THE CONTROL OF CHAOS THEORY AND APPLICATIONS

THECONTROLOFCHAOS:THEORYANDAPPLICATIONSS.BOCCALETTI!,C.GREBOGI,Y.-C.LAI#,H.MANCINI!,D.MAZA!!DepartmentofPhysicsandAppliedMathematics,InstituteofPhysics,UniversidaddeNavarra,Irunlarreas/n,31080Pamplona,SpainInstituteforPlasmaResearch,DepartmentofMathematics,andInstituteforPhysicalScienceandTechnology,UniversityofMaryland,CollegePark,MD20742,USA#Dept.ofMath.andElectricalEngineering,CenterforSystemsScienceandEngineeringResearch,ArizonaStateUniversity,Tempe,AZ85287AMSTERDAM}LAUSANNE}NEWYORK}OXFORD}SHANNON}TOKYOS.Boccalettietal./PhysicsReports329(2000)103}197103PhysicsReports329(2000)103}197Thecontrolofchaos:theoryandapplicationsS.Boccaletti!,C.Grebogi,Y.-C.Lai#,H.Mancini!,D.Maza!!DepartmentofPhysicsandAppliedMathematics,InstituteofPhysics,UniversidaddeNavarra,Irunlarreas/n,31080Pamplona,SpainInstituteforPlasmaResearch,DepartmentofMathematics,andInstituteforPhysicalScienceandTechnology,UniversityofMaryland,CollegePark,MD20742,USA#Dept.ofMathandElectricalEngineering,CenterforSystemsScienceandEngineeringResearch,ArizonaStateUniversity,Tempe,AZ85287,USAReceivedJune1999;editor:I.ProcacciaContents1.Introduction1061.1.Thecontrolofchaos:exploitingthecriticalsensitivitytoinitialconditionstoplaywithchaoticsystems1061.2.FromtheOtt}Grebogi}Yorkeideasandtechniquetotheothercontrolmethods1071.3.Targetingdesirablestateswithinchaoticattractors1081.4.Thecontrolofchaoticbehaviors,andthecommunicationwithchaos1091.5.Theexperimentalvericationsofchaoscontrol1101.6.OutlineoftheReport1102.TheOGYmethodofcontrollingchaos1112.1.Thebasicidea1112.2.Aone-dimensionalexample1112.3.Controllingchaosintwodimensions1142.4.Poleplacementmethodofcontrollingchaosinhighdimensions1212.5.Discussion1273.Theadaptivemethodforcontrolofchaos1283.1.Thebasicidea1283.2.Thealgorithmforadaptivechaoscontrol1293.3.Applicationtohigh-dimensionalsystems1314.Theproblemoftargeting1364.1.Targetingandcontrollingfractalbasinboundaries1364.2.Theadaptivetargetingofchaos1455.Stabilizingdesirablechaotictrajectoriesandapplication1495.1.Stabilizingdesirablechaotictrajectories1495.2.Theadaptivesynchronizationofchaosforsecurecommunication1776.Experimentalevidencesandperspectivesofchaoscontrol1796.1.Introduction1796.2.Nonfeedbackmethods1816.3.ControlofchaoswithOGYmethod1826.4.Controlofelectroniccircuits1846.5.Controlofchemicalchaos1856.6.Controlofchaosinlasersandnonlinearoptics1866.7.Controlofchaosin#uids1876.8.Controlofchaosinbiologicalandbiomechanicalsystems1896.9.Experimentalcontrolofchaosbytimedelayfeedback1906.10.Otherexperiments192Acknowledgements192References1930370-1573/00/$-seefrontmatter(2000ElsevierScienceB.V.Allrightsreserved.PII:S0370-1573(99)00096-4AbstractControlofchaosreferstoaprocesswhereinatinyperturbationisappliedtoachaoticsystem,inordertorealizeadesirable(chaotic,periodic,orstationary)behavior.Wereviewthemajorideasinvolvedinthecontrolofchaos,andpresentindetailtwomethods:theOtt}Grebogi}Yorke(OGY)methodandtheadaptivemethod.Wealsodiscussaseriesofrelevantissuesconnectedwithchaoscontrol,suchasthetargetingproblem,i.e.,howtobringatrajectorytoasmallneighborhoodofadesiredlocationinthechaoticattractorinbothlowandhighdimensions,andpointoutapplicationsforcontrollingfractalbasinboundaries.Inshort,wedescribeproceduresforstabilizingdesiredchaoticorbitsembeddedinachaoticattractoranddiscusstheissuesofcommunicatingwithchaosbycontrollingsymbolicsequencesandofsynchronizingchaoticsystems.Finally,wegiveareviewofrelevantexperimentalapplicationsoftheseideasandtechniques.(2000ElsevierScienceB.V.Allrightsreserved.PACS:05.45.#bS.Boccalettietal./PhysicsReports329(2000)103}1971051.Introduction1.1.Thecontrolofchaos:exploitingthecriticalsensitivitytoinitialconditionstoplaywithchaoticsystemsAdeterministicsystemissaidtobechaoticwheneveritsevolutionsensitivelydependsontheinitialconditions.Thispropertyimpliesthattwotrajectoriesemergingfromtwodi!erentclosebyinitialconditionsseparateexponentiallyinthecourseoftime.Thenecessaryrequirementsforadeterministicsystemtobechaoticarethatthesystemmustbenonlinear,andbeatleastthreedimensional.Thefactthatsomedynamicalmodelsystemsshowingtheabovenecessaryconditionspossesssuchacriticaldependenceontheinitialconditionswasknownsincetheendofthelastcentury.However,onlyinthelastthirtyyears,experimentalobservationshavepointedoutthat,infact,chaoticsystemsarecommoninnature.Theycanbefound,forexample,inChemistry(Belouzov}Zhabotinskireaction),inNonlinearOptics(lasers),inElectronics(Chua}Matsumotocircuit),inFluidDynamics(Rayleigh}BeHnardconvection),etc.Manynaturalphenomenacanalsobecharacterizedasbeingchaotic.Theycanbefoundinmeteorology,solarsystem,heartandbrainoflivingorganismsandsoon.Duetotheircriticaldependenceontheinitialconditions,andduetothefactthat,ingeneral,experimentalinitialconditionsareneverknownperfectly,thesesystemsareinstrinsicallyun-predictable.Indeed,thepredictiontrajectoryemergingfromabonaxdeinitialconditionandtherealtrajectoryemergingfromtherealinitialconditiondivergeexponentiallyincourseoftime,sothattheerrorintheprediction(thedistancebetweenpredictionandrealtra