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.

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.

FORMULAS OF VALUES OF SINGULAR POINT AND THE INTEGRABILITY CONDITIONS FOR A CLASS OF CUBIC SYSTEM, M(3)≥7

The values of the singular point in complex autonomous differential systems have been introduced and discussed in [1] in order that the focal values and the saddle values in real plane autonomous differential systems can be treated in the same way. One problem unsloved up to now is that for any second order autonomous differential system with a cubic polynomial on its right-hand side, what the maximum order of fineness of its weak focal point, weak saddle point in real domain and weak critical singular point in complex domain is(M(3)=?). Inequality M(3)≥5 can be obtained from the results in[2] and[3].

LIMIT CYCLE F0R CUBIC SYSTEM OF KOLMOGOROV TYPE WITH A CONIC ALGEBRAIC TRAJECTORY

In this paper we study the existence of limit cycle for cubic system (E)_3, of Kolmogorov typewith a conic algebraic trajectoryF_2(x,y)=ax^2+2bxy+cy^2+dx+ey+f=0 It has been proved in my former papers that (E)_3 doesn't have any limit cycle on the whole planeIf b^2-ac≠0, Now we are investigating the case where b^2-ac=0. We prove the sufficient andnecessary formula (2) or (13) witb which (E)_3 must have a parabolic trajectory F_2(x,y)=0. Thenthere will not be any limit cycle on the full plane. On the basis of this, we conclude: The cubic system of Kolmogorov type with a non-degenerated quadratic algebraic trajectory onthe whole plane has no limit cycle.

An apodized cubic phase mask used in a wavefront coding system to extend the depth of field

The point spread function(PSF)caused by a wavefront coding system with a cubic phase mask has big side-lobes which leads to bad image restoration.This paper proposes a novel apodized cubic phase mask to suppress the side-lobes of the PSF.Simulated annealing algorithm is used to optimize the cubic and the truncation parameter of the phase mask.The system with the novel phase mask has better performance in the modulation transfer function(MTF)especially in low-and-medium spatial frequency region.The simulation results show that the restored images with the novel phase mask are superior to the one with the classic cubic phase mask in contrast and ringing effect.The experimental results show that the side-lobes of the PSF are suppressed by using the apodized cubic phase mask.

THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lidnard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system （2） in the following when parameter al changes.

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.