Limit Cycle Containing Nine Critical Points in its Interior for a Class of Cubic Systems

Discuss a class of real planar cubic systems with a critical point O (0,0) of nine orders and obtain the conditions for its limit cycle surrounding the origin, and prove that when small pertubations of coefficients are made, the critical point O (0,0) of nine orders is split into nine real simple critical points and the limit cycle surrounding the origin becomes the limit cycle containing nine critical points in its interior.

ANovel Non-Isolated Cubic DC-DC Converter with High Voltage Gain for Renewable Energy Power Generation System

In recent years,switched inductor(SL)technology,switched capacitor(SC)technology,and switched inductor-capacitor(SL-SC)technology have been widely applied to optimize and improve DC-DC boost converters,which can effectively enhance voltage gain and reduce device stress.To address the issue of low output voltage in current renewable energy power generation systems,this study proposes a novel non-isolated cubic high-gain DC-DC converter based on the traditional quadratic DC-DC boost converter by incorporating a SC and a SL-SC unit.Firstly,the proposed converter’s details are elaborated,including its topology structure,operating mode,voltage gain,device stress,and power loss.Subsequently,a comparative analysis is conducted on the voltage gain and device stress between the proposed converter and other high-gain converters.Then,a closed-loop simulation system is constructed to obtain simulation waveforms of various devices and explore the dynamic performance.Finally,an experimental prototype is built,experimental waveforms are obtained,and the experimental dynamic performance and conversion efficiency are analyzed.The theoretical analysis’s correctness is verified through simulation and experimental results.The proposed converter has advantages such as high voltage gain,low device stress,high conversion efficiency,simple control,and wide input voltage range,achieving a good balance between voltage gain,device stress,and power loss.The proposed converter is well-suited for renewable energy systems and holds theoretical significance and practical value in renewable energy applications.It provides an effective solution to the issue of low output voltage in renewable energy power generation systems.

Metastable face-centered cubic ruthenium-based binary alloy for efficient alkaline hydrogen oxidation electrocatalysis

Metastable nanostructured electrocatalyst with a completely different surface environment compared to conventional phase-based electrocatalyst often shows distinctive catalytic property.Although Ru-based electrocatalysts have been widely investigated toward hydrogen oxidation reaction(HOR)under alkaline electrolytes,these studies are mostly limited to conventional hexagonal-close-packed(hcp)phase,mainly arising from the lack of sufficient synthesis strategies.In this study,we report the precise synthesis of metastable binary RuW alloy with face-centered-cubic(fcc)phase.We find that the introduction of W can serve as fcc phase seeds and reduce the formation energy of metastable fcc-RuW alloy.Impressively,fcc-RuW exhibits remarkable alkaline HOR performance and stability with the activity of 0.67 mA cm_(Ru)^(-2)which is almost five and three times higher than that of hcp-Ru and commercial Pt/C,respectively,which is attributed to the optimized binding strength of adsorbed hydroxide intermediate derived from tailored electronic structure through W doping and phase engineering.Moreover,this strategy can also be applied to synthesize other metastable fcc-RuCr and fcc-RuMo alloys with enhanced HOR performances.

Synthesizing active and durable cubic ceria catalysts(<6 nm)for fast dehydrogenation of bio-polyols to carboxylic acids coproducing green H_(2)

Dehydrogenation is considered as one of the most important industrial applications for renewable energy.Cubic ceria-based catalysts are known to display promising dehydrogenation performances in this area.Large particle size(>20 nm)and less surface defects,however,hinder further application of ceria materials.Herein,an alternative strategy involving lactic acid(LA)assisted hydrothermal method was developed to synthesize active,selective and durable cubic ceria of<6 nm for dehydrogenation reactions.Detailed studies of growth mechanism revealed that,the carboxyl and hydroxyl groups in LA molecule synergistically manipulate the morphological evolution of ceria precursors.Carboxyl groups determine the cubic shape and particle size,while hydroxyl groups promote compositional transformation of ceria precursors into CeO_(2) phases.Moreover,enhanced oxygen vacancies(Vo)on the surface of CeO_(2) were obtained owing to continuous removal of O species under reductive atmosphere.Cubic CeO_(2) catalysts synthesized by the LA-assisted method,immobilized with bimetallic PtCo clusters,exhibit a record high activity(TOF:29,241 h^(-1))and Vo-dependent synergism for dehydrogenation of bio-derived polyols at 200℃.We also found that quenching Vo defects at air atmosphere causes activity loss of PtCo/CeO_(2) catalysts.To regenerate Vo defects,a simple strategy was developed by irradiating deactivated catalysts using hernia lamp.The outcome of this work will provide new insights into manufacturing durable catalyst materials for aqueous phase dehydrogenation applications.

A novel rapid cooling assembly design in a high-pressure cubic press apparatus

In traditional high-pressure–temperature assembly design, priority has been given to temperature insulation and retention at high pressures.This limits the efficiency of cooling of samples at the end of experiments, with a negative impact on many studies in high-pressure Earth andplanetary science. Inefficient cooling of experiments containing molten phases at high temperature leads to the formation of quench textures,which makes it impossible to quantify key compositional parameters of the original molten phase, such as their volatile contents. Here,we present a new low-cost experimental assembly for rapid cooling in a six-anvil cubic press. This assembly not only retains high heatingefficiency and thermal insulation, but also enables a very high cooling rate (∼600 ℃/s from 1900 ℃ to the glass transition temperature).Without using expensive materials or external modification of the press, the cooling rate in an assembly (∼600 ℃/s) with cube lengths of38.5 mm is about ten times faster than that in the traditional assembly (∼60 ℃/s). Experiments yielding inhomogeneous quenched melttextures when the traditional assembly is used are shown to yield homogeneous silicate glass without quench textures when the rapid coolingassembly is used.

