Xppaut Roscoff

7/27/2019 Xppaut Roscoff

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Introduction to XPPAUTMathieu [email protected]

Session A3: Mathematics for oscillations

24-25 July

mailto:[email protected]

mailto:[email protected]



Introduction to XPPAUT

‘XPP-’ ‘-AUT’...



Part I

(short) Introduction to

! (parametrised families of) ODEs! their Bifurcations

! direct simulation of ODEs

! numerical continuation study



Part I







Part I







Part I







Part I







• ODEs: variables, parameters, set of 1st order equations, ...

• Equilibria: set the Right-Hand-Side (RHS) to 0

• Stability: linearise and compute eig. of Jac. matrix. Different

topological type: (un)stable focus, (un)stable node, saddle ...

• Bifurcations: change a parameter and the type of the solution

changes (from eq. to periodic, from periodic to quasi-

per., ...) ... structural stability is lost at bifurcation point.

• Example of (local) bifurcations:

Theory

• eq. 2 eq. : fold (LP), pitchfork (BP), transcritical (BP)

• eq. 2 per. : Hopf (HB)

• per. 2 per. : fold of periodic orbits, period-doubling, torus











Theory














Theory














Theory














Theory






Starting with 1D systems

x (state) variables

parameters

x = f (x,α)

α




x (state) variables

parameters

x = f (x,α)

αEquilibria: find all such that f( , )=0x x (α is fixed!)

α




x (state) variables

parameters

Stability: sign of f ’( )x

x = f (x,α)


α




x (state) variables

parameters

Stability: sign of f ’( )x

f’( )<0: is stable

f’( )>0: is unstable

x x

x x

x = f (x,α)


α



1D systems ... let’s vary !




•Nothing changes in terms of ‘behaviour’:

! structural stability




•Nothing changes in terms of ‘behaviour’:

! structural stability

• Sudden change of stability when one varies !:

! bifurcation





Example 1: transcritical bifurcation

" Equilibria : x = 0, x = α

x = x(α− x)ODE :





" Stability : f 0(x) = α− 2x

x = x(α− x)ODE :






f

0

(0) = α

, f

0

(α

) =−α

x = x(α− x)ODE :






f

0

(0) = α

, f

0

(α

) =−α

SO 1) it depends on !

x = x(α− x)ODE :






f

0

(0) = α

, f

0

(α

) =−α

SO 1) it depends on !

2) always opposite stability BUT at 0!

x = x(α− x)ODE :



Bifurcation diagram

•

•

•

•

•





Bifurcation diagram

•

•

•

•

•

! = 0.4 > 0 x=! is stable & x=0 is unstable



Bifurcation diagram

•

•

•

•

•

! = 0 one eq. point, half stable, half unstable



Bifurcation diagram

•

•

•

•

•

! = 0 one eq. point, half stable, half unstable

! Transcritical bifurcation point



Example 2: fold bifurcation

ODE : x = α− x2






ODE :

" Equilibria : x = ±√

α when α > 0

x = α− x2




ODE :


α when α > 0

x = 0 when α

= 0

x = α− x2




ODE :


α when α > 0

x = 0 when α = 0

when α < 0no equilibrium

x = α− x2






ODE :

" Equilibria :

SO 1) it depends on !>0

x = ±√

α when α > 0

x = 0 when α = 0


x = α− x2

" Stability : f 0(x) = −2x

f ld if i




ODE :

" Equilibria :

SO 1) it depends on !>0

2) always opposite stability BUT at 0!

x = ±√

α when α > 0

x = 0 when α = 0


x = α−x2

" Stability : f 0(x) = −2x

Bif i di



Bifurcation diagram

•

•

•

x = α− x2

Bif i di



Bifurcation diagram

•

•

! = -0.5 < 0 no equilibrium point!

•

x = α− x2

Bif ti di



Bifurcation diagram

•

•

! = 0.3 > 0 x=

"! is stable & x=-

"! is unstable

•

x = α− x2











• IF there is an equilibrium point, it is either stable or unstable

• stable equilibrium: the system converges towards it as t#+#

• unstable equilibrium: the system converges towards it as t#-#

Summary: 1D systems ...













HoweverThe system is still nonlinear, so an exact

solution is unlikely to exist!

Same for equilibria!







HoweverThe system is still nonlinear, so an exact

solution is unlikely to exist!

Same for equilibria!

$ Need to be able to simulate the system ...

