辅助函数法应用举例

本文给出用辅助函数法解题的若干例子。由此可以看出辅助函数法应用的一斑。例1 已知acosθ bsinθ=c,acosφ bsinφ=c((θ-φ)/2≠kπ,k为整数)。求证a/cos(θ φ)/2=b/sin(θ φ)/2=c/cos(θ-φ)/2 证明作辅助函数f=(x,y)=ax by-c,则点P(cosθ,sinθ),Q(cosφ,sinφ)在直线f(x,y)=0上,此时直线方程为ax by=c,由两点式可得 (y-sinθ)/(x-cosθ) =(sinθ-sinφ)/(cosθ-cosφ) ∴xcos[(θ φ)/2] ysin[(θ φ)/2] =cos[(θ-φ)/2],     

Bogdanov-Takens系统极限环和同宿轨线及分岔

讨论Bogdanov-Takerrs系统极限环、同宿轨线及其关于参数分岔的曲线定量分析。给出这些问题的近似解析表达式的参数增量法；利用时间变换，将极限环和同宿轨线表示为广义谐函数的解析表达式；画出参数与极限环关于振幅稳定性特征指数、极限环与同宿轨线的相图，以及参数的分岔图等曲线。     

勿忘(0)~(1/2)=0

我们知道 0 =0 ,但在解题过程中 ,却常常忽视了 ,这反映了我们考虑问题的片面性 .例 1　当α、β取什么范围内的值时 ,式子sin2αcosβ有意义 ?错解　由 sin2 α≥ 0知　 sin2 αcosβ≥ 0等价于 cosβ≥ 0 .即β∈ [2 kπ - π2 ,2 kπ π2 ](k∈ Z) .分析　因为学生牢记“实数的平方为非负数”,即α∈ R时 ,sin2 α≥ 0 .所以由sin2 αcosβ≥ 0推导出 cosβ≥ 0 .事实上 ,这漏掉了另一种情况 :sinα =0 ,cosβ∈ R时原式也有意义 .即α =kπ,且β∈ R,k∈ Z.正解　原式有意义等价于 cosβ≥ 0或sinα =0 .解得β∈ [2 kπ - π2 ,2 k…     

两个不等式的三角证法及其推广

在一些刊物上 ,常讨论下述不等式 :设a >1,b>1,c>1,则a3b2 - 1+ b3c2 - 1+ c3a2 - 1≥ 92 3(1)a5b2 - 1+ b5c2 - 1+ c5a2 - 1≥2 56 15 (2 )本文就这两个不等式给出统一的三角证法 ,并给以推广 .首先给出下面引理 .引理　若θ∈ 0 ,π2 ,k是正整数 ,则sin2 θ·cos2k - 1θ≤ 2 (2k - 1) 2k - 1(2k + 1) 2k +1.当且仅当secθ =2k + 12k - 1时 ,等号成立 .证　由均值不等式得 :(2k - 1) (sin2 θ +cos2 θ) =(2k - 1)sin2 θ2 +(2k - 1)sin2 θ2 +cos2 θ+cos2 θ +… +cos2 θ(2k - 1)项≥ (2k +1)·2k + 1(2k - 1) 24 (sin2 θ·cos2k - 1…     

2000年普通高等学校春季招生考试(北京、安徽卷)数学(理工农医类)

本试卷分第 卷 (选择题 )和第 卷 (非选择题 )两部分 .共 15 0分 .考试时间 12 0分钟 .第 卷 (选择题共 60分 )参考公式 :三角函数和差化积公式sinθ sinφ=2 sinθ φ2 cosθ-φ2sinθ-sinφ=2 cosθ φ2 sinθ-φ2cosθ cosφ=2 cosθ φ2 cosθ-φ2cosθ-cosφ=-2 sinθ φ2 sinθ-φ2正棱台、圆台的侧面积公式S台侧 =12 ( c′ c) l其中 c′、c分别表示上、下底面周长 ,l表示斜高或母线长台体的体积公式V台体 =13 ( S′ S′S S) h其中 S′、S分别表示上、下底面积 ,h表示高一、选择题 :本大题共 14小题 ;第 ( 1)— ( 10 )题每小题…     

研究三角函数性质的法宝——“三个一”

高一学生分析问题时最缺乏的就是目标意识,有的同学拿到三角函数性质的题目,想半天都没有一个明确的解题方向,其实所有这类问题都是首先将目标三角函数化为“三个一”:y=Asin(ωx+φ)+k的形式,即一个角的一种函数名称的一次式的形式,因为课本中三角函数的每一种性质都是由“三个一”型三角函数而展开讨论的,我们只有将目标三角函数化归成这种模型,才能使用课本结论灵活解题·例1求函数y=sin3xsin3cxos+22cxos3xcos3x+sin2x的最小值.分析只需将目标三角函数化简为“三个一”:y=Asin(ωx+φ)+k的形式即可·解法1因为sin3xsin3x+cos3xcos3x=(sin3xsinx)sin2x+(cos3xcosx)cos2x=21[(cos2x-cos4x)]sin2x+21[(cos2x+cos4x)cos2x]=21[cos2x+(cos2x-sin2x)cos4x]=21(cos2x+cos2xcos4x)=21cos2x(1+cos4x)=cos32x,∴y=cos32xcos22x+sin2x=cos2x+sin2x=2sin(2x+4π).当sin(2x+π4)=-1时,y...     

