Simplicial Multivalued Maps and the Witness Complex Zachary Alexander + , Elizabeth Bradley ∗ , James D. Meiss ∗∗ , and Nikki Sanderson o University of Colorado, Boulder *Professor, Computer Science, **Professor, Applied Mathematics, o graduate student, Mathematics + data scientist, Microsoft Erwin Schr¨ odinger International Institute for Mathematical Physics February 23, 2015 1 / 23
Overview • What we did: Develop a new computational topology tool. • Why it’s useful: Computer-based proofs of existence of regular and chaotic invariant sets of dynamical systems. • How we did it: Discretization of state space, novel outer approximation of dynamics, computation of discrete Conley index. 2 / 23
Set-up • Underlying dynamical system is a flow ϕ : R × R n → R n or a map f : R n → R n from the state space to itself. • Only knowledge of dynamics is Γ = { x 0 , . . . x T − 1 } , a finite time series from a state space trajectory, with x i ∈ R n . • Goal: characterize properties of f | Λ (i.e. number and type of orbits, topological entropy) • Issue: need to locate isolating neighborhoods of f to compute Conley index. 3 / 23
How We Did It How do we get this information from a (*scalar) time series? 4 / 23
How We Did It How do we get this information from a (*scalar) time series? 1) (*Delay coordinate embedding) 2) Discretization of state space 3) Construct outer approximation of dynamics as multivalued map 4) Compute discrete Conley index using this multivalued map 4 / 23
Multivalued Maps *Only have access to finite time series; need to locate isolating neighborhoods of f to compute Conley index. 5 / 23
Multivalued Maps *Only have access to finite time series; need to locate isolating neighborhoods of f to compute Conley index. A grid on X is a family of nonempty compact sets A such that: 1 Geometrical realization of A , denoted |A| , is equal to X . 2 Each set in A is equal to the closure of its interior. 3 Distinct sets of A can only intersect in their boundaries or not at all. 4 Each compact S ⊂ X is covered by a finite subset of A . • Previous work: cubical grid, cubical multivalued map (Kacyznski, Mischaikow, Mrozek) *Computationally expensive: throw away lots of cells, number of boundary cells scales badly with dimension 5 / 23
Multivalued Maps STEP 1: GENERALIZE Definition (Cellular Multivalued Map, CMM) A cellular multivalued map F A : |A| → |A| is an outer approximation of f defined on the geometrical realization of an arbitrary grid A by � F A ( x ) := { A ∈ A : A ∩ f ( B ) � = ∅} . B ∈A : x ∈| B | 6 / 23
Intermediary Ideas To get a discretization of our state space into cells, we utilize the following constructions from computational geometry: 7 / 23
Intermediary Ideas To get a discretization of our state space into cells, we utilize the following constructions from computational geometry: The Voronoi diagram of a set L , denoted V ( L ) = { V ℓ : ℓ ∈ L } , is the covering of R n by the cells V ℓ := { x ∈ R n : d ( x , ℓ ) ≤ d ( x , ℓ ′ ) , ∀ ℓ ′ ∈ L } . 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7 / 23
