Plane Equilibrium Research Articles

The first six propositions of Archimedes' on equilibrium of planes 1

ABSTRACT Modern commentators have doubts about the authenticity and cogency of the early propositions of Archimedes’ On Equilibrium of Planes Book 1. Ernst Mach famously said that the proof of Prop. 6, the so-called law of the lever, assumes what is to be proven. Comparing the initial text in Heiberg’s modern edition (1881, 1913) to the first propositions in Eutocius’ commentary on EP 1, J. L. Berggren ([1976]. ‘Spurious Theorems in Archimedes’ Equilibrium of Planes: Book I’, Archive for History of Exact Sciences 16.2 (1976), 87–103.) claimed that the propositions up through Proposition 3 of the standard modern edition are schoolbook additions written by an ancient author inferior to Archimedes. The present paper argues for the logical connectedness of Postulates 1–5 to Props. 1–6, by means of a detailed examination of the course of the argument and a re-examination of Eutocius’ remarks. The paper reinterprets the role of the empirical in the early propositions and offers a reading of the contribution of Archimedes’ mechanics to the method of EP 1.

Development of a tokamak magnetohydrodynamic code with the discontinuous Galerkin and Weighted Essentially Non-Oscillatory methods

In this study, we present the development of a new initial-value magnetohydrodynamic (MHD) code for toroidal geometry using discontinuous Galerkin (DG) and Weighted Essentially Non-Oscillatory (WENO) methods. The code utilizes a triangular mesh based on the flux of the fixed boundary equilibrium in the poloidal plane, which is uniformly divided in the toroidal direction. By solving the conservative perturbed generalized Lagrange multiplier (GLM) MHD model, the code can simulate both ideal and resistive toroidal plasmas. We validate the code by performing the linear calculations of the internal kink mode and tearing mode, and our results show good agreement with calculations conducted using an eigen-value MHD code. Furthermore, nonlinear simulations of the resistive internal kink mode and tearing mode are also carried out. We discuss the challenges and limitations of the code, as well as the future efforts to improve its performance. The development of this new initial-value MHD code opens up new possibilities for studying and understanding the dynamics of toroidal plasmas, contributing to the advancement of fusion research.

Offset-Control Plane Coexisting Behaviors in Two-Memristor-Based Hopfield Neural Network

Memristor synapse with activated synaptic plasticity can be taken as an adaptive connection synaptic weight. To demonstrate its kinetic effects, this article presents an improved Hopfield neural network with two memristive self-connection synaptic weights. The two-memristor-based Hopfield neural network (TM-HNN) has a plane equilibrium set related to two memristor initial conditions and its stability distributions are analyzed by two non-zero roots of the eigenvalue polynomial. Afterwards, the parameter-related bifurcation behaviors are investigated using bifurcation plots and phase portraits. Emphatically, the kinetic effects of memristor synapses are demonstrated by taking the memristor initial conditions as two invariant measures. The theoretical and numerical results show that the TM-HNN can exhibit the wondrous offset-control plane coexisting behaviors and its plane coexisting attractors can be controlled by switching the two memristor initial conditions. Besides, a digital hardware device is developed and the offset-control plane coexisting attractors are experimentally reproduced to verify the numerical ones.

Initial-offset boosted dynamics in memristor-sine-modulation-based system and its image encryption application

The initial-offset boosted dynamics has attracted much attentions for its potential applications in synchronous control and secure communication. This paper builds a five-dimensional (5-D) memristor-sine-modulation-based (MSMB) system to generate initial-offset boosted dynamics. The 5-D MSMB system is built by adding a negative feedback loop and one positive feedback loop to a three-dimensional linear system, that is, both the memristor- and sine-modulation methods are used. Through theoretical analysis, the plane equilibrium set with its initial-related stability and the periodical evolutionary dynamics together with its mechanism are revealed. Furthermore, initial-offset boosted dynamics induced by the memristor initial conditions (MICs) and the non-memristor initial conditions (NMICs) is uncovered by numerical simulations. Several sets of coexisting initial-offset boosted attractors are also captured by Microcontroller-based digital implementation, which are well matched with the numerical results. Finally, three sets of the pseudo random sequences generated from the MSMB system are applied to image encryption with good encryption performance.

