneous retailers (as regards the type of expectations’ formation). On the basis of analyzing the stabilities of four fixed points in the three-dimensional dynamic system, local stable regions of Nash equilibrium are obtained. Effects ofSon the stable regions and profit are studied. Simulation results show that the adjustment of price speed has an obvious impact on the complexity of competition. The performances of the model in different period are measured by using the index of average profit. The results show that unstable behavior in economic system is often an unfavorable outcome. So this paper discusses the application of parameters control method when the model is in chaos and then allows the oligarchs to eliminate the negative effects.

Визуализация фазового пространства некоторых классов трехмерных динамических систем

Визуализация фазового пространства некоторых классов трехмерных динамических систем

A computer-assisted proof of existence of a periodic solution

We consider a three-dimensional dynamical system proposed in \textit{Physica D}, \textbf{164}, (2002), 168–186. It is a conservative system and is unusual in that most of the solutions are unbounded. The paper presented a conjecture that an unstable periodic orbit determines directions of unbounded orbits of helical form. In the present paper we prove existence and local uniqueness of the conjectured periodic orbit by a method of numerical verification.

Time-history simulation of civil architecture earthquake disaster relief- based on the three-dimensional dynamic finite element method

Earthquake action is the main external factor which influences long-term safe operation of civil construction, especially of the high-rise building. Applying time-history method to simulate earthquake response process of civil construction foundation surrounding rock is an effective method for the anti-knock study of civil buildings. Therefore, this paper develops a civil building earthquake disaster three-dimensional dynamic finite element numerical simulation system. The system adopts the explicit central difference method. Strengthening characteristics of materials under high strain rate and damage characteristics of surrounding rock under the action of cyclic loading are considered. Then, dynamic constitutive model of rock mass suitable for civil building aseismic analysis is put forward. At the same time, through the earthquake disaster of time-history simulation of Shenzhen Children’s Palace, reliability and practicability of system program is verified in the analysis of practical engineering problems.

Асимптотичні хвильові розв’язки моделі середовища з осциляторами Ван дер Поля

A one-dimensional mathematical model for a complex medium with van der Pol oscillators has been studied. Using the Bogolyubov–Mitropolsky method, the wave solutions for a weakly nonlinear model are derived, with their amplitudes being described by a three-dimensional dynamical system analyzed in more details by numerical and qualitative methods. In particular, periodic, multiperiodic, and chaotic trajectories are found in the phase space of the dynamical system. Bifurcations of those regimes were considered using the Poincar´e section technique. Exact solutions are derived in the case where the three-dimensional system for amplitudes is reduced to the two-dimensional one.

Mutual correlation and spectral analysis of excitations from reference tracks on different ruts

Issues related to assignment of kinematic excitation from random path profile on dynamic three-dimensional system of a moving machine using spectral modelling method are considered. During calculation for three reference backgrounds, self- and mutual statistical characteristic of excitation on left and right ruts for different moving velocities are obtained.

Effect of Purmorphamine on Osteogenic Differentiation of Human Mesenchymal Stem Cells in a Three-Dimensional Dynamic Culture System

We have previously demonstrated that purmorphamine exerts an osteoinductive effect on human bone marrow mesenchymal stem cells (hBM-MSCs) in a two-dimensional (2D) static culture system. In the present study, we have evaluated the effect of purmorphamine on osteogenic differentiation of human mesenchymal stem cells (hMSCs) in a three-dimensional (3D) dynamic culture system vs. a 2D static culture condition. In this study, hMSCs were seeded in either 3D collagen scaffolds using a perfusion bioreactor or on 2D culture plates. One day later, the medium was replaced with osteogenic medium supplemented with dexamethasone, l‐ascorbic acid and β-glycerophosphate in the presence or absence of purmorphamine. Histological staining, flow cytometry, and real-time PCR were conducted for evaluation of osteogenesis in the study groups. The expressions of RUNX-2, alkaline phosphatase (ALP), osteocalcin (OC), collagen I (Col I), and bone sialoprotein (BSP) were compared between the 3D and 2D culture systems at 14 and 21 days post-induction of differentiation. Based on our results, alizarin red staining showed deposition of the mineralized matrix in the 2D static and 3D dynamic cultures. Our flow cytometric analysis indicated that the number of ALP+/OC− cells increased in the presence of purmorphamine in the 2D culture rather than the 3D system at day14. One week later, the numbers of OC+/ALP− and OC+/Stro-1− cells increased significantly in the 3D dynamic model compared with the 2D cultures. Expression of RUNX-2 was upregulated in the presence of purmorphamine in both cultures at day 14. The purmorphamine response gene, Gli-1, was upregulated during both the early and late culture periods in the 3D dynamic system, while similar up-regulation was observed only during the early culture period in 2D culture. No difference was observed in expressions of collagen type I and BSP between groups. The present study indicates that purmorphamine as agonist of Shh signaling pathway affects on osteogenic differentiation program of hMSCs in static and dynamic culture systems.