Intermediary Ideas The α -diagram of a set L , denoted A α ( L ) = { A ℓ : ℓ ∈ L } , is the collection of cells A ℓ := { x ∈ R n : V ℓ ∩ B α ( ℓ ) } . (The geometrical realization of an α -diagram is called an α -grid .) The α -complex , K α , is the nerve of an α -diagram: Let σ = � l 0 , . . . , l k � , where { l 0 , . . . , l k } is any finite subset of L . Consider the intersection of cells A σ := � l i ∈ σ A l i . A simplex σ is in the α -complex, K α , if A σ � = ∅ . (The geometrical realization of an α -complex is called an α -shape .) 8 / 23
SMM and the WC In particular, we can define a CMM on the α -grid: � F A ( x ) := { A ∈ A : A ∩ f ( B ) � = ∅} . B ∈A : x ∈| B | F A α A 1 A 2 ℓ 1 A 2 ℓ 2 x A 3 A 3 A 4 ℓ 3 ℓ 4 f ( A 3 ) A 6 A 5 A 5 ℓ 5 A 6 ℓ 6 A 7 ℓ 7 f ( x ) f ( A 1 ) ℓ 8 ℓ 9 A 9 A 9 A 8 A 8 9 / 23
SMM and the WC F K ( σ ) := { τ : A τ ⊂ F A ( A σ ) } . F A α A 1 A 2 ℓ 1 A 2 ℓ 2 x A 3 A 3 A 4 ℓ 3 ℓ 4 f ( A 3 ) A 6 A 5 A 5 ℓ 5 A 6 ℓ 6 A 7 ℓ 7 f ( x ) f ( A 1 ) ℓ 8 ℓ 9 A 9 A 8 A 9 A 8 F K 〈 ℓ 1 〉 〈 ℓ 1 〉 , 〈 ℓ 3 ℓ 3 , 〈 ℓ 2 〉 ℓ 〈 ℓ 2 〉 〉 ℓ 2 , 〈 ℓ 2 〉 4 〈 ℓ 3 〉 〈 ℓ 3 〉 〈 ℓ 4 〉 〈 ℓ 3 , ℓ 5 〉 〈 ℓ 3 F K ( 〈 ℓ 1 ,ℓ 3 〉 ) , 〈 ℓ 6 ℓ 〉 4 〈 ℓ 3 ,ℓ 5 ,ℓ 6 〉 , F K ( 〈 ℓ 3 〉 ) ℓ 7 〉 〈 ℓ 5 〉 〈 ℓ 5 〉 〈 ℓ 6 〉 〈 ℓ 6 〉 〈 ℓ 5 ,ℓ 6 ,ℓ 8 〉 F K ( 〈 ℓ 1 〉 ) 〈 ℓ 6 , ℓ 9 〉 〈 ℓ 〈 ℓ 7 〉 5 , ℓ 〈 ℓ 7 , ℓ 9 〉 8 〉 〈 ℓ 6 ,ℓ 8 ,ℓ 9 〉 〈 ℓ 8 〉 〈 ℓ 9 〉 〈 ℓ 8 〉 〈 ℓ 8 , ℓ 9 〉 〈 ℓ 9 〉 10 / 23
SMM and the WC When the nerve K of a grid A is a geometrical simplicial complex, an SMM, in turn, induces a CMM on the geometrical realization |K| : � F K ( x ) := {|F K ( σ ) |} . σ ∈K : x ∈| σ | Theorem (Alexander, Bradley, Meiss, S. ) If a CMM on an α -grid is acyclic, then the corresponding SMM on the α -complex induces a CMM on the α -shape that induces the same map on homology as the CMM on the α -grid. 11 / 23
SMM and the WC • to use all of the points of Γ = { x 0 , . . . , x T − 1 } as the centers of α -cells would be computationally expensive and not necessarily illuminating • instead use a much smaller set of points, called landmarks , L = { l 1 , . . . , l k } • don’t know f on all points of state space • define a CMM on the α -grid based on a witness relation between the points of Γ and the landmarks of L . 12 / 23
SMM and the WC We never actually compute the α -diagram A α ( L ) corresponding to the set of landmarks L ! 13 / 23
SMM and the WC We never actually compute the α -diagram A α ( L ) corresponding to the set of landmarks L ! Yet we construct a CMM on the α -grid using the temporal ordering of Γ and a witness relation: Definition (Witness Relation) We use a fuzzy version of the strong-witness relation (Carlsson, de Silva): W ǫ (Γ , ℓ i ) := { x ∈ Γ : d ( x , ℓ i ) ≤ d ( x , L ) + ǫ } . 13 / 23
