LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.

On the(1,3)Distributions of Limit Cycles of Plane Quadratic Systems

In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3) distri-butions of limit cycles, we expand the variable regions of parameters ( s , r) and clearly. show them in figure, (c) we study the (1, 3) distributions of limit cycles of one kind quadratic systems with two singular points at the infinite: and (d) we give a generalmethod to discuss the ( 1 ,3) distibutions`of limit cycles of system (1.1) whatever there isone, two or three singular points at the infinite.

BIFURCATION OF LIMIT CYCLE FOR QUADRATIC SYSTEM WITH INVARIANT PARABOLA

In a previous paper, we have proved that a planar quadratic system with invariant parabola r has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions un'der which a PQSp with three non-degenerate singular points can be transformed into twO different definite forms. In this wayl we obtain all the bifurcations of such a system.

Exact Expressions of Energy Spectrum and Partition Function for Quantum Quadratic Systems

Based on the linear quantum transformation theory,we present a new approach to obtain the explicit expressions of energy spectrum and simplify the derivations of partition functions for general multi-mode boson and fermion quadratic systems.

Dynamic Event-Triggered Quadratic Nonfragile Filtering for Non-Gaussian Systems:Tackling Multiplicative Noises and Missing Measurements

This paper focuses on the quadratic nonfragile filtering problem for linear non-Gaussian systems under multiplicative noises,multiple missing measurements as well as the dynamic event-triggered transmission scheme.The multiple missing measurements are characterized through random variables that obey some given probability distributions,and thresholds of the dynamic event-triggered scheme can be adjusted dynamically via an auxiliary variable.Our attention is concentrated on designing a dynamic event-triggered quadratic nonfragile filter in the well-known minimum-variance sense.To this end,the original system is first augmented by stacking its state/measurement vectors together with second-order Kronecker powers,thus the original design issue is reformulated as that of the augmented system.Subsequently,we analyze statistical properties of augmented noises as well as high-order moments of certain random parameters.With the aid of two well-defined matrix difference equations,we not only obtain upper bounds on filtering error covariances,but also minimize those bounds via carefully designing gain parameters.Finally,an example is presented to explain the effectiveness of this newly established quadratic filtering algorithm.

Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Enhancement of the group delay in quadratic coupling optomechanical systems subjected to an external force

We theoretically study the effect of the quadratic coupling strength on optomechanical systems subjected to a continuous external force. Quadratic coupling strength originates from strong coupling between the optical and the mechanical degrees of freedom. We show that the quadratic coupling strength reduces the amplitude of the dispersion spectra at the resonance in both blue-and red-sideband regimes. However, it increases(decreases) the amplitude of the absorption spectrum in the blue-(red-)sideband regime. Furthermore, in both sideband regimes, the effective detuning between the pump and the cavity deviates with the quadratic coupling strength. Thereby, appropriate selection of the quadratic coupling strength results in an important magnification(in absolute value) of the group delay for both slow and fast light exiting from the optomechanical cavity.

Nonparametric Statistical Feature Scaling Based Quadratic Regressive Convolution Deep Neural Network for Software Fault Prediction

The development of defect prediction plays a significant role in improving software quality. Such predictions are used to identify defective modules before the testing and to minimize the time and cost. The software with defects negatively impacts operational costs and finally affects customer satisfaction. Numerous approaches exist to predict software defects. However, the timely and accurate software bugs are the major challenging issues. To improve the timely and accurate software defect prediction, a novel technique called Nonparametric Statistical feature scaled QuAdratic regressive convolution Deep nEural Network (SQADEN) is introduced. The proposed SQADEN technique mainly includes two major processes namely metric or feature selection and classification. First, the SQADEN uses the nonparametric statistical Torgerson–Gower scaling technique for identifying the relevant software metrics by measuring the similarity using the dice coefficient. The feature selection process is used to minimize the time complexity of software fault prediction. With the selected metrics, software fault perdition with the help of the Quadratic Censored regressive convolution deep neural network-based classification. The deep learning classifier analyzes the training and testing samples using the contingency correlation coefficient. The softstep activation function is used to provide the final fault prediction results. To minimize the error, the Nelder–Mead method is applied to solve non-linear least-squares problems. Finally, accurate classification results with a minimum error are obtained at the output layer. Experimental evaluation is carried out with different quantitative metrics such as accuracy, precision, recall, F-measure, and time complexity. The analyzed results demonstrate the superior performance of our proposed SQADEN technique with maximum accuracy, sensitivity and specificity by 3%, 3%, 2% and 3% and minimum time and space by 13% and 15% when compared with the two state-of-the-art methods.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Enhancing Proficiency in Quadratic Equations and Functions Through the MILAPlus Strategy