N i l i l i f ODE



Numerical simulation of ODEs

x = f (x,α) (1)

N i l i l i f ODE




x = f (x,α) (1)

Idea $ discretise time :

$ ‘march’ in time :

Compute numerically approximate sol. to (1)?

x(t0) = x0 x(t1) = ...?

t0 t1 = t0 + h, h = “dt”

N i l i l ti f ODE




x = f (x,α) (1)

Idea $ discretise time :

$ ‘march’ in time :

Compute numerically approximate sol. to (1)?

x(t0) = x0 x(t1) = ...?

t0 t1 = t0 + h, h = “dt”

Example 1: Euler scheme

$ Error:

uses the slope of f(x(t0)) at x(t0)

to approximate x(t1)

(Taylor)

x(t1) ≈ x(t0) + f (x(t0),α)h

O(h2)

N i l i l ti f ODE




x = f (x,α) (1)

Example 1: Euler scheme

$ Error:

uses the slope of f(x(t0)) at x(t0)to approximate x(t1)

(Taylor)

x(t1) ≈ x(t0) + f (x(t0),α)h

O(h2)

Example 2: 2nd order Runge-Kutta scheme

$ Error: (Taylor)O(h3)

takes a half Euler step to the point (t0+0.5h, x(t0)+0.5k )

[see figure next]1





2D systems (nD n>2)



2D systems ... (nD, n>2)

vector of (state) variables

vector of parameters

x

α

x = f (x,α)

Equilibria: find all such that f ( , )=0x∗

α∗

x∗ (α∗ is fixed!)



2D systems (nD n>2)



2D systems ... (nD, n>2)

vector of (state) variables

vector of parameters

x

α

x = f (x,α)

Stability: Jacobian matrix J = ∂ f i

∂ xj∗

! stability depends on the eigenvalues of J

Equilibria: find all such that f ( , )=0x∗

α∗

x∗ (α∗ is fixed!)

2D systems : nullclines




x = f (x,y,α)y = g(x,y,α)






Equilibria:(fix !=! )

0

f (x∗

, y∗

,α0) = 0 & g(x∗

, y∗

,α0) = 0






Equilibria:(fix !=! )

0

f (x∗

, y∗

,α0) = 0 & g(x∗

, y∗

,α0) = 0

! Geometrically:

{f (x,y,α0) = 0}∩{g(x,y,α0) = 0}






y

(example) FitzHugh-Nagumo

v = v−

v3

/3−

w + I y = ε(v + a − bw)

I = 1.5ε = 0.08

a = 0.7

b = 0.8

! f is cubic

! g is linear




y


v = v−

v3

/3−

w + I y = ε(v + a − bw)

I = 1.5ε = 0.08

a = 0.7

b = 0.8

! f is cubic

! g is linear






y


v = v−

v3

/3−

w + I y = ε(v + a − bw)

ε = 0.08

a = 0.7

I = 0.4

b = 1.8

! f is cubic

! g is linear




y


v = v−

v3

/3−

w + I y = ε(v + a − bw)

ε = 0.08

a = 0.7

I = 0.4

b = 1.8◦◦

◦

3 eq.

! f is cubic

! g is linear

Linear a roximation: 2D case




sourcesink nodal sink nodal source hyperbolic centre

•

•

C •

•

•• •• ••

•

•Re

Im






•

•

C •

•

•• •• ••

•

•

- nonlinear dynamics -

Re

Im






•

•

C •

•

•• •• ••

•

•

focus


Re

Im






•

•

C •

•

•• •• ••

•

•

focus node


Re

Im










•

•

C •

•

•• •• ••

•

•

focus node saddle


(un)stable eigenspaces

! nonlinear equivalent: (un)stable manifolds

W s,u

Re

Im



Bifurcations



Bifurcations

•Transcritical, fold, ... : similar to 1D systems

Bifurcations



Bifurcations

•Transcritical, fold, ... : similar to 1D systems1D : Bifurcation when f’(x) goes through 02D : Bifurcation when an eigenvalue goes through 0

Bifurcations



Bifurcations


• Truly 2D effect : oscillatory dynamics

(complex eigenvalues)

1D : Bifurcation when f’(x) goes through 02D : Bifurcation when an eigenvalue goes through 0

Bifurcations



Bifurcations


• Truly 2D effect : oscillatory dynamics

(complex eigenvalues)

1 - ‘Damped’ nonlin. oscillations: focus eq.

2 - ‘Sustained’ nonlin. oscillations: limit cycle

1D : Bifurcation when f’(x) goes through 02D : Bifurcation when an eigenvalue goes through 0



Hopf bifurcation



stable focus

Hopf bifurcation

• 2D system : possibility for complex eig.

