Recap

Terminologyrecap

• Variableorstate• Differentialequation• Initialcondition• Trajectory• Parameter• Steadystate• Transientbehaviour• Perturbation• Ordinarydifferentialequations(ODE)• 3dimensionalODE

Terminologyrecap

• Phasespace/statespace• Vectorfield• Fixedpoint(stable/unstable,focus/node)• Nullcline• Saddles,separatrix• Bistability

DynamicalSystems

Yujiang.wang@ncl.ac.ukLecture3of5

Overview• Whataredynamicalsystems?• Howtointerpretadifferentialequation• Howtoanalyse differentialequationsystems• Howtosolvedifferentialequationsystems• Stabilityanalysis,multistability• Oscillatorysolutions• Parametervariations,bifurcations• Choiceofcoolstuff:Chaos,turbulence,spatio-temporalsystems,slow-fastsystems,transients,andmore.

Numericallysolvingdifferentialequations

Numericallysolvingdifferentialequations

• Why?– Veryrarethatwecananalyticallysolveequations– Implementationspeed– Convenience

• Whynot?– Sometimeslongsimulationtimes– Inaccuracies– Variationsbetweendifferentsolvers

Numericallysolvingdifferentialequations:methods

• Euler• Heun• Runge-kutta

Eulermethod

Exampleequation:dydt=f(t,y)Solvesinafixedstep(h=1)iterativemanner

Letssayinitialcondition,y0 = 1 and f(t,y)=y

Ifwestartwithy=0 attime(t)=0,howmuchdoesy changebetweent=0 andt=4?

Eulermethod

Exampleequation:dydt=f(y)Solvesinafixedstep(h=1)iterativemanner

Letssayinitialcondition,y0 = 1 and f(y)=y

Time->

Eulermethod

• Notveryaccurateifh istoolarge

Eulermethod

• Notveryaccurateifh istoolarge

• Alternatively:Notveryaccurateifthechangeiny,relativetothechangeint(i.e.h)istoolarge

Eulermethod

Eulermethod:anotherexample

Eulermethod:anotherexample

• Inphasespace

Eulermethod:code

Eulermethod• Knownasafixedstepsolversinceh isaconstant(intheseexamplesalwaysh=1)

• Easytoimplement• Predictableruntimes(scaleslinearlywithnumberoftimesteps)

• Easilyadaptedtoincorporatedelayse.g.wheredydt=f(y,t-τ)

• Easilyadaptedtoincorprate noisee.g.dydt=f(y,t)+w• Canbeslowandinaccuratecomparedtoothersolvers...

• Usesinformationfromtwopoints– Changeiny aty(t)– Changeinyatpredictedy(t+Δt)– dydt= y(t) + h/2 (f(y,t) + f(t+Δt,y+h f(y,t)))

Heun’s method

Heun’s method• Usesinformationfromtwopoints

– Changeiny aty(t)– Changeiny atpredictedy(Δt)– dydt= y(t) + h/2 (f(y,t) + f(t+Δt,y+h f(y,t)))

Heun’s method

SameasEuler’smethod• Usesinformationfromtwopoints

– Changeiny aty(t)– Changeiny atpredictedy(Δt)– dydt= y(t) + h/2 (f(y,t) + f(t+Δt,y+h f(y,t)))

Heun’s method• Usesinformationfromtwopoints

– Changeiny aty(t)– Changeiny atpredictedy(Δt)– dydt= y(t) + h/2 (f(y,t) + f(t+Δt,y+h f(y,t)))

Euler’smethod• Usesinformationfromonepoint

– Changeiny aty(t)

ComparingEulerandHeun methods

Euler Heun

Heun’s methodcode

Heun’s method

• Knownasasecondordermethod• IsmorecomputationallyexpensivethanEuler’smethodforthesamestepsize(twofunctionevaluations)

• OutperformsEuler’smethodforthesamestepsize

• Canincorporatenoise(morecomplicatedthough)• Canincorporatedelays(again,morecomplicated)• Stillnotthebestthough...

Runge Kutta• Fourthordersolver• InHeun’s methodthemeanbetweenthestartandendpointsistaken

• InRunge Kutta differentweightsaregiventodifferentpoints

Runge Kutta code

Runge Kutta’s method• Knownasafourthordermethod• IsmorecomputationallyexpensivethanEuler’smethodforthesamestepsize(fourfunctionevaluations)

• OutperformsEuler’smethodforthesamestepsize

• OutperformsHeun’s method• Canincorporatedelays(complicated)• Difficult&complicatedtoincludenoise(ongoingresearch)

Errortolerance

• Canestimatetheerrormadeateachstep(inaniterativemanner)

• Thiserroriscalledtheabsoluteerror• Canthencalculatetherelativeerror(absoluteerror/currentstate)

• WecantelltheMatlab solverswhaterrorswecantolerate:

• options =odeset(’RelTol',1e-3,’AbsTol',1e-6)ode45(@odefunc,timespan,initialConditions,options,…)

Variablestepsolvers

Finalwordsofcaution

• Whensimulatinganewsystemitisalwaysworthcheckingtheresultswithdifferentsolversanderrortolerancesettings

• Especiallywhenexpectyourdynamicstochangeslowly,butwithsuddenfastburstsofactivity

• NumericalsolutionsareALWAYSONLYANAPPROXIMATION