Introduction to Set Oriented Numerics
Roberto Castelli
BCAM
Bilbao, 1st February 2011
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
1 / 29
Set Oriented Numerics
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Main Goal: Study the long term
behavior of complex Dynamical
Systems
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Set Oriented Numerics
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Computation of several short term trajectories instead of single long
term trajectory
Approximation of Global structure
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Invariant Sets : global attractors, Invariant manifolds
Invariant measures, almost invariant sets
Transport operators
Multiobjective optimization (Pareto set)
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
2 / 29
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
3 / 29
Invariant Sets
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
4 / 29
Invariant Sets
Relative Global Attractor
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
5 / 29
Invariant Sets
Relative Global Attractor
xk+1 = f (xk ),
k = 0, 1, 2, . . .
Discrete dynamical system
Consider
f : Rn → Rn diffeomorphism.
Invariant set
A ⊂ Rn is Invariant if f (A) = A
Definition Relative Global attractor
Let Q ⊂ Rn be a compact set. Define global attractor relative to Q by
\
AQ =
f j (Q)
j≥0
Properties
AQ ⊂ Q
f −1 (AQ ) ⊂ AQ but not necessarily f (AQ ) ⊂ AQ .
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
6 / 29
Invariant Sets
Relative Global Attractor
Computation of Relative Global Attractor
The Subdivision-Selection Algorithm
From the initial set B0 = Q, define a sequence B1 , B2 , B3 of finite
collections of compact subsets of Rn , such that
I diam(Bk ) = maxB∈B diam(B) → 0,
k→∞
k
I Bk approaches AQ
I
Inductively Bk is obtained from Bk−1 in two steps
1 Subdivision: define a new collection B̃k such that
[
[
B=
B
B∈B̃k
B∈Bk−1
diam(B̃k ) ≤ θk diam(Bk−1 ),
θk ∈ (0, 1)
2 Selection: define Bk as
Bk = {B ∈ B̃k : ∃B̃ ∈ B̃k such that f −1 (B) ∩ B̃ 6= ∅}
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
7 / 29
Invariant Sets
Relative Global Attractor
Computation of Relative Global Attractor
The Subdivision-Selection Algorithm
From the initial set B0 = Q, define a sequence B1 , B2 , B3 of finite
collections of compact subsets of Rn , such that
I diam(Bk ) = maxB∈B diam(B) → 0,
k→∞
k
I Bk approaches AQ
I
Inductively Bk is obtained from Bk−1 in two steps
1 Subdivision: define a new collection B̃k such that
[
[
B=
B
B∈B̃k
B∈Bk−1
diam(B̃k ) ≤ θk diam(Bk−1 ),
θk ∈ (0, 1)
2 Selection: define Bk as
Bk = {B ∈ B̃k : ∃B̃ ∈ B̃k such that f −1 (B) ∩ B̃ 6= ∅}
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
7 / 29
Invariant Sets
Relative Global Attractor
Computation of Relative Global Attractor
The Subdivision-Selection Algorithm
From the initial set B0 = Q, define a sequence B1 , B2 , B3 of finite
collections of compact subsets of Rn , such that
I diam(Bk ) = maxB∈B diam(B) → 0,
k→∞
k
I Bk approaches AQ
I
Inductively Bk is obtained from Bk−1 in two steps
1 Subdivision: define a new collection B̃k such that
[
[
B=
B
B∈B̃k
B∈Bk−1
diam(B̃k ) ≤ θk diam(Bk−1 ),
θk ∈ (0, 1)
2 Selection: define Bk as
Bk = {B ∈ B̃k : ∃B̃ ∈ B̃k such that f −1 (B) ∩ B̃ 6= ∅}
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
7 / 29
Invariant Sets
Relative Global Attractor
Example: Hénon map
Consider the Hénon map
xk+1 = 1 − ax2k + yk /5
yk+1 = 5bxk
a = 1, b = 0.54
Covering of the Relative Global Attractor of Q = [−2, 2] × [−8, 8] at
different subdivision steps.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
8 / 29
Invariant Sets
Relative Global Attractor
Convergence Result
The abstract subdivision algorithmSconverges to the relative global
attractor AQ . Denote with Qk = B∈Bk B, k > 0
Theorem
Dellnitz and Hohmann (1997)
Let AQ be a global attractor relative to the compact set Q and B0 a finite
collection of closed subsets with Q0 = ∪B∈B0 B = Q Then
lim h(AQ , Qk ) = 0
k→∞
where h(B, C) denotes the Hausdorff distance between two compact sets
B, C ⊂ Rn
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
9 / 29
Invariant Sets
Invariant manifold
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
10 / 29
Invariant Sets
Invariant manifold
The stable manifold
Let p be a hyperbolic fixed point for f . Define Local Stable Manifold of p
Wεs (p) = {x :k f k (x) − p k< ε, for every k ∈ N}
(Local unstable: k
The sets
W u (p) =
(−k))
[
f k (Wεu (p)),
[
W s (p) =
k∈N
f −k (Wεs (p))
k∈N
are the global (un)-stable manifolds of p.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
11 / 29
Invariant Sets
Invariant manifold
The stable manifold
Let p be a hyperbolic fixed point for f . Define Local Stable Manifold of p
Wεs (p) = {x :k f k (x) − p k< ε, for every k ∈ N}
(Local unstable: k
The sets
W u (p) =
(−k))
[
f k (Wεu (p)),
[
W s (p) =
k∈N
f −k (Wεs (p))
k∈N
are the global (un)-stable manifolds of p.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
11 / 29
Invariant Sets
Invariant manifold
Computing the unstable manifold
Let p a fixed point for f and Uε (p) a ε-neighborhood of p.