•

•

C

Re

Im

stable focus

•



Hopf bifurcation



unstable focus

Hopf bifurcation


•

•

C

Re

Im

! focus eq. destabilises when eigenvalues cross Im

stable focus

• •

unstable focus

Hopf bifurcation



unstable focus

Hopf bifurcation


•

•

C

Re

Im

! focus eq. destabilises when eigenvalues cross Im

stable focus

• •

unstable focus

a stable

limit cycleis created!

Hopf : bifurcation diagram



α

Measure of

the solution

αH




α

Measure of

the solution

αH

Before : stable equilibrium




α

Measure of

the solution

αH

After : unstable equilibrium




α

Measure of

the solution

αH

After : stable limit cycle




α

Measure of

the solution

αH


Represented

in this diagram bymax. and min. values




α

Measure of

the solution

αH


Represented

in this diagram bymax. and min. values

From stationary to periodic when varying !!


3D i li i



- 3D visualisation -

α

x

y

Transient dynamicsLong-term dynamics (attractors)

Numerical continuation : idea



• Goal is to compute families (branches) of solutions of

nonlinear equations of the form:

F(x) = 0, F : Rn+1 → Rn

! under-determined system (one more unknowns than equations)

! away from singularities, solution set = 1-dim. manifold

embedded in (n+1)-dim. space



Parameter continuation



• Suppose we have one solution to the problem and wish to vary

one component to find a new solution ...

• In the I.F.T. holds at this point, then locally there is a branch

of solutions parameterised by $: (x($), $).

F(x0) = 0, x0 = (x0,λ0) ∈ Rn × R

• Small change in the parameter $ new point that is not a

solution of the problem but close to one!

x#

1 = (x0,λ0 + δ s),

F(x#1 ) 6= 0,

F(x#1 ) 1

Parameter continuation



• SO initial guess for the new solution is

• New solution computed to a desired accuracy by using Newton’s method

on the augmented problem

• Note: additional equation is to ensure unique solution for Newton’s

method

x#1 = (x0,λ0 + δ s)

F(x) = 0,

λ−

(λ0 + δλ) = 0



Problem at a fold !!



Keller’s pseudo-arclength continuation



• Problem at a fold: the “parameter” chosen to do the

continuation cannot parameterise the solution curve

• Solution: parameterise by something that do not have this problem

!Arclength s along the curve !

• The problem to be solved becomes F(x) = 0,

(λ−

λ0)λ0 + (x−

x0)x0−

δ s = 0

Arclength measured along the tangent space !

Periodic orbit continuation



• We look for periodic solutions of the problem :

! We seek for solutions of period 1 i.e. such that :

x = F(x(t),λ)

x0 = T F(x(t),λ)

x(1) = x(0)

! The true period T is now an additional parameter

• Note: the above equations do not uniquely specify x and T ...

... translation invariance !

! Necessity of a Phase condition

• Example: Poincaré orthogonality condition

• In practice: Integral phase condition

Z 1

0

xk(t)∗x0k−1

dt = 0

(xk(0)− xk−1(0))∗x0k−1(0)) = 0

• Fix the interval of periodicity by the transformation t # t/T

such that :




• We look for periodic solutions of the problem :

! We seek for solutions of period 1 i.e. such that :

x = F(x(t),λ)

x0 = T F(x(t),λ)

x(1) = x(0)

! The true period T is now an additional parameter

• Note: the above equations do not uniquely specify x and T ...

... translation invariance !

! Necessity of a Phase condition

• Example: Poincaré orthogonality condition

• In practice: Integral phase condition

Z 1

0

xk(t)∗x0k−1

dt = 0

(xk(0)− xk−1(0))∗x0k−1(0)) = 0

• Fix the interval of periodicity by the transformation t # t/T

such that :




• ... we then solve a large G(X) = 0 augmented by the

arclength-continuation equation:

• Discretisation of the periodic orbits: using orthogonalcollocation (piecewise polynomials on mesh intervals)

! Solve exactly at mesh points (boundaries of mesh intervals)

Z 1

0

(xk(t)− xk−1(t))∗xk−1dt + (T k − T k−1) T k−1

+(λk − λk−1)λk−1 − δ s = 0

! Inside mesh intervals: well-chosen collocation points (good

convergence properties)

• Well-posedness: n+1 unknowns (n components of x and period T)

for n+1 conditions ( periodicity + phase)

! varying 1 parameter will give a 1-parameter family of per. orbits



Part II: XPPAUT

- main features & capabilities -

Equation file: FitzHugh-Nagumo






Differential Equations




Parameters: initialisation




State variables: initial conditions




Numerical accuracy

XPP-part



• one ‘simple’ example: the FitzHugh-Nagumo equation ... let’s

learn how to use XPP on this example!