Accounting for Quadratic and Cubic Invariants in Continuum Mechanics–An Overview

The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields.These equations contain derivatives of the first order with respect to time.The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero.Derivatives are used to derive the wave equation.The differential wave equation is second order in time.Therefore,increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation.This is important for calculating the generation of sound waves and water hammer waves.Therefore,the Euler continuity equation with finite time increments is of interest.The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor.This is a new branch in hydrodynamics.Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.

Effect of Size and Initial Water Content on the Effective Diffusion Coefficient during Convective Drying of Sweet Potato Cut into Cubic and Cylindrical Shapes

In this article, we investigated the influence of size and initial water content on the effective diffusion coefficient of sweet potatoes samples cut into cubic and cylindrical shapes. The sizes of the cubic samples are 0.5, 1, 1.5, 1.75, 2, 2.5 and 3 cm edge with a respective initial water content of 2.7, 3.76, 3.48, 2.68, 3.28, 2.17 and 2.29 kg/kgms. For cylindrical samples, the radius is set at 0.5 cm and sample heights are 1, 1.5, 2, 2.5, 3, 3.5 and 4 cm with respective water contents of 2.2, 3.19, 2.85, 2.1, 2.17, 2.39 and 2.03 kg/kgms. The effective diffusion coefficients of cubic samples are of the order of 10−10 and 10−9 m2∙s−1 grew with sample edge. As for the cylindrical samples, the effective diffusion coefficients were of the order of 10−9 m2∙s−1 and there was no linear correlation between cylinder height and their effective diffusion coefficient. As for the examination of the initial water content on the effective diffusion coefficient, it turned out that the initial water content had no influence on the effective diffusion coefficient of the sweet potato samples.

Global Analyses of a Class of Cubic Systems

In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P3（x,y）,=y+Q3（x,y）（0.1） where P3 and Q3 are homogeneous polynomials of degree 3 in x,y.Through this work,we draw an overall outline of such systems.

The Global Bifurcation of a Cubic System

In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.

THE E^(1)_(3) TYPE OF CUBIC SYSTEMS WITH TWO INTEGRAL STRAIGHT LINES

The existence and uniqueness of limit cycle for the E 1 3 type of cubic systems with two integral straight lines has been studied in this paper. It is found that the system has no limit cycle when the two integral straight lines intersect each other; it has a unique limit cycle when the two integral straight lines are paralleled. The sufficient and necessary conditions are also given to guarantee the existence of the unique limit cycle.

LOCAL TOPOLOGICAL STRUCTURE OF THE PLANE CUBIC SYSTEM IN THE UNDETERMINED SIGN CASE

In this paper we discuss the topological structure near the singular point O (0,0) of the plane cubic system in the undetermined sign case, and give their coefficient conditions.

GLOBAL BIFURCATIONS IN A PERTURBED CUBIC SYSTEM WITH Z_2-SYMMETRY

In this paper we consider global and local bifurcations in disturbed planar Hamiltonianvector fields which are invariant under a rotation over π. All calculation formulas of bifurcationcurves have been obtained. Various possible distributions and the existence of limit cycles andsingular cycles in different parameter regions have been determined. It is shown that for a planarcubic differential system there are infinitely many parameters in the three-parameter space suchthat Hilbert number H(3)≥11.

A SPECIAL CUBIC SYSTEM CLOSE RELATED TOTHE GENERAL QUADRATIC SYSTEM

We study the number and distribution of critical points as we Ⅱ as algebraic solutions of a cubic system close related to the general quadratic system.

A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.

THE UNIQUENESS OF LIMIT CYCLE AND CRITICAL POINT FOR A CLASS OF CUBIC SYSTEM

In this paper,we consider an accompany system concerning some class of cubic system. We then prove that the system has at most one limit cycle. Finally,we obtain the topological structure of both the critical points at infinity and the singular points lying on invariant lines.

ON THE BERLINSKII'S THEOREM FOR CUBIC SYSTEMS

In [1]-[3], the Berlinskii's theorem of the distribution of critical points for quadratic differential systems is extended to the general n-th differential systems with n2 finite critical points. In this paper, we prove that 5 - 4 distribution of critical points for cubic system is impossible by using the method of basic triangle and index formula. Then we discuss the possible distributions of cubic systems with eight, seven or six finite critical points.

1∶2 INTERNAL RESONANCE OF COUPLED DYNAMIC SYSTEM WITH QUADRATIC AND CUBIC NONLINEARITIES

The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.

Existence of Limit Cycles for a Cubic Kolmogorov System with a Hyperbolic Solution

This paper is concerned with a cubic Kolmogorov system with a solution of central quadratic curve which neither contacts with the coordinate axes, nor passes through the origin. The conclusion is that such a system may possess limit cycles.

Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System

By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.