Connecting curves in higher dimensions

Connecting curves have been shown to organize the rotational structure of strange attractors in three-dimensional dynamical systems. We extend the description of connecting curves and their properties to higher dimensions within a special class of differential dynamical systems. The general properties of connecting curves are derived and selection rules stated. Examples are presented to illustrate these properties for dynamical systems of dimension n = 3, 4, 5.

A fundamental model exhibiting nonlinear oscillatory dynamics in solid oxide fuel cells

In this paper, we address the phenomenon of temporal, self-sustained oscillations which have been observed under quite general conditions in solid oxide fuel cells. Our objective is to uncover the fundamental mechanisms giving rise to the observed oscillations. To this end, we develop a model based on the fundamental chemical kinetics and transfer processes which take place within the fuel cell. This leads to a three-dimensional dynamical system, which, under typical operating conditions, is rationally reducible to a planar dynamical system. The structural dynamics of the planar dynamical system are studied in detail. Self-sustained oscillations are shown to arise through Hopf bifurcations in this planar dynamical system, and the key parameter ranges for the occurrence of such oscillations are identified.

Simulations of a graphene excitable laser for spike processing

We propose and simulate a novel excitable laser employing passively \(Q\)-switching with a graphene saturable absorber for spike processing. Our approach combines the picosecond processing and switching capabilities of both linear and nonlinear optical device technologies to integrate both analog and digital optical processing into a single hardware architecture capable of computation without the need for analog-to-digital conversion. We simulate the laser using the Yamada model—a three-dimensional dynamical system of rate equations—and show behavior that is typical of spiking processing algorithms simulated in small circuits of excitable lasers.

Destabilizing a stable crisis: Employment persistence and government intervention in macroeconomics

The basic Keen model is a three-dimensional dynamical system describing the time evolution of the wage share, employment rate, and private debt in a closed economy. In the absence of government intervention this system admits, among others, two locally stable equilibria: one with a finite level of debt and nonzero wages and employment rate, and another characterized by infinite debt and vanishing wages and employment. We show how the addition of a government sector, modelled through appropriately selected functions describing spending and taxation, prevents the equilibrium with infinite debt. Specifically, we show that, by countering the fall in private profits with sufficiently high government spending at low employment, the extended system can be made uniformly weakly persistent with respect to the employment rate. In other words, the economy is guaranteed not to stay in a permanently depressed state with arbitrarily low employment rates.

Chaotic characteristics of three-dimensional function determined by cross-section geometric shape

The calculation and simulation results show that f(x,y,z)=sin(k(x2+y2+z2)), f(x,y,z)=k(1-(x2+y2+z2))e(-(x+y+z+u)2), f(x,y,z)=k((x2+y2+z2)/3)(1-(x2+y2+z2)/3) can easily constructe a three-dimensional (3D) discrete dynamic system by combining other two polynomial functions generated randomly. Through calculating Lyapunov exponent and drawing the bifurcation diagram, the characteristics of chaos of the function are confirmed, and according to the bifurcation diagram of parameters and the Lyapunov exponent curve more chaotic mapping functions are found. Analysis shows that the cross-section geometric shape can determine the chaotic characteristics of 3D function, and the cross-sections are all the median convex or middle concave surfaces, which can constructe chaotic dynamic systems easily. In the future, the mathematical description model and some basic theorems are to be further investigated and their results will be used to solve practical problems such as turbulence.