We define an outer approximation of our dynamical system as follows: Definition (Cellular Witness Map) Suppose α and ǫ are as above. The witness map F W : |A α ( L ) | → |A α ( L ) | for the witness relation is the cellular multivalued map � F W ( x ) := { A j ∈ A α ( L ) : ∃ x t ∈ W ǫ (Γ , l i ) s . t . x t +1 ∈ W ǫ (Γ , l j ) } . A i ∈A α ( L ): x ∈ A i 14 / 23
We define an outer approximation of our dynamical system as follows: Definition (Cellular Witness Map) Suppose α and ǫ are as above. The witness map F W : |A α ( L ) | → |A α ( L ) | for the witness relation is the cellular multivalued map � F W ( x ) := { A j ∈ A α ( L ) : ∃ x t ∈ W ǫ (Γ , l i ) s . t . x t +1 ∈ W ǫ (Γ , l j ) } . A i ∈A α ( L ): x ∈ A i We define witness complex to be the clique complex for the edge-set determined by pairwise intersections of the sets of witnesses to a landmark. 14 / 23
SMM and the WC For Γ , L , α and ǫ as specified below, we get that W ǫ (Γ , L ) = K α ( L ). Theorem (Alexander, Bradley, Meiss, S.) K α ( L ) ⊆ W ǫ (Γ , L ) : For a time series Γ , a set of landmarks L, and α, ǫ > 0 , let K α ( L ) be the α -complex and W ǫ (Γ , L ) be the witness complex. If there exists a δ > 0 with δ ≤ ǫ/ 2 such that Γ is δ -dense of A α ( L ) , then K α ( L ) ⊆ W ǫ (Γ , L ) . Theorem (Alexander, Bradley, Meiss, S.) W ǫ (Γ , L ) ⊆ K α ( L): Suppose K α ( L ) and W ǫ (Γ , L ) are as above. Let M = max x ∈ Γ d ( x , L ) and β = min i � = j d ( l i , l j ) . If α > 0 is selected β such that M + ǫ ≤ α ≤ 2 and K α ( L ) is a clique complex, then √ W ǫ (Γ , L ) ⊆ K α ( L ) . 15 / 23
SMM and the WC We also get that: Theorem (Alexander, Bradley, Meiss, S.) Suppose that |A α ( L ) | is compact and f is Lipschitz on |A α ( L ) | 2 min { 1 , 1 with constant c. If there exists a δ > 0 with δ ≤ ǫ c } such that Γ is δ -dense on |A α ( L ) | , then F W is an outer approximation of f . Since the nerve of A α ( L ) is K α ( L ) = W ǫ (Γ , L ), the CMM F W induces a SMM on the witness complex: F W : W ǫ (Γ , L ) → W ǫ (Γ , L ) . 16 / 23
SMM and the WC • We check the acyclicity of the CMM F W • We then use the SMM F W to construct an explicit chain selector ϕ : W ǫ (Γ , L ) → W ǫ (Γ , L ) • It follows that | ϕ | : |W ǫ (Γ , L ) | → |W ǫ (Γ , L ) | is a continuous selector of the CMM F W So we can compute the Conley index [ F ( N , E ) ∗ ] = [ ϕ ∗ ] and prove the existence of non-trivial invariant sets of dynamical systems! 17 / 23
Example: Finding fixed point of H´ enon map 0.4 0.2 y 0 −0.2 −0.4 −1.5 −1 −0.5 0 0.5 1 1.5 x For f ( x , y ) = ( y + 1 − 1 . 4 x 2 , 0 . 3 x ) with initial condition z 0 = ( − 0 . 4 , 0 . 3) near attractor Λ, we generate a trajectory Γ of length T = 10 5 . We select 216 landmarks L on a hexagonal grid √ with spacing β = 0 . 05. Choosing α = β/ 2 and ǫ = 0 . 005, we construct CMM F W using the fuzzy witness relation. 18 / 23
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