Heteroclinic cycles in coupled cell systems

Heteroclinic cycles in coupled cell systems Michel Field University of Houston, USA, & Imperil College, UK Reserch supported in prt by Leverhulme Foundtion nd NSF Grnt DMS-0071735 Some of the reserch reported on here is joint with Peter Ashwin, Exeter Extrordinire comme les mthémtiques vous ident à vous connâitre - Smuel Beckett, Molloy Heteroclinic cycles p.1/60

Invrint subspces Certin clsses of dynmicl system nturlly hve invrint spces. Tht is, subspces of phse spce tht re invrint for the dynmics of ll systems in the clss. Heteroclinic cycles p.2/60

Invrint subspces Certin clsses of dynmicl system nturlly hve invrint spces. Tht is, subspces of phse spce tht re invrint for the dynmics of ll systems in the clss. Symmetric (+ reversible or Hmiltonin) systems. Heteroclinic cycles p.2/60

Semiliner feedbck systems A semiliner feedbck (SLF) system is set of ODEs ẋ i = f i (x i ) + x i F i (x 1,..., x i,..., x n ), where x i R n i, 1 i n nd f i (0) = 0. The evolution of x i ccording to ẋ i = f i (x i ) is modified by the liner feedbck term F i which depends typiclly nonlinerly on the remining vribles. F i x i x i F i Heteroclinic cycles p.3/60

Lotk-Volterr popultion models re SLF systems with F i liner. Mny (not ll) exmples of symmetric systems re lso of this type t lest if we restrict to cubic trunctions (F i will typiclly be qudrtic). Note tht x i = 0 will lwys be n invrint subspce for the dynmics of n SLF system. Since intersections of invrint subspces re invrint, x i1 =... = x is = 0 is lso invrint, 1 i 1 <... < i p n. If we dd symmetry nd/or reversibility to the mix, this will result in more invrint subspces which my or my not ply role in the formtion of heteroclinic cycles. Heteroclinic cycles p.4/60

Dynmics & Intersections The presence of invrint subspces llows the existence of robust (stble) non-trnsverse intersections of invrint mnifolds of equilibri nd limit cycles. In the simplest cses, these will be sddle connections see figure. Generlly, intersections will be singulr (ll this is understood nd covered by the theory of equivrint trnsverslity). Z 2 Vector field Diffeomorphism Heteroclinic cycles p.5/60

Heteroclinic cycles In both equivrint dynmics nd SLF models (in prticulr, popultion models), it is possible to hve robust cycles of non-trnsverse sddle connections. First observed by My & Leonrd (1975) (popultion dynmics), lter by Dos Reis (1978) (equivrint dynmics on surfces) nd then by Guckenheimer nd Holmes (1988) (equivrint bifurction theory). p 001 p 111 Dynmics on flow invrint ttrcting sphere. p 100 p 010 Symmetry group: Note the ttrcting heteroclinic cycle Σ Z 2 3.Z 3 Σ Heteroclinic cycles p.6/60

Exmples & Models If we look t dynmics on R n equivrint with respect to Z n 2 Z n (or just Z n 2), there re infinite fmilies of robust ttrcting heteroclinic cycles. We describe the cse n = 4. We do this by showing dynmics on the positive orthnt of S 3 invrint under the flow of Z 4 2 Z 4 -equivrint dynmics on S 3. (A popultion dynmicist might look t the invrint simplex x 1 +... + x 4 = 1 in the positive orthnt of R 4 ). We ssume symmetry here but the phenomen we show re chrcteristic of SLF systems nd hve nothing to do with symmetry. Indeed, from our perspective, ll of our exmples should be viewed more s phenomenologicl models. Heteroclinic cycles p.7/60

Edge cycles V 4 Edge Cycle V 24 V 1 V 1234 V 13 V 3 V 2 Heteroclinic cycles p.8/60

Fce cycles V 4 V 14 Fce Cycle V 24 V 34 V 1 V 13 V 3 V 12 V 2 V 23 Heteroclinic cycles p.9/60

Phenomenolgy If we tke the N-dimensionl simplex N, N 3, nd choose 1 p N 2, we cn construct (ttrcting) heteroclinic cycles connecting equilibri (or periodic orbits or chotic sets) on the p-dimensionl fces of N. For exmple, if N = 5, p = 3, there exist ttrcting heteroclinic cycles... 123 234 345 451 512 123... These cycles my connect equilibri, periodic orbits or chotic sets ( cycling chos ). Heteroclinic cycles p.10/60

