Download Numerical techniques: Deterministic Dynamical Systems

Numerical techniques:
Deterministic Dynamical Systems
Henk Dijkstra
Institute for Marine and Atmospheric research Utrecht,
Department of Physics and Astronomy, Utrecht, The Netherlands
Bifurcation diagram for
dx
= λ − x2
dt
# degrees of freedom:
d=1
€
Attracting fixed points
trajectories
(partial) bifurcation diagram
Steady solutions and their stability
dx
= λ − x2
dt
€
saddle
node
steady
€
stable
attracting
T
repelling
unstable
Bifurcation diagram
_2
0=λ− x
Linear Stability
Other elementary (co-dim 1) bifurcations
dx
= x
dt
2
x
transcritical
dx
3
= λx − x
dt
pitchfork
x
λ
Solution for all values of the parameter
(Reflection) Symmetry in the problem
Hopf bifurcation
x˙ = λx − ωy − x(x 2 + y 2 )
y˙ = λy + ωx − y(x 2 + y2 )
y
# degrees of freedom:
d=2
supercritical
x
λ
2
y
2
y
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
x
steady state
-0.6
-0.4
-0.2
0
0.2
0.4
x
limit cycle
0.6
Autonomous systems: fixed points
Arclength
parametrization
Euler-Newton continuation
Starting Point:
Compute initial tangent:
Solve Extended system:
With Euler guess:
The Newton - Raphson process
G(x) = 0
Scalar function:
G(x)
Newton-Raphson
x
G(x) = 0 ⇒ G′ (xk )∆xk+1 = −G(xk )
y = G0 (xk )x + b
and hence
y = G0 (xk )(x
Then:
G(xk ) = G0 (xk )xk + b
xk ) + G(xk )
0 = G0 (xk )(xk+1
xk ) + G(xk )
Linear stability
Dynamical system:
The linear stability problem of a fixed
point
leads to a generalized eigenvalue problem
Linear stability
=
r
+i
i
; x = x̂r + ix̂i
How to detect bifurcation points?
Transcritical, Saddle-node, Pitchfork:
A single real eigenvalue crosses the imaginary axis
Hopf:
A complex conjugated pair of eigenvalue crosses the
imaginary axis
Periodic orbit near Hopf bifurcation?
(t) = e
rt
(x̂r cos
it
x̂i sin
i t)
AUTO demonstration pp2:
predator-prey system
control parameter p_1