This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadratic equations and functions through a study carried out at Quezon National High School.The research involved 116 Grade 9 students and utilized a quantitative approach,incorporating both pre-assessment and post-assessment measures.The research utilizes a quasi-experimental design,examining the academic performance of students before and after the introduction of MILAPlus.The pre-assessment establishes a baseline,and the subsequent post-assessment measures the impact of the instructional strategy.Statistical analyses,including t-tests,assess the significance of differences in mean scores and mean percentage scores,providing quantitative insights into the effectiveness of MILAPlus.Findings from the study revealed a statistically significant improvement in both mean scores and mean percentage scores after the utilization of MILAPlus,indicating enhanced proficiency in quadratic equations and functions.The Mean Proficiency Scores(MPS)also showed a substantial increase,demonstrating a marked improvement in overall proficiency levels among Grade 9 students.In light of the results,recommendations were given including the continued utilization of MILAPlus as an instructional strategy and aligning its development with prescribed learning competencies.Emphasizing the consistent adherence to policies and guidelines for MILAPlus implementation is suggested for sustaining positive effects on students’long-term performance in mathematics.This research contributes valuable insights into the practical application and effectiveness of MILAPlus within the context of Grade 9 mathematics education at Quezon National High School.

A METHOD PROVING THE UNIQUENESS OF THE LIMIT CYCLE OF THE QUADRATIC SYSTEM

We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.

TWO PROBLEMS OF PLANAR QUADRATIC SYSTEMS

The main results of this paper are as follows: （ⅰ） The important formulas, given by Bautin, of three focal quantities for the specific form of quadratle system （E2） have been generalized to the general form of （E2）. （ⅱ） By using the method in [13], a kind of （E2） possessing at least four limit cycles is given. Theorem 2 herein contains the results in [11--13] on （1,3）-distribution of limit cycles of （E2）.

POINCAR BIFURCATION FOR QUADRATIC SYSTEMS WITH A CENTER REGION AND AN UNBOUNDED TRIANGULAR REGION

In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincaré bifurcation, that inside the center region quadratic system perturbed by quadratic polynomial perturbation may generate three limit cycles.

QUADRATIC SYSTEMS WITH A 3RD-ORDER (OR 2ND-ORDER) WEAK FOCUS

In this paper, we prove that a planar quadratic systems with a 3rd-order weak focus has at most one limit cycle, and a planar quadratic system with a 2nd-order weak focus has at most two limit cycles.

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,and that if this system has a separatrix cycle passing through the weak saddle,then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.

BIFURCATION PROBLEM OF CRITICAL POINTSFOR QUADRATIC DIFFERENTIAL SYSTEMS

In this paper foe bifurcation of critical points for the quadratic systems of type(II)and (III) is investigated. and an answer to the problem given in[1] is given.

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.

HOMOCLINIC CYCLES OF A QUADRATIC SYSTEM DESCRIBED BY QUARTIC CURVES

This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system. We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. Finally, the corresponding global phase diagrams are drawn.

THE HOPF BIFURCATION OF QUADRATIC SYSTEM (I)_(n=0)

δlm is the parameter space of quadratic system (I)n=0. A partition of parameters corresponding to the existence and nonexistence of the limit cycle of the system is given in detail. The Hopf bifurcation surfaces of (I)n=0 are obtained, and the sketch of Hopf bifurcation surfaces of (I)n=0 are drawn.