Cycling chos exmple: N = 3, p = 1 This exmple is built on the following system of ODEs defined on C 2 : z 1 = z 1 ( z 1 2 + z 2 2 )z 1 + β z 2 1 z 2 + γz 3 2, z 2 = z 2 ( z 1 2 + z 2 2 )z 2 + βz 1 z 2 2 + γ z 3 1., β, γ R. Choose 0 so tht there is globlly ttrcting invrint sphere for the dynmics. Heteroclinic cycles p.11/60

Dynmics, projection on y 1, y 2 We show the projection of the phse portrit onto the (y 1, y 2 )-plne when β = 1.5, γ = 0.29. Heteroclinic cycles p.12/60

Model system We couple three of these systems together to obtin the Z 3 -equivrint system Z 1 = F (Z 1 ) + τ Z 2 2 Z 1, Z 2 = F (Z 2 ) + τ Z 3 2 Z 2, Z 3 = F (Z 3 ) + τ Z 1 2 Z 3. Here τ is rel prmeter nd we hve written the eqution of the bsic cell in the form Z = F (Z), where Z = (z 1, z 2 ) C 2. We write the rel coordintes for the ith. cell s (x i1, y i1, x i2, y i2 ). Heteroclinic cycles p.13/60

Cycling chos: τ = 2.2 Heteroclinic cycles p.14/60

Periodic chos: τ = 1.97 Heteroclinic cycles p.15/60

Progrm Trnsltion from vrious types of heteroclinic behviour tht occur in symmetric systems to nlogous behviour in coupled cell systems (no symmetry). Heteroclinic cycles p.16/60

Coupled cell systems: Cell types We shll be looking t finite collection of different cell types. We write these A, B, C,.... Ech cell hs finite number of inputs nd n output. Outputs A B C... Inputs Heteroclinic cycles p.17/60

Cells: Inputs Output (type A) Inputs A b1 b2 b2 c c c c d e 2 inputs type (from cells of type A) 1 input type b1 (from cell of type B) 2 inputs type b2 (from cells of type B) 4 inputs type c (from cells of type C) 1 input type d (from cell of type D) 1 input type e (from cell of type E) A given cell type my receive inputs from cells of vrious types. In the figure, cell of type A receives inputs from cells of types A, B, C, D nd E. Heteroclinic cycles p.18/60

Ptchcord rules We interconnect cells using ptchcords. A type ptchcord goes from the output of cell of type A to the input of cell. If there re type 1, 2,... inputs, then we color code ptchcords so s to indicte which type of input the cord should be ptched into. There re no restrictions on the number of outputs we tke from cell. No more thn one ptchcord is plugged into given input. Normlly we regrd ptchcords s dynmiclly neutrl. However, ptchcords could include, for exmple, dely line. Heteroclinic cycles p.19/60

Exmple Type A: red Type B: green b1 b2 b1 b2 b1 b1 b1 Heteroclinic cycles p.20/60

Ptching the inputs. Type A: red Type B: green b1 b2 b1 b2 b1 b1 b1 Heteroclinic cycles p.21/60

Ptching the b inputs. Type A: red Type B: green b1 b2 b1 b2 b1 b1 b1 Heteroclinic cycles p.22/60

Another Ptching. Type A: red Type B: green b1 b2 b1 b2 b1 b1 b1 Heteroclinic cycles p.23/60

Coupled cell systems For us, coupled cell system will consist of A (finite) number of cells, finite number of cell types. Heteroclinic cycles p.24/60

Coupled cell systems For us, coupled cell system will consist of A (finite) number of cells, finite number of cell types. The cells will be ptched together ccording to the input-output rules sketched bove. If cell hs multiple inputs of prticulr type it is immteril which input the ptchcord is plugged into (= locl symmetry). Heteroclinic cycles p.24/60

Invrint subspces Given: coupled cell system. We re interested initilly in synchronised solutions of the system. These correspond to certin types of invrint subspce of the phse spce. We illustrte the ides with two very simple exmples. Heteroclinic cycles p.25/60

Five cell system I Type A: red Type B: green b1 b2 b1 b2 b1 b1 b1 The only invrint subspce of synchronous solutions corresponds to ll the type A cells being synchronised nd ll the type B cells being synchronised. (This property is true for ll coupled cell networks trivil synchronised stte.) Heteroclinic cycles p.26/60

Five cell system II Type A: red Type B: green b1 b2 b1 b2 B1 B2 B3 b1 b1 b1 It is possible for just the B cells to be synchronised nd for just B1 (or B3) nd B2 to be synchronised. Heteroclinic cycles p.27/60

Reptching rules Ech clss of synchronised solutions determines reptching rule tht neither destroys the invrint subspce nor the dynmics on the invrint subspce. The reptching rule defines n equivlence reltion on the set of cells or, equivlently, defines prtition of the cells ( blnced prtition). Conversely, blnced prtition (or set of dmissible reptching rules) determines unique invrint subspce of synchronised solutions. Reptching yields unique miniml network with number of cells equl to the number of elements of the prtition this is trivil. Dynmics of the miniml network determine dynmics on the ssocited invrint subspce. We illustrte this for the previous exmple. Heteroclinic cycles p.28/60

