《非线性与线性微分系统的等价关系研究》.pdf

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1、、，ol-44．N0-3JournalofShanghaiNormalUniversity(NaturalSciences)Jun．，2015D0I：10．3969／J．1SSN．100—5137．2015．03．010OntherelationshipbetweennonlinearandlineardifferentialsystemsZHOUZhengxin，MAOFangfang，GUOYingying，WANGFei，CHENYayuan(SchoolofMathematicalSciences，YangzhouUni

2、versity，Yangzhou225002，China)Abstract：Inthisarticle，weestablishtherelationshipbetweenthequadrmictime—varyingdiferentialsystemsandthelinearsystems，givingthesufficientconditionsforthequadraticsystemstohavethereflectingfunctionintheformoffractiona1function．Weusetheobtai

3、nedresultstodiscussthequal—itativebehaviorofthesolutionsofthequadraticdiferentialsystemsandthetime—varyingKolmogrovequations．Keywords：reflectingfunction；periodicsolution；time-varyingsystemCLCnumber：O175．12Documentcode：AArticleID：1000，5137(2015)03—0322—08MR(2000)：34A1

4、21IntroductionTodiscussthepropertiesofthesolutionsofthediferentialsystem=x(t，)isveryimportantnotonlyforthetheoryofordinarydifferentialequationsbutalsoforpracticalreason．Whenthissystemisanautonomoussystem，itattractsmanyscholars，andmanyexcellentresultsareachieved『l_4】．

5、However，weknowlittleaboutthequalitativebehaviorofthesolutionsofnon．autonomoussystems．Thebook[引showsthat，alotofbiologicalmodelscanbeexpressedbythetime—varyingdiferentialsystem(1)．Specially，whensystem(1)isa2a；一periodicsystem，i．e．，x(t+2w，)=x(t，)(isapositiveconstant)，tos

6、tudythebehaviorofthesolutionsof(1)，wecanuse，asintroducedin[6—7]，thePoincardmapping．ButitisverydificulttofindoutthePoincardmappingformanysystemsthatarenotintegrableinfiniteterms．MironenkolJfirstestablishedthetheoryofreflectingfunctions．Sincethen，anewmethodtoestablisht

7、hePoincardmappinghasbeenfound．Receiveddate：2014．．12．10Foundationitem：ThisworkissupportedbytheNSFofChina(61374010)andtheNSFofJiangsuProvince(BK2012682)and2014一SSFofYangzhouUniversity．Correspondingauthor：ZHOUZhengxin，E—mall：zxzhou@yzu．edu．cnVo1．44，No．3ZHOUZX，MAOFGUOYYe

8、tal：Ontherelationshipbetweennonlinearandlinear．一323Considerthediferentialsystem(1)whichhasacontinuouslydifferentiableright-handside