Incremental integral reconstitution for detecting initial condition effects

The five-dimensional (5-D) two-memristor-based jerk (TMJ) system can emerge the initial condition-dependent extreme multi-stability due to two zero roots caused by its plane equilibrium set. However, using conventional stability analysis methods, we cannot explain the kinetic mechanism of initial condition effects on the 5-D TMJ system. To this end, this paper establishes a three-dimensional (3-D) incremental integral reconstructed (IIR) model using an integral transformation method, and converts the internal initial conditions and plane equilibrium set of the original system into the explicit initial condition parameters (ICPs) and one or no equilibrium point of the reconstructed model respectively. With the explicit ICPs and determined equilibrium point of the reconstructed model, the memristor and non-memristor initial condition effects on the original system are thereby detected by the stability distributions and then confirmed by the kinetic distributions. The results show that the memristor and non-memristor initial conditions have powerful influence on the kinetic effects of the 5-D TMJ system, and its kinetic mechanism of initial condition effects can be explained effectively based on the 3-D IIR model. Besides, with analog circuit and digital circuit platform, the circuit simulations and hardware experiments verify the kinetic effects of the ICPs on the 3-D IIR model.

Передний мини-инвазивный экстраперитон бальный доступ к позвоночнику на уровне t12-s1

Anterior mini-invasive approach to the spine that enables to achieve all levels from T12 to SI is described. This approach can be used both in injuries and in degenerative pathology of the spine. Surgical results were studied roentgenologically and that gave us the possibility to assess the performed osteosynthesis and the equilibrium in the sagittal plane. The suggested approach is the continuation of classical anterior approaches and it provides significant advantages for the performance of various surgical interventions at all levels of the lumbar spine without damaging the muscles. Theoretically it possesses neither neurologic risk nor the problem of blood loss which occur when intervertebral transplantation is performed via the posterior approach. Anterior extraperitoneal mini-approach enables to adjust the size of the graft and to perform the correction of sizable deformities using either rigid or semirigid graft. It can also be the only choice in case of considerable loss of posterior bone mass, weakness of the posterior graft and infection in the zone of the posterior approach.

Persistence and Bifurcation Analysis of a Plankton Ecosystem with Cross-Diffusion and Double Delays

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.

Infinitely Many Necklace-Shaped Coexisting Attractors in a Nonautonomous Memcapacitive Oscillator

Extreme multistability usually emerges in a mem-element’s circuit or system that possesses a line or plane equilibrium set closely associated with the internal initial state of the mem-element. To extend the investigation of extreme multistability, this paper proposes a nonautonomous memcapacitive oscillator, discovering a new type of extreme multistability due to the infinitely many discrete equilibrium points therein. This memcapacitive oscillator is constructed by connecting a simple memcapacitor-resistor circuit with a sinusoidal voltage. With its normalized model, the infinitely many discrete equilibrium points are computed and the infinitely many necklace-shaped coexisting attractors that were not yet reported are disclosed by numerical methods. Since the number and stability of the equilibrium points vary with time, the attraction basins with complex ripple structures are formed in the memcapacitive oscillator, resulting in the appearance of a special type of extreme multistability. Furthermore, PSIM circuit simulations and microcontroller-based hardware experiments are performed to verify the numerical results.

Dynamic Analysis in a Plankton Ecosystem Involving Two Delays

In this paper, a phytoplankton and zooplankton relationship system with two delays is investigated whose coefficients are related to one of the two delays. Firstly, the dynamic behaviors of the system with one delay are given and the stability of positive equilibrium and the existence of periodic solutions are obtained. Using the fact that the system may occur, the stable switching phenomenon is verified. Under certain conditions, the periodic solutions will exist in a wide range as the delay gets away from critical values. Fixing the delay [Formula: see text] in the stable interval, it is revealed that the effect of [Formula: see text] can also cause the vibration of system. This explains that two delays play an important role in the oscillation behavior of the system. Furthermore, using the crossing curve methods, the stable changes of the positive equilibrium in two-delays plane are given, which generalizes the results of systems for which the coefficients do not depend on delay. Some numerical simulations are provided to verify the theoretical results.

Extreme Multistability and Antimonotonicity in a Shinriki Oscillator with Two Flux-Controlled Memristors

This paper proposes a novel memristive chaotic circuit which originated from a Shinriki oscillator with two flux-controlled memristors of different polarities. This two-memristor-based Shinriki oscillator (TMSO) having a special plane equilibrium is prone to exhibiting the initial-dependent phenomenon of extreme multistability. To investigate its internal dynamics, a third-order dimensionality reduction model is established by utilizing the constitutive relationship of its memristor’s flux and charge. The uncertain plane equilibrium is transfered into some deterministic model that can accurately predict the dynamical evolution of the system, where interesting phenomena of asymmetric bifurcations, extreme multistability and antimonotonicity are detected and analyzed by evaluating the position and stability of the equilibria in the flux–charge model. The simulation is carried out via Multisim to validate the analysis model, and the comparison of the phase trajectories, before and after dimensionality reduction, shows that this oscillator is good for research and practical use.