Reptching 1 Type A: red Type B: green b1 b2 b1 b2 B1 B2 B3 b1 b1 b1 Heteroclinic cycles p.29/60

Reptching 2 Type A: red Type B: green b1 b2 b1 b2 B1 B2 B3 b1 b1 b1 Heteroclinic cycles p.30/60

Reptching 3 Type A: red Type B: green b1 b2 b1 b2 B1 B2 B3 b1 b1 b1 Heteroclinic cycles p.31/60

Miniml network A = F(A;A,B,B) b1 b2 B = G(B;A,A,B) Note: F(x;y,u,v) = / F(x;y,v,u) but G(u;y,z,u) = G(u;z,y,u). B1 b1 This coupled cell system determines dynmics on the invrint subspce of A nd B synchronized solutions. Every solution (A(t),B(t)) of this system extends uniquely to synchronized solution of the originl system nd conversely. Heteroclinic cycles p.32/60

Heteroclinic cycles We describe results on the existence of heteroclinic cycles in (symmetric) coupled cell systems. First, observe tht we cn get stble connections between A, B synchronised sttes nd B, C synchronised sttes. A, C synchronized B, C synchronized C synchronized A synchronized synchronized B Heteroclinic cycles p.33/60

Heteroclinic cycles ctd This gives us wy of constructing robust heteroclinic cycles between groups of synchronised cells. Bsed on our erlier models we cn mke trnsition between edge, fce,... heteroclinic cycles nd corresponding heteroclinic cycle between synchronised sttes. V 4 V 14 V 34 V 1 V 3 V 12 V 2 V 23 Heteroclinic cycles p.34/60

Heteroclinic cycles ctd : : : (,,, * ) ( *,,, * ) (,, *, * ) Σ ( *,,, ) (,, *, ) ( *, *,, ) (, *, *, ) (, *,, ) Heteroclinic cycles p.35/60

Cycles between synchronous clusters A A B B B C C Here we might expect to see cycle of the form... AB BC CA AB... Heteroclinic cycles p.36/60

Exmple A cycle between 3 pirs of synchronous sttes Heteroclinic cycles p.37/60

Embedded cycles A B C SLF system with heteroclinic cycle A B C A B C A Asymmetric coupled cell system which dmits heteroclinic cycle. Heteroclinic cycles p.38/60

Dynmics x 1 x 2 x 3 x 4 x 5 x 6 x 7 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 t Heteroclinic cycles p.39/60

Cndidte cycle A B C A B C A This system reptches preserving A, B, C synchronous subspce to Heteroclinic cycles p.40/60

Reptched system A B C A B C A This system dmits heteroclinic cycle. Does the originl system? Yes. Stbility properties? Wht other combintoril wys re there of generting/nlysing heteroclinic cycles in coupled cell systems? Heteroclinic cycles p.41/60

Simplest exmple? A1 = F(A1;A2,B1), A2 = F(A2;A1,B1), B1 = F(B1;A2,B2), B2 = F(B2;A2,B1). A1 A2 B1 B2 Synchronized sttes: {A1,A2}, {B1,B2}, {A1,A2 B1,B2}, {A1,A2,B1,B2} Heteroclinic cycles p.42/60

Possible heteroclinic cycle {A1,A2} synchronized subspce p {A1,A2 B1,B2} synchronized subspce {B1,B2} synchronized subspce Heteroclinic cycles p.43/60

Some nlysis Suppose we hve one-dimensionl cell dynmics. The vector field F governing the evolution of the cells is x 1 = f(x 1 ; x 2, y 1 ), x 2 = f(x 2 ; x 1, y 1 ), y 1 = f(y 1 ; x 2, y 2 ), y 2 = f(y 2 ; x 2, y 1 ). Here x vribles correspond to A cells, y vribles to B cells, nd f : R 3 R is smooth function (no constrints). If the system hs n equilibrium t p = (0, 0, 1, 1), then f(0; 0, 1) = f(1; 0, 1) = 0. Heteroclinic cycles p.44/60

Locl representtion of F Ner p we consider f of the form f(u; v, w) = (u 1) 2 g 1 (u; v, w) + u 2 g 2 (u; v, w), where g 1, g 2 re smooth. Suppose tht g 1 (u, v, w) = αu + βv + γ(w 1), g 2 (u, v, w) = (u 1) + bv + c(w 1), where α, β, γ nd, b, c re constnt ner (0, 0, 1) nd (1, 0, 1) respectively. We compute the lineristion DF (p) t p. Heteroclinic cycles p.45/60