Remark: if ε is small enough, then Wεu (p) is the global attractor relative
to Uε (p):
\
Wεu (p) =
f k (Uε (p))
k≥0
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
12 / 29
Invariant Sets
Invariant manifold
Computing the unstable manifold
Initialization - Continuation Algorithm
Aim: Compute the unstable manifold of a point p into a (large ) compact
set Q, ( p ∈ Q)
1 Given Q, compute P0 , P1 , . . . Pl nested sequence of fine partitions of
u (p) ∩ C
Q. Select the element C ∈ Pl such that p ∈ C and AC = Wloc
2 Initialization Starting from B0 = {C}, refine the approximation of
u (p) ∩ C by subdivision, yielding B (0) ⊂ P
Wloc
l+k
k
(j−1)
3 Continuation From Bk
compute
(j)
(j−1)
Bk = {B ∈ Pl+k : f (B 0 ) ∩ B 6= ∅, for some B 0 ∈ Bk
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
}
13 / 29
Invariant Sets
Invariant manifold
Computing the unstable manifold
Initialization - Continuation Algorithm
Aim: Compute the unstable manifold of a point p into a (large ) compact
set Q, ( p ∈ Q)
1 Given Q, compute P0 , P1 , . . . Pl nested sequence of fine partitions of
u (p) ∩ C
Q. Select the element C ∈ Pl such that p ∈ C and AC = Wloc
2 Initialization Starting from B0 = {C}, refine the approximation of
u (p) ∩ C by subdivision, yielding B (0) ⊂ P
Wloc
l+k
k
(j−1)
3 Continuation From Bk
compute
(j)
(j−1)
Bk = {B ∈ Pl+k : f (B 0 ) ∩ B 6= ∅, for some B 0 ∈ Bk
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
}
13 / 29
Invariant Sets
Invariant manifold
Computing the unstable manifold
Initialization - Continuation Algorithm
Aim: Compute the unstable manifold of a point p into a (large ) compact
set Q, ( p ∈ Q)
1 Given Q, compute P0 , P1 , . . . Pl nested sequence of fine partitions of
u (p) ∩ C
Q. Select the element C ∈ Pl such that p ∈ C and AC = Wloc
2 Initialization Starting from B0 = {C}, refine the approximation of
u (p) ∩ C by subdivision, yielding B (0) ⊂ P
Wloc
l+k
k
(j−1)
3 Continuation From Bk
compute
(j)
(j−1)
Bk = {B ∈ Pl+k : f (B 0 ) ∩ B 6= ∅, for some B 0 ∈ Bk
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
}
13 / 29
Invariant Sets
Invariant manifold
Computing the unstable manifold
Initialization - Continuation Algorithm
Aim: Compute the unstable manifold of a point p into a (large ) compact
set Q, ( p ∈ Q)
1 Given Q, compute P0 , P1 , . . . Pl nested sequence of fine partitions of
u (p) ∩ C
Q. Select the element C ∈ Pl such that p ∈ C and AC = Wloc
2 Initialization Starting from B0 = {C}, refine the approximation of
u (p) ∩ C by subdivision, yielding B (0) ⊂ P
Wloc
l+k
k
(j−1)
3 Continuation From Bk
compute
(j)
(j−1)
Bk = {B ∈ Pl+k : f (B 0 ) ∩ B 6= ∅, for some B 0 ∈ Bk
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
}
13 / 29
Invariant Sets
Lorenz system

 ẋ = σ(y − x)
ẏ = ρx − y − xz

ż = −βz + xy
Invariant manifold
(0, 0, 0) is fixed point
σ = 10, ρ = 28, β = 8/3
Covering of the two-dimensional stable manifold of the origin
Left: l = 9, k = 6, j = 4 initial box Q = [−70, 70] × [−70, 70] × [−80, 80],
Right:l = 21, k = 0, j = 10, initial box Q = [−120, 120] × [−120, 120] × [−160, 160]
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
14 / 29
Invariant Sets
Invariant manifold
Stable manifold Lorenz system
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
15 / 29
Invariant Sets
Invariant manifold

α = 18, β = 33
 ẋ = α(y − m0 x − 13 m1 x3 )
ẏ = x − y + z
m0 = −0.2, m1 = 0.01
Chua Circuit ,

ż = −βy
q
q
m0
0
,
0,
∓
−3
Fixed points P± = ± −3 m
m1
m1
(a)
(b)
Figure: (a) Approximation of the relative global attractor for the Chua system (b)
Union of the coverings of the unstable manifold of the fixed points P± .