用(sinα±cosα)~2=1±2sinαcosα解题

由平方关系sin2α+cos2α=1不难得到(sinα±cosα)2=1±2sinαcosα.它揭示了sinα+cosα、sinα-cosα、sinαcosα三者之间的密切关系,知其一必能求出另二.在一些解方程、求最值问题中,恰当运用此关系有助于简化运算、发现解题途径.例1已知sinα+cosα=1/5(0<α<π),求tanα的值.分析本题可先求出sinα-cosα的值,再和sinα+cosα=15联立方程组求出sinα,cosα     

在不定积分计算中不能忽视"区间I"

有这么一道求不定积分的题目 :例 ∫ 1 -sin2θdθ在以往的教学乃至某些考研资料中发现有这样做的 :解 ∫ 1 -sin2θdθ=∫ (sinθ-cosθ) 2 dθ=∫ |sinθ-cosθ|dθ=± (cosθ+sinθ) +C初看似乎没错 ,但仔细推敲就会发现有问题。实际上只有当θ∈ [2kπ -3π4,2kπ + π4]时 (k是整数 ) ,cosθ-sinθ 0 ,才有(cosθ+sinθ)′=cosθ -sinθ=｜sinθ-cosθ｜从而cosθ+sinθ在这些区间上才是 |sinθ-cosθ|的一个的原函数。而当θ∈ [2kπ + π4,2kπ + 5π4]时 ,sinθ-cosθ 0 ,(-cosθ-sinθ)′=sinθ -cosθ=｜sinθ-cosθ｜从而与上面…     

具有扫描的混合环路的数学分析

本文研究这样的锁相环路,其环路的滤波器为理想滤波器,特性为(s α)/s,而扫描速度为R,鉴相器特性为sinφ k sin 2φ。因而相应的环路方程为:     

巧解三角题一例

例设a,b,θ满足方程组:(选自《全国中学数学竞赛题解》1978科学普及出版社P44天津(复赛)第一题) 该《题解》的处理方法是把前二个等式变换后划去参数sinθ、cosθ、分别代入第三个等式及变形后的第三个等式,得到关于a、b的二元二次方程组。方程中项数少的有4项,多的则达6项,可以设想其计算量的多而且杂。倘若我们把sin0,cos0,a,b互相视为“参数”或用sinθ、cosθ代换(或部分代换)a、b,或求出sinθ,cosθ,或通过巧妙的变换,化二次为一次运算,通过成为简单的计算而得解。下解此题。     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

Bifurcations of limit cycles from infinity for a class of quintic polynomial system

In this work, we use an indirect method to investigate bifurcations of limit cycles at infinity for a class of quintic polynomial system, in which the problem for bifurcations of limit cycles from infinity be transferred into that from the origin. By the computation of singular point values, the conditions of the origin (correspondingly, infinity) to be the highest degree fine focus are derived. Consequently, we construct a quintic system with a small parameter and eight normal parameters, which can bifurcates 1 to 8 limit cycles from infinity respectively, when let normal parameters be suitable values. The positions of these limit cycles without constructing Poincaré cycle fields can be pointed out exactly.     

The limit cycles of a second-order system of differential equations: The method of small forms

In this paper we investigate the existence of limit cycles of a system of the second-order differential equations with a vector parameter.We propose a method for representing a solution as a sum of forms with respect to the initial value and the parameter; we call this technique the method of small forms. We establish the conditions under which a sufficiently small neighborhood of the equilibrium point contains no limit cycles. We construct a polynomial, whose positive roots of odd multiplicity define the lower bound for the number of cycles, and simple positive roots (other positive roots do not exist) define the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.We prove theorems, whose conditions guarantee that a positive root of odd multiplicity defines a unique limit cycle, but a positive root of even multiplicity defines exactly two limit cycles.We propose a method for defining the type of the stability of limit cycles.     

一类异宿环分支问题中极限环的唯一性

在具余维２奇点的四维系统的两参数开折的研究中出现一类三点异宿环的扰动分支,对此异宿环产生极限环的唯一性一直未得到完整的解决,本文圆满地解决了这一问题,并获得了全局分支中极限环的唯一性。     

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.     

The number and stability of limit cycles for planar piecewise linear systems of node–saddle type

The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles.     

Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation

In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of quadratic systems with small and large limit cycles. The expressions for the first five Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to Lienard system and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.     

On the limit cycles of perturbed discontinuous planar systems with 4 switching lines

Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results.     

Analysis of limit cycle behavior in DC–DC boost converters

This paper deals with limit cycle behaviors in DC–DC boost converters with a proportional-integral (PI) voltage compensator, which is a popular design solution for increasing output voltage in power electronics. Extensive cycle-by-cycle numerical simulations are used to capture all limit cycle behaviors. It is found that there exist two types of limit cycle behaviors rather than only one type in a boost converter. For each type of limit cycle, its underlying mechanism is revealed by circuit analysis. Moreover, the critical condition is derived to predict the occurrence of the limit cycle behaviors in terms of Routh stability criterion, and the analytical expressions for the limit cycles I and II are given based on the averaged model approach. Finally, these theoretical results are verified by numerical simulations and circuit experiments.