Henk Dijkstra

Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht, The Netherlands

Numerical techniques: Deterministic Dynamical Systems

Transition behavior from (proxy) data: Oxygen Isotope Ratio (ice cores)

time (kyr)

Transition behavior from (model) data: FAMOUS model

Time series of the MOC (in Sv, 1 Sv = 106 m3s-1) at 26N and a depth 1000m in the Atlantic

Control Simulation

Hosing Simulation

Equilibrium Simulations

Latitude

Depth

/m

−30 −20 −10 0 10 20 30 40 50 60

−5000

−4500

−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

−10

−5

0

5

10

15

Elementary bifurcations (can be obtained with variation of one parameter)

Saddle-node bifurcation (limit point, turning point)

Transcritical bifurcation

Pitchfork bifurcation

Hopf bifurcation

Bifurcation diagram for

€

dxdt

= λ − x 2

d = 1# degrees of freedom:

Attracting fixed points

trajectories (partial) bifurcation diagram

Determine all fixed points of the dynamical system:

Exercise 1: Saddle node

and next their linear stability€

dxdt

= λ − x 2

Steady solutions and their stability

Bifurcation diagram

stable

unstable

saddle node

Linear Stability

T

€

0 = λ − x_ 2

€

dxdt

= λ − x 2 steady

repelling

attracting

Other elementary (co-dim 1) bifurcations

dxdt

= λx − x3

λ

x

dx

dt= �x� x2

transcritical pitchfork

Solution for all values of the parameter (Reflection) Symmetry in the problem

Hopf bifurcation

-0.5

0

0.5

1

1.5

2

-0.6 -0.4 -0.2 0 0.2 0.4

y

x

λ

y

x

˙ x = λx − ωy − x(x2 + y2 )˙ y = λy +ωx − y(x 2 + y2 )

limit cyclesteady state

-0.5

0

0.5

1

1.5

2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

y

x

d = 2# degrees of freedom:

supercritical

example

Numerical Bifurcation Theory

System of PDEs:

Operators containing parameters

Discretization (N)

x: state vector

Dynamical system:

Exercise 2: Burgers equation

Use central differences to derive the corresponding dynamical system. What is the

state vector?

@u

@t

+ u

@u

@x

= ⌫

@

2u

@x

2

u(0, t) = 1;@u

@x

(1, t) = 0

Demonstration MatCont

↵ = 360 ; µ2 = 6.25

Autonomous systems: fixed points

Arclength parametrization

Euler-Newton continuation

Compute initial tangent:

Solve Extended system:

With Euler guess:

Starting Point:

The initial tangentDifferentiate: to s:

If is not a bifurcation point, then this matrix has rank N

The Newton - Raphson process

Scalar function: G(x) = 0

Newton-Raphson

x

G(x)

G(x) = 0 ⇒ G′(xk)∆x

k+1 = −G(xk)

y = G

0(xk)x+ b G(xk) = G

0(xk)xk + b

and hence

y = G

0(xk)(x� x

k) +G(xk)

0 = G

0(xk)(xk+1 � x

k) +G(xk)Then:

Exercise 3: The Newton - Raphson process

Formulate the NR process for the extended system:

Solve Extended system:

Detection of bifurcation points

1. Direct indicators f(s)

2. Solve linear stability problem

Use secant iteration:

det(�x

(s)) = 0

�̇ = 0

1. Orthogonal to tangent 2. Use imperfections

Branch switching

Determining isolated branches

(d) Residue continuation

use homotopy parameter

Linear stability

Dynamical system:

Show that the linear stability problem of a fixed point

Exercise 4:

leads to a generalized eigenvalue problem

Why is B often singular?

Numerical linear algebra

Solution methods: !

1. QZ 2. Jacobi-Davidson QZ

3. Arnoldi 4. Simultaneous Iteration

Cayley Transform

C = (A� �B)�1(A� µB)

� = �µ = 100

Cx = �x

� =� � µ

� � �

Linear stability

�(t) = e

�rt(x̂r cos�it� x̂i sin�it)

Transcritical, Saddle-node, Pitchfork: A single real eigenvalue crosses the imaginary axis

Hopf: A complex conjugated pair of eigenvalue crosses the

imaginary axis

How to detect bifurcation points?

Periodic orbit near Hopf bifurcation?

� = �r + i�i ; x = x̂r + ix̂i

Exercise 5

Formulate a generalized eigenvalue problem as a fixed point problem to trace branches of eigenvalues of a

linear stability problem. !

Computation of Periodic Orbits (autonomous systems)

1. Boundary value problem

2. Fixed points of Poincare map Poincare section

Stability of Periodic Orbits: I

Fixed point (periodic orbit) Linear stability

1

0

Quasi-periodic behavior

Additional periodic orbitsPeriod doubling

AB

C

Example:

Stability of Periodic Orbits: II

A Cyclic Fold Cyclic Pitchfork BPeriod Doubling

CNaimark-Sacker

(Torus)

Useful tools□ auto, http://indy.cs.concordia.ca/auto/ !

□ xppaut www.math.pitt.edu/~bard/xpp/xpp.html ■ Solves ODEs,DDEs,also AUTO built in

!□ winpp

■ Windows version of xppaut but used LOCBIF instead of AUTO !

□ matcont allserv.rug.acbe/~ajdhooge/research.html ■ Continuation software in Matlab July 9th 2004 (lastest version) !!

□ DDE-BIFTOOL ■ Matlab package for numerical bifurcation analysis of delay equations