常微分方程课件-第13讲

1,Lecture13,2,NonlinearDifferentialEquations(非线性微分方程),ChapterVI(第六章),6.1Stability(稳定性),6.2MethodsofVFunctions(V函数方法),6.3SingularPoints(奇点),6.4Limit-cyclesandPhasePortraits(极限环和平面图貌相图),3,NonlinearDifferentialEquations(非线性微分方程),ChapterVI(第六章),6.1Stability(稳定性),6.2MethodsofVFunctions(V函数方法),6.3SingularPoints(奇点),6.4Limit-cyclesandPhasePortraits(极限环和平面图貌相图),4,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),5,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),6,6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),Considerthefollowingsystemofnfirst-orderDEs,Itsvectorformis,where,7,cfC/P195,8,DefinitionC/P249;E/P553,9,Theorem1*C/P250;E/P554(Existence-uniquenesstheorem)(存在唯一性定理),10,Theorem2*C/P250(Theoremofcontinuationandcontinuityofsolutions)(解的延拓与连续性定理),11,Theorem3*C/P250(Theoremofdifferentiability)(可微性定理),12,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),13,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),14,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),15,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),16,6.1.2Liapunovstability(李雅普诺夫稳定性),1.AnexampleC/PP250-252;E/PP471-472,Considerfirst-ordernonlinearDE,17,18,19,20,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),21,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),22,2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系)C/P252,Anundisturbedsolution,23,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),24,6.1.2Liapunovstability(李雅普诺夫稳定性),1.Anexample(一个例子),2.Therelationshipbetweenthezerosolutionandthenonzeroparticularsolution(零解与非零特解之间的关系),3.Definitionsofstability(稳定性的定义),25,Definition1C/P253/(cfC/P236),3.Definitionsofstability(稳定性的定义),26,Definition1C/P253/(cfC/P236),3.Definitionsofstability(稳定性的定义),27,Definition1C/P253/(cfC/P236),3.Definitionsofstability(稳定性的定义),28,Definition1C/P253/(cfC/P236),3.Definitionsofstability(稳定性的定义),29,Definition1C/P253/(cfC/P236),3.Definitionsofstability(稳定性的定义),30,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),31,6.1Stability(稳定性),6.1.1Theexistence-uniquenesstheoremforsystemsofordinarydifferentialequations(常微分方程组的存在唯一性定理),6.1.2Liapunovstability(李雅普诺夫稳定性),6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),32,6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),1.StabilityoflinearsystemofDEs(线性微分方程组的稳定性),2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),3.TheHurwitzcriterion(赫尔维茨准则),33,6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),1.StabilityoflinearsystemofDEs(线性微分方程组的稳定性),2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),3.TheHurwitzcriterion(赫尔维茨准则),34,1.StabilityoflinearsystemofDEsC/P256;Cf(C/P235),35,Theorem11C/PP235-236,36,Theorem1C/P256,37,6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),1.StabilityoflinearsystemofDEs(线性微分方程组的稳定性),2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),3.TheHurwitzcriterion(赫尔维茨准则),38,6.1.3Stabilitydeterminedaccordingtolinearapproximation(按线性近似决定稳定性),1.StabilityoflinearsystemofDEs(线性微分方程组的稳定性),2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),3.TheHurwitzcriterion(赫尔维茨准则),39,2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),Autonomoussystem(自治系统),40,2.TherelationshipbetweenstabilityoflinearsystemofDEsandstabilityofnonlinearsystemofDEs(线性方程组的稳定性与非线性方程组的稳定性之间的关系),Theorem2C/P256,Thelinearizedsystemof(6.13)(6.13)的线性化系统,41,RemarkC/P256,42,Example1C/P257;E/P523,43,Example1C/P257;E/P523,44,Result1,45,46,Result2,47,48,49,QuestionC/P259,50,Theorem3(HurwitzCriterion)C/P259,51,Example2C/P259,Solution,52,53,Example3C/P260,54,ExerciseC/P261/4(1),55,NonlinearDifferentialEquations(非线性微分方程),ChapterVI(第六章),6.1Stability(稳定性),6.2MethodsofVFunctions(V函数方法),6.3SingularPoints(奇点),6.4Limit-cyclesandPhasePortraits(极限环和平面图貌相图),56,NonlinearDifferentialEquations(非线性微分方程),ChapterVI(第六章),6.1Stability(稳定性),6.2MethodsofVFunctions(V函数方法),6.3SingularPoints(奇点),6.4Limit-cyclesandPhasePortraits(极限环和平面图貌相图),57,6.2MethodsofVFunctions(V函数方法),6.2.1LiapunovTheorem(李雅普诺夫定理),6.2.2TheConstructionofQuadraticFormVFunctions(二次型V函数的构造),58,6.2MethodsofVFunctions(V函数方法),6.2.1LiapunovTheorem(李雅普诺夫定理),1.Somebasicconcepts(一些基本概念),2.Liapunovtheorem(李雅普诺夫定理),59,1.Somebasicconcepts(一些基本概念),Definition2C/P264,60,Example1C/P264,61,ExerciseC/P277/1(1,3,5),62,Exercise,63,Example3C/P264,64,Definition3C/P21+P263;E/P471,65,Definition4C/P264,66,Example4C/P269,67,6.2MethodsofVFunctions(V函数方法),6.2.1LiapunovTheorem(李雅普诺夫定理),1.Somebasicconcepts(一些基本概念),2.Liapunovtheorem(李雅普诺夫定理),68,6.2MethodsofVFunctions(V函数方法),6.2.1LiapunovTheorem(李雅普诺夫定理),1.Somebasicconcepts(一些基本概念),2.Liapunovtheorem(李雅普诺夫定理),69,2.Liapunovtheorem(李雅普诺夫定理),70,2.Liapunovtheorem(李雅普诺夫定理)