Jcobin Mtrix We find tht DF (p) = α β γ 0 β α γ 0 0 b c 0 b c. This mtrix hs eigenvlues given by Heteroclinic cycles p.46/60

Eigenvlues of Jcobin 1. α + β + + c ± ( + c α β) 2 + 4γb 2 Eigenspce: {A1, A2 B1, B2}. 2. α β. (Eigenspce lies in {B1, B2}.) 3. c. (Eigenspce lies in {A1, A2}.) If α, β,, c < 0, α > β, < c, nd γb < 0, then p is of index three nd W u (p) {B1, B2}. Note tht for sufficiently negtive γb we my require the pir of eigenvlues for {A1, A2 B1, B2} to be complex. Heteroclinic cycles p.47/60

We my similrly choose f ner q = (1, 1, 2, 2) so tht p is of index three with W u (q) {A1, A2}. Finlly, we cn choose f on the complement of neighbourhoods of p, q so tht there re pir connections from p to q in {B1, B2} nd pir of connections from q to p in {A1, A2}. {A1,A2} synchronized subspce (Dimension: 3) p q {A1,A2 B1,B2} synchronized subspce (Dimension: 2) {B1,B2} synchronized subspce (Dimension: 3) Heteroclinic cycles p.48/60

A finl exmple A1 B1 C1 A2 B2 C2 Asymmetric system of identicl cells. Synchronous sttes {B1, B2}, {C1, C2}, {B1, B2 C1, C2}, {A1, A2 B1, B2 C1, C2} nd trivil synchronised stte. In this cse one cn show there is 1-dimensionl heteroclinic cycle B BC C BC. Heteroclinic cycles p.49/60

Dynmics on {B1, B2 C1, C2} subspce A1 B1 C1 A2 B2 C2 Heteroclinic cycles p.50/60

Dynmics on {B1, B2 C1, C2} subspce A1 B1 C1 A2 Heteroclinic cycles p.51/60

Dynmics on {B1, B2 C1, C2} subspce A1 B1 C1 A2 A1 = F(A1;B1,C1), A2 = F(A2;B1,C1), B1 = F(B1;C1,A1), C1 = F(C1;B1,A2). Heteroclinic cycles p.52/60

Dynmics on {A1, A2 B1, B2 C1, C2} subspce A1 B1 C1 A2 B2 C2 Heteroclinic cycles p.53/60

Dynmics on {A1, A2 B1, B2 C1, C2} subspce A1 B1 C1 A2 B2 C2 Heteroclinic cycles p.54/60

Dynmics on {A1, A2 B1, B2 C1, C2} subspce A1 B1 C1 A2 B2 C2 Heteroclinic cycles p.55/60

Dynmics on {A1, A2 B1, B2 C1, C2} subspce A1 B1 C1 Heteroclinic cycles p.56/60

Dynmics on {A1, A2 B1, B2 C1, C2} subspce A1 B1 C1 This hs the symmetry A1 < > B1 nd Blue < > Red If Blue=Red, we get system with D 3 symmetry A1 = F(A1;B1,C1), B1 = F(B1;C1,A1); C1 = F(C1;B1,A1). Heteroclinic cycles p.57/60

Invrint subspces Suppose tht coupled cell system C = {A i 1, B j 1,...} hs n invrint subspce E corresponding to clss of synchronised solutions. We hve n ssocited prtition P = {C k } of C defined by grouping synchronised cells. Unsynchronised cells define singletons in the prtition. Groups of cells of identicl type but different synchronistion lie in different elements of the prtition. Ech element of prtition will consist of cells of the sme type. Heteroclinic cycles p.58/60

Invrint subspces LEMMA Let C i, C j P (they my be the sme). Let α be the type of cells in C i. Let d C j. The totl number of inputs into d from cells in C i depends only on i, j. Proof. If there is vrition in the number then we cn choose symmetric inputs for the cells in C j nd so E could not be n invrint subspce comprised of synchronous solutions... [If not, we cn find pir of cells in C j which receive different number of inputs from cells of type α in some C k (k i). Choose the initil stte of the cells in C k to be different from the initil sttes of the cells in C i.] Heteroclinic cycles p.59/60

Reptching rules We cll prtition of C stisfying the conditions of the Lemm for ll i, j blnced prtition. If P = {C k } is blnced prtition of C, then there is n ssocited invrint subspce of synchronous solutions. Clusters re given explicitly by the prtition. The conditions of the lemm give reptching rules. Fix C i, C j P. We cn permute outputs of cells in C i tht go to C j without restriction. The resulting network still hs n invrint subspce with sme cells synchronised s the originl network. This gives rise to unique miniml network defining the dynmics of the clss of synchronised solutions. Heteroclinic cycles p.60/60