American Institute of Mathematical Sciences

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July  2018, 23(5): 2021-2041. doi: 10.3934/dcdsb.2018193

From coupled networks of systems to networks of states in phase space

Oskar Weinberger 1,, and  Peter Ashwin 2,

1. 

Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 100 44 Stockholm, Sweden
2. 

Centre for Systems, Dynamics and Control, and Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK

* Corresponding author: oskar.weinberger@gmail.com

Received  May 2017 Revised  August 2017 Published  July 2018 Early access  May 2018

Dynamical systems on graphs can show a wide range of behaviours beyond simple synchronization - even simple globally coupled structures can exhibit attractors with intermittent and slow switching between patterns of synchrony. Such attractors, called heteroclinic networks, can be well described as networks in phase space and in this paper we review some results and examples of how these robust attractors can be characterised from the synchrony properties and how coupled systems can be designed to exhibit given but arbitrary network attractors in phase space.

Keywords: Dynamical systems, heteroclinic networks, coupled systems.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193
References:
[1]

N. Agarwal and M. J. Field, Dynamical equivalence of network architecture for coupled dynamical systems Ⅰ: asymmetric inputs, Nonlinearity, 23 (2010), 1245-1268.  doi: 10.1088/0951-7715/23/6/001.  Google Scholar

[2]

N. Agarwal and M. J. Field, Dynamical equivalence of network architecture for coupled dynamical systems Ⅱ: general case, Nonlinearity, 23 (2010), 1269-1289.  doi: 10.1088/0951-7715/23/6/002.  Google Scholar

[3]

M. Aguiar, P. Ashwin, A. Dias and M. Field, Dynamics of coupled cell networks: Synchrony, heteroclinic cycles and inflation, Journal of Nonlinear Science, 21 (2011), 271-323.  doi: 10.1007/s00332-010-9083-9.  Google Scholar

[4]

M. A. D. Aguiar, A. P. S. Dias, M. Golubitsky and M. da C A Leitee, Bifurcations from regular quotient networks: A first insight, Physica D, 238 (2009), 137-155.  doi: 10.1016/j.physd.2008.10.006.  Google Scholar

[5]

M. A. D. Aguiar, S. B. S. D. Castro and I. S. Labouriau, Dynamics near a heteroclinic network, Nonlinearity, 18 (2005), 391–414, URL http://stacks.iop.org/0951-7715/18/i=1/a=019. doi: 10.1088/0951-7715/18/1/019.  Google Scholar

[6]

P. Ashwin and P. Chossat, Attractors for robust heteroclinic cycles with continua of connections, Journal of Nonlinear Science, 8 (1998), 103–129, URL http://dx.doi.org/10.1007/s003329900045. doi: 10.1007/s003329900045.  Google Scholar

[7]

P. Ashwin and C. Postlethwaite, On designing heteroclinic networks from graphs, Physica D, 265 (2013), 26-39.  doi: 10.1016/j.physd.2013.09.006.  Google Scholar

[8]

P. Ashwin, S. Coombes and R. Nicks, Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience, The Journal of Mathematical Neuroscience, 6 (2016), Art. 2, 92 pp, URL https://doi.org/10.1186/s13408-015-0033-6. doi: 10.1186/s13408-015-0033-6.  Google Scholar

[9]

P. Ashwin and M. Field, Heteroclinic networks in coupled cell systems, Archive for Rational Mechanics and Analysis, 148 (1999), 107-143.  doi: 10.1007/s002050050158.  Google Scholar

[10]

P. Ashwin and Ö. Karabacak, Robust heteroclinic behaviour, synchronization, and ratcheting of coupled oscillators, Dynamics, Games and Science. II, Springer Proc. Math., Springer, Heidelberg, 2 (2011), 125–140, URL http://dx.doi.org/10.1007/978-3-642-14788-3_10. doi: 10.1007/978-3-642-14788-3_10.  Google Scholar

[11]

P. Ashwin, Ö. Karabacak and T. Nowotny, Criteria for robustness of heteroclinic cycles in neural microcircuits, The Journal of Mathematical Neuroscience, 1 (2011), Art. 13, 18 pp, URL http://dx.doi.org/10.1186/2190-8567-1-13. doi: 10.1186/2190-8567-1-13.  Google Scholar

[12]

P. Ashwin and C. Postlethwaite, Designing heteroclinic and excitable networks in phase space using two populations of coupled cells, Journal of Nonlinear Science, 26 (2016), 345–364, URL http://dx.doi.org/10.1007/s00332-015-9277-2. doi: 10.1007/s00332-015-9277-2.  Google Scholar

[13]

P. Ashwin and C. Postlethwaite, Quantifying noisy attractors: From heteroclinic to excitable networks, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1989–2016, URL http://dx.doi.org/10.1137/16M1061813. doi: 10.1137/16M1061813.  Google Scholar

[14]

P. Ashwin and M. Timme, Unstable attractors: Existence and robustness in networks of oscillators with delayed pulse