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
16 / 29
GAIO Implementation
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
17 / 29
GAIO Implementation
Implementation – GAIO package
GAIO: Global Analysis Invariant Object
Boxes : Generalized rectangle B(c, r) ⊂ Rn
B(c, r) = {y ∈ Rn : |yi − ci | ≤ ri , i : 1 . . . n}
identified by a centre and vector of radii, c, r ∈ Rn .
Subdivision : by bisection along one of the coordinate direction: from
B(c, r) to B1 (c+ , r1 ), B2 (c− , r1 )
ri
i 6= j
ci
i 6= j
±
1
ri =
, ci =
ri /2 i = j
ci ± ri /2 i = j
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
18 / 29
GAIO Implementation
Implementation – GAIO package
Storage of boxes: Binary tree
In Fig: representation of three subdivision steps in three dimension.
The spatial coordinates of a collection are completely determined by the
tree structure of the boxes and the initial rectangle B0 .
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
19 / 29
GAIO Implementation
Box Map and Box Intersection
The Subdivision and the Continuation algorithm are based on the
theoretical selection step
Bk = {B 0 ∈ B̃k : f (B) ∩ B 0 6= ∅ for some B ∈ B̃k }
How to define the image of a box?
Box image FB (B) = {B 0 ∈ B : f (B) ∩ B 0 6= ∅}
How to compute f (B) and then FB (B) ?
The method is to choose a finite set T ⊂ B of test points and define
F̃B (B) = {B 0 ∈ B : f (T ) ∩ B 0 6= ∅}
Clearly
F̃B (B) ⊂ FB (B)
but possibly
F̃B (B) 6= FB (B)
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
20 / 29
GAIO Implementation
Box Map and Box Intersection
The Subdivision and the Continuation algorithm are based on the
theoretical selection step
Bk = {B 0 ∈ B̃k : f (B) ∩ B 0 6= ∅ for some B ∈ B̃k }
How to define the image of a box?
Box image FB (B) = {B 0 ∈ B : f (B) ∩ B 0 6= ∅}
How to compute f (B) and then FB (B) ?
The method is to choose a finite set T ⊂ B of test points and define
F̃B (B) = {B 0 ∈ B : f (T ) ∩ B 0 6= ∅}
Clearly
F̃B (B) ⊂ FB (B)
but possibly
F̃B (B) 6= FB (B)
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
20 / 29
GAIO Implementation
Box Map and Box Intersection
The Subdivision and the Continuation algorithm are based on the
theoretical selection step
Bk = {B 0 ∈ B̃k : f (B) ∩ B 0 6= ∅ for some B ∈ B̃k }
How to define the image of a box?
Box image FB (B) = {B 0 ∈ B : f (B) ∩ B 0 6= ∅}
How to compute f (B) and then FB (B) ?
The method is to choose a finite set T ⊂ B of test points and define
F̃B (B) = {B 0 ∈ B : f (T ) ∩ B 0 6= ∅}
Clearly
F̃B (B) ⊂ FB (B)
but possibly
F̃B (B) 6= FB (B)
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
20 / 29
GAIO Implementation
Choice of test-points
In low dimensional phase space (d ≤ 3)
I
N points on the edges of the boxes + the center
I
on uniform grid within the box
In higher dimension
I
randomly distributed
Remark: Rigorous choice of test point in such a way that no boxes are lost
due to the discretization could be done if the Lipschitz constant of the
map f is known.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
21 / 29
GAIO Implementation
Choice of test-points
In low dimensional phase space (d ≤ 3)
I
N points on the edges of the boxes + the center
I
on uniform grid within the box
In higher dimension
I
randomly distributed
Remark: Rigorous choice of test point in such a way that no boxes are lost
due to the discretization could be done if the Lipschitz constant of the
map f is known.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
21 / 29
Application: detecting connecting orbit
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
22 / 29
Application: detecting connecting orbit
The Hat Algorithm
Goal: Detect (homo-)heteroclinic orbit connecting two hyperbolic fixed
points.
