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