• Equation file: equations, parameters, initialisation, numericalaccuracy, ...

• launch it, run it ...

• accuracy: choose the scheme, set the time step, theintegration time, the tolerances, ...

• change: initial conditions, parameter(s) (sweep)

• equilibria: can integrate until transient has passed & get

approx. of equilibria / alternatively , use the in-built routine tofind eq. & assess stability & approximation of (un)stable mflds

• par. dependence: move par. & find ass. eq. manually ... find

interesting transitions like eq 2 per i.e. one has gone through

a Hopf bifurcation! ... Can do that more systematically!!

XPP-part







•accuracy: choose the scheme, set the time step, the

integration time, the tolerances, ...







XPP-part

















XPP-part















XPP-part





















Let’s launch it and try!



We’re now going to learn XPP

with the FitzHugh-Nagumoequation ...

AUT-part



• initialise: grab an equilibrium point

• launch AUTO: start the continuator ...

• set up: accuracy, problem type, cont. par., ...

• compute: 1-par. family of equilibria (branch)

• detect: possible bif. like fold (LP) or Hopf (HB) ...

• branch-switch at HB: extend the problem to compute the

emanating family of periodic orbits ...











FHN: 1-par. continuation results

b=1 8



! !"#$ !"$ !"%$ &−#

−&

!

&

#

b=1.8

I

v

•

•HB

HB

stable

unstable

stable

•

•

LP

LP

∗

∗

HO

HO







Part III: XPPAUT

- practice it yourself -



Hindmarsh-Rose equation file



D.E.





Hindmarsh-Rose equation file



D.E.

Par.

I.C.

Num.













Simulating the system for different values of I, youshould obtain time series that look like these ones



You should obtain a bifurcation

diagram that looks like this one



diagram that looks like this one

! " #$ %# %&

−%

−!'"(

!'(

#'"(

)

I

x••

•

•

HB

HB HB

HB





References

XPPAUT

http://www.math.pitt.edu/~bard/xpp/xpp.html




% Bard Ermentrout (University of Pittsburgh, USA):

XPPAUT

















References ...

ODEs & Bifurcations



ODEs & Bifurcations

(‘Tutorials’ & ‘Software’ sections)

http://www.bio.vu.nl/thb/research/project/globif/

(MathDoctorMitchell’s Channel)

References ...

ODEs & Bifurcations

http://www.youtube.com/user/MathDoctorMitchell










http://www.elmer.unibas.ch/pendulum/lroom.htm










http://www.dynamicalsystems.org/



http://www.bifurcation.de/


http://www.scholarpedia.org/article/Main_Page
























































































































































































































































































































































ODEs & Bifurcations

(‘Tutorials’ & ‘Software’ sections)


(MathDoctorMitchell’s Channel)


















































































































































































































































































































































































Beyond XPPAUT!



Command-line AUTO: the ‘proper way’ to do continuation!

Beyond XPPAUT!




! Allows to continue: equilibria, limit cyclesAND boundary-value problems

Beyond XPPAUT!





Other continuation packages: MatCont Matlab based !CoCo

Beyond XPPAUT!

http://cocotools.sourceforge.net/


http://matcont.sourceforge.net/






Other continuation packages: MatCont Matlab based !

!! mostly for ODEs !!

CoCo

Beyond XPPAUT!










Beyond ODEs:


CoCo

Beyond XPPAUT!










Beyond ODEs:

%codes exist for PDEs ( ) and DDEs ( )


CoCo

LOCA Knut

http://gitorious.org/knut

http://gitorious.org/knut

http://www.cs.sandia.gov/LOCA/

http://www.cs.sandia.gov/LOCA/







References

AUTO - 07p



p



References

AUTO - 07p

http://indy.cs.concordia.ca/auto/



% Eusebius Doedel (Concordia University, Montreal):

p

References

AUTO - 07p







p

References

AUTO - 07p

http://www.springerlink.com/content/978-1-4020-6355-8/%23section=334012&page=1








p

http://www2 imperial ac uk/~jswlamb/LDSG/grad0506/auto course htm

http://www2.imperial.ac.uk/~jswlamb/LDSG/grad0506/auto_course.htm









Thanks for your attention!



Thanks for your attention!

... questions?

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