Consider
ẋ = f (x, λ), λ ∈ Λ
and xλ , yλ two one-parameter families of hyperbolic fixed points.
If for λ = λ̄ there exist a heteroclinic connection γ, then
γ ∈ W u (xλ̄ ) ∩ W s (yλ̄ )
.Discretization: Analysis of
(j)
(j)
Ik (λ) = Uk (xλ ) ∩ Sk (yλ )
k subdivision steps
j iteration steps
when λ ∈ Λ̃ and k vary.
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
23 / 29
Application: detecting connecting orbit
The Hat Algorithm
(j)
(j)
I
Fix λ ∈ Λ̃ and compute Uk (xλ ) and Sk (yλ ) for k small. ( rough
covering)
I
Refine the covering up to the maximum k = m(λ) s.t Ik (λ) 6= ∅
I
Change λ and repeat the procedure
I
Look at the maximum of m(λ).
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
24 / 29
Application: detecting connecting orbit
The Hat Algorithm: example
Heteroclinic connection between two Lyapunov orbits in the CRTBP.
xλ , yλ Lyapunov orbits, λ Jacobi constant
(a) Graph of the function m(λ)
(b) Heteroclinic connection for J = 3.16988
Figure: The Hat Algorithm applied to detect heteroclinic connections between
two Lyapunov orbits in the Earth-Moon CRTBP
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
25 / 29
Bibliography
OUTLINE
Invariant Sets
Relative Global Attractor
Invariant manifold
GAIO Implementation
Application: detecting connecting orbit
Bibliography
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
26 / 29
Bibliography
REFERENCES-1
Conley, C.: Isolated invariant sets and Morse index. American Mathematical
Society, (1978).
Hsu, H.: Global analysis by cell mapping. Int. J. Bif. Chaos 2, pp. 727-771, (1992)
Dellnitz, M., Hohmann, A., The computation of unstable manifolds using
subdivision and continuation. In: Nonlinear Dynamical Systems and Chaos, pp
449-459. Birkhäuser, PNLDE 19, (1996).
Dellnitz, M., Junge, O.: On the approximation of complicated dynamical behavior.
SIAM Journal on Numerical Analysis 36(2), pp. 491-515, (1999).
Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO - set oriented
numerical methods for dynamical systems. In: Ergodic theory, analysis, and efficient
simulation of dynamical systems, pp. 145-174, 805-807. Springer, Berlin (2001).
Dellnitz, M., Junge, O.: Set oriented numerical methods for dynamical systems. In:
Handbook of dynamical systems, Vol 2. pp 221-264. North-Holland, Amsterdam
(2002).
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
27 / 29
Bibliography
REFERENCES-2
APPLICATIONS
Dellnitz, M., Schtze, O., Sertl, S.: Finding Zeros by Multilevel Subdivision
Techniques, IMA Journal of Numerical Analysis, 22(2), pp. 167-185, (2002)
Schütze, O., Mostaghim, S., Dellnitz, M., Teich, J.: Covering Pareto Sets by
Multilevel Evolutionary Subdivision Techniques Proceedings of EMO 2003, Faro,
Portugal
Dellnitz, M., Junge, O., Post, M., Thiere, B.: On Target for Venus - Set Oriented
Computation of Energy Efficient Low Thrust Trajectories. Celestial Mechanics and
Dynamical Astronomy, Special Issue Celmec IV, 95(1-4), pp.357-370, Springer,
(2006)
Dellnitz, M., Ober-Blöbaum, S., Post, M., Schütze, O., Thiere, B.: A
multi-objective approach to the design of low thrust space trajectories using optimal
control. Celestial Mechanics and Dynamical Astronomy, 105(1), pp. 33-59 , (2009).
Baier, R., Dellnitz, M., Hessel-von Molo, M., Kevrekidis, I.G., Sertl, S.: The
Computation of Invariant Sets via Newton’s Method Submitted to SINUM
Matsumoto, T., Chua, L.O., Korumo, M.: The double scroll. IEEE Transactions on
Circuits and systems, 32(8), (1985)
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
28 / 29
Bibliography
Thank you
⇒ Relative Global attractor
⇒ GAIO Implementation
⇒ Invariant Manifolds
⇒ Connecting Orbits
1st Febraury 2011
Introduction to set Oriented Numerics
Roberto Castelli
29 / 29