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Henk Dijkstra Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht, The Netherlands Numerical techniques: Deterministic Dynamical Systems
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Page 1: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Henk Dijkstra

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

Numerical techniques: Deterministic Dynamical Systems

Page 2: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

time (kyr)

Page 3: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 4: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Saddle-node bifurcation (limit point, turning point)

Transcritical bifurcation

Pitchfork bifurcation

Hopf bifurcation

Page 5: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Bifurcation diagram for

dxdt

= λ − x 2

d = 1# degrees of freedom:

Attracting fixed points

trajectories (partial) bifurcation diagram

Page 6: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Determine all fixed points of the dynamical system:

Exercise 1: Saddle node

and next their linear stability€

dxdt

= λ − x 2

Page 7: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 8: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 9: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 10: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Numerical Bifurcation Theory

System of PDEs:

Operators containing parameters

Discretization (N)

x: state vector

Dynamical system:

Page 11: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 12: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Demonstration MatCont

↵ = 360 ; µ2 = 6.25

Page 13: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Autonomous systems: fixed points

Arclength parametrization

Page 14: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Euler-Newton continuation

Compute initial tangent:

Solve Extended system:

With Euler guess:

Starting Point:

Page 15: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

The initial tangentDifferentiate: to s:

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

Page 16: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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:

Page 17: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Exercise 3: The Newton - Raphson process

Formulate the NR process for the extended system:

Solve Extended system:

Page 18: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Detection of bifurcation points

1. Direct indicators f(s)

2. Solve linear stability problem

Use secant iteration:

det(�x

(s)) = 0

�̇ = 0

Page 19: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

1. Orthogonal to tangent 2. Use imperfections

Branch switching

Page 20: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Determining isolated branches

(d) Residue continuation

use homotopy parameter

Page 21: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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?

Page 22: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Numerical linear algebra

Solution methods: !

1. QZ 2. Jacobi-Davidson QZ

3. Arnoldi 4. Simultaneous Iteration

Page 23: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Cayley Transform

C = (A� �B)�1(A� µB)

� = �µ = 100

Cx = �x

� =� � µ

� � �

Page 24: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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

Page 25: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Exercise 5

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

linear stability problem. !

Page 26: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Computation of Periodic Orbits (autonomous systems)

1. Boundary value problem

2. Fixed points of Poincare map Poincare section

Page 27: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Stability of Periodic Orbits: I

Fixed point (periodic orbit) Linear stability

1

0

Quasi-periodic behavior

Additional periodic orbitsPeriod doubling

AB

C

Example:

Page 28: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

Stability of Periodic Orbits: II

A Cyclic Fold Cyclic Pitchfork BPeriod Doubling

CNaimark-Sacker

(Torus)

Page 29: Numerical techniques: Deterministic Dynamical  · PDF fileNumerical techniques: Deterministic Dynamical Systems. ... auto,   ... Continuation software in Matlab July 9th 2004

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


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