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1、2NonlinearDynamicsofEconomicCyclesComplexdynamicsofeconomicsystemscanbestudiedbyapplyingtheconceptsandtechniquesofnonlineardynamicsandchaos.Somemodelsofbusinesscycles,suchasKaldor’snonlinearinvestment-savingsfunc-tionsandGoodwin’snonlinearaccelerator-multiplier,canbereducedto
2、thevanderPolequationwhichdescribesrelaxationoscillations.Byintroducinganexogenousdriver,theforcedvanderPolequationcanbeadoptedasaprototypemodelforcomplexeconomicdynamics.Numer-icalsolutionsofthismodelcanelucidatethefundamentalpropertiesofcomplexeconomicsystemswhichexhibitawea
3、lthofnonlinearbe-haviorssuchasmultistabilityaswellascoexistenceoforderandchaos.Unstableperiodicorbitsaretheskeletonofchaoticattractorsincom-plexeconomicsystems.2.1EmpiricalEvidenceofNonlinearityandChaosinEconomicDataRecently,thereisagrowinginterestinnonlineardynamicsandchaosi
4、neconomics.Actualeconomictimeseriesarerarelycharacterizedbyregular(periodic,sinusoidal)dynamicstypicaloflinearsystems.In-stead,varioustypesofirregular(aperiodic,non-sinusoidal)formsoflarge-amplitudefluctuationsineconomictimeseriesareoftenobserved,whichcannotbeadequatelyexplain
5、edbylinearanalysis.Thesignifi-cantfluctuationsindicatedbymanyeconomicvariablesrelativetotheirmeanvaluessuggestthatmosteconomicsystemsarefarawayfromtheequilibrium,i.e.,inherentlynonlinear.102NonlinearDynamicsofEconomicCyclesChaoticmotionscanariseinnonlineareconomicsystemsiftheti
6、meseriesisaperiodicanddisplayssensitivedependenceoninitialcondi-tions(Puu1989,Lorenz1993).Empiricalevidenceofcomplexbehav-iorsofnonlineardeterministicsystemscanbeobtainedbycalculatingstatisticalquantitiessuchasLyapunovexponents,entropies,fractaldi-mensions,andcorrelationdimen
7、sions.Thesequantitativemeasuresofchaosaredefinedforinfinitelylargedatasets.Inpractice,largeamountofdatapointsareoftenunavailableinmacroeconomictimeseries.Incontrasttothelaboratoryexperimentswherealargeamountofdatapointscaneasilybeobtained,mosteconomictimeseriesconsistsofmonthly
8、,quarterly,orannualdata,withtheexceptionofsomefinan-cialdatawithdaily