Hopf Bifurcation and Stability Crossing Curve in a Planktonic Resource–Consumer System with Double Delays

In this paper, a planktonic resource–consumer system with two delays is investigated and the coefficients depend on [Formula: see text] one of the two delays. Firstly, the property of solution and the existence of equilibrium are given. The dynamical analysis of the system including stability and Hopf bifurcation by using the delays as parameters is carried out. Both the single delay and two delays can cause the system to produce Hopf bifurcation and the stable switching phenomena may exist. Furthermore, using the crossing curve methods, we obtain the stable changes of equilibrium in two-delay parameter plane, which generalizes the results of the system that the coefficients do not depend on delay. Furthermore, the numerical simulation results show that the theoretical analyses are correct when the delays change.

Parametric finite-volume method for Saint Venant’s torsion of arbitrarily shaped cross sections

We extend our recent finite-volume based approach to the solution of Saint Venant’s torsion problems of bars and beams comprised of rectangular sections to enable analysis of arbitrary cross sections characterized by curved boundaries. This is accomplished by incorporating parametric mapping based on transfinite grid generation to enable discretization of the bar cross section by quadrilateral rather than rectangular subvolumes employed in the original version. The construction of the local stiffness matrix that relates the surface-averaged subvolume warping functions to the corresponding tractions is carried out in the reference plane such that the subvolume equilibrium in the physical plane is satisfied in a surface-averaged sense. This produces explicit expressions for the stiffness matrix elements that may be readily coded. Orthotropic subvolumes are intrinsic in the method’s construction so that bars with heterogeneous and composite microstructures may be analyzed. The convergence and accuracy of the parametric finite-volume method are assessed and verified upon comparison with exact elasticity solutions for cross sections with convex and concave boundaries. Examples involving structural applications of prismatic bars with curved boundaries illustrate the utility of the developed methodology. These include cross sections that resemble biological constructs with homogeneous and graded regions aimed at enhancing torsional rigidities, as well as homogeneous and graded elliptical cross sections with orthotropic shear moduli aimed at reducing and eliminating warping. We demonstrate for the first time that by laminating an elliptical cross section with alternating stiff and soft isotropic layers in a manner that mimics the required orthotropic moduli at the homogenized level, warping can be practically eliminated with sufficient microstructural refinement.

Reconstitution for interpreting hidden dynamics with stable equilibrium point

Abstract Extreme multistability induced by line or plane equilibrium point sets can be successfully reconstituted in the integral domain to achieve mechanism understanding and physical observation. Inspired by this, hidden dynamics with stable equilibrium point, a peculiar multistability, may also be interpreted in the integral domain. To this end, a three-dimensional (3-D) quadratic jerk system with hidden chaotic and periodic behaviors is selected as a representative paradigm to disclose the reconstituted hidden dynamics. Through integral and linear transformations, a 3-D reconstituted system with standalone initials-related constant parameters is constructed. Similar to the original system, the reconstituted system possesses one stable equilibrium point but can reconstitute the previous dynamical behaviors while initiating from the origin point. The immediate benefit is that the reconstituted hidden dynamical behaviors are relatively easier to be observed in a physical circuit. Finally, the physical circuit is designed and circuit simulations are performed by Power SIMulation (PSIM) to validate the numerical results.

A New Method for Plane Equilibrium Injection-production Adjustment of Water-drive Reservoir

In view of the lack of theoretical research on plane equilibrium injection-production adjustment in water drive reservoir, the production prediction method under the condition of variable pressure difference of production well is deduced according to Buckley Leverett non-piston displacement theory. Taking the same degree of extraction as the goal, the adjustment method of plane equilibrium injection-production per unit time is put forward. The evaluation coefficient of the optimal equilibrium injection-production scheme is defined, which can guide the optimization of the equilibrium injection-production scheme. Through theoretical model analysis, it is considered that the adjustment method of plane equilibrium injection-production is reasonable and reliable. Seven kinds of adjustment schemes are designed for BZ oilfield in Bohai Sea. Of equilibrium displacement time. When the equilibrium displacement time is more than 10 years, the increment of recovery degree slows down to about 1.8%. Equilibrium injection-production adjustment was carried out for 1195 sand body in BZ oilfield. The sand body increased oil by 25m3/d per day, and the natural decline rate decreased by 1.1%.

Dynamics of a hyperchaotic map with spherical attractor

This paper presents a novel three-dimensional hyperchaotic map with the spherical attractor. The typical dynamic properties of the new chaotic system are evaluated by attractor diagrams, Lyapunov exponent (LE), bifurcations and permutation entropy (PE) complexity. The results show that it has rich dynamical behaviors, including hyperchaos, large Lyapunov exponent and typical bifurcations. There are also some unusual features, such as coexisting attractors and plane equilibrium. To improve its performance, we extend the attractor space by adding the staircase wave function to the new system. Compared with the original system, the expanded system has a larger complexity and more complex topology. By controlling the system parameters, it can effectively improve the PE complexity of the system. Finally, the digital circuit of the new system is realized based on DSP technology, and the feasibility of the system model is verified. The proposed hyperchaotic map has advantages in complexity and rich dynamic characteristics, which has good application prospects in secure communication.

Chaotic flows with special equilibria

In this review, we categorize chaotic systems with special properties in their equilibrium into eight groups: Systems with no equilibrium (NE), systems with stable equilibrium (SE), systems with line equilibrium (LE), systems with curve equilibrium (CE), systems with plane equilibrium (PE), systems with surface equilibrium (ES), systems with unstable equilibrium (US), and systems which belong to more than one of the above categories (Chameleon systems). We introduce the pioneer works from each group. Also, we review some recent important papers related to those groups.

Memristor initial-boosted coexisting plane bifurcations and its extreme multi-stability reconstitution in two-memristor-based dynamical system

Initial-dependent extreme multi-stability and offset-boosted coexisting attractors have been significantly concerned recently. This paper constructs a novel five-dimensional (5-D) two-memristor-based dynamical system by introducing two memristors with cosine memductance into a three-dimensional (3-D) linear autonomous dissipative system. Through theoretical analyses and numerical plots, the memristor initial-boosted coexisting plane bifurcations are found and the memristor initial-dependent extreme multi-stability is revealed in such a two-memristor-based dynamical system with plane equilibrium. Furthermore, a dimensionality reduction model with the determined equilibrium is established via an integral transformation method, upon which the memristor initial-dependent extreme multi-stability is reconstituted theoretically and expounded numerically. Finally, physically circuit-implemented PSIM (power simulation) simulations are carried out to validate the plane offset-boosted coexisting behaviors.

Two-memristor-based chaotic system and its extreme multistability reconstitution via dimensionality reduction analysis

Abstract This paper presents a two-memristor-based chaotic system by introducing two memristors into an existing chaotic jerk system. Such a memristive system possesses plane equilibrium therein, leading to the emergence of extreme multistability. Due to the existence of two zero eigenvalues for the plane equilibrium, it is extremely difficult to quantitatively explore the dynamical mechanism based on the original system model. For this reason, an equivalent dimensionality reduction model is obtained using state variable mapping (SVM) method. Consequently, the initials-dependent extreme multistability in the two-memristor-based chaotic system is reconstituted by the initial parameters-related dynamics in the dimensionality reduction model, upon which the exact prediction, analysis and control of extreme multistability is thereby executed through traditional quantitative analyses. Furthermore, PSIM circuit simulations based on a physical circuit are performed to confirm the extreme multistability. It is demonstrated that the initials-dependent extreme multistability can be reconstituted through dimensionality reduction analysis, which is efficient for exploring the inner mechanisms and further seeking possible applications of this special phenomenon.

Memristor initial boosting behaviors in a two-memristor-based hyperchaotic system

Abstract By substituting two resistive couplings with two memristive couplings, a two-memristor-based hyperchaotic system is proposed. This hyperchaotic system has a plane equilibrium with two zero eigenvalues and three nonzero ones, and the equilibrium plane is divided into three regions with different stabilities, including unstable saddle, stable node, and stable node-focus. Through using the local attraction basin, bifurcation diagrams and Lyapunov exponent spectra, dynamical behaviors are analyzed in the unstable saddle region, from which the bi-stability phenomenon is observed in such a hyperchaotic system. More particularly, the memristor initial boosting behaviors are found, which indicates that the attractor offset boosting is controlled by the memristor initial values. The memristor initial boosting is multi-dimension, nonlinearity and non-monotonic, completely different from the variable offset boosting. Subsequently, PSIM circuit simulations are performed to verify the numerical results.

Influence of Poynting-Robertson Drag and Oblateness on Existence and Stability of Out of Plane Equilibrium Points in Spatial Elliptic Restricted Three Body Problem

This paper studies existence and stability of the out of plane equilibrium points \(L_{6,7}\) analytically and numerically in the elliptical restricted three body problem, where both the primaries are radiating oblate spheroid, incorporating the effects of Poynting-Robertson (P-R) drag of the radiating primaries as well. It is observed that consideration of PR-drag forces results in the non-zero y-coordinate of the out of plane equilibrium points. The stability of the equilibrium points is studied in presence of PR-drag forces. Also the stability is further analyzed in the case when the PR-drag forces are neglected. We have explored the existence of out of plane equilibrium points and their stability around the binary systems: Luyten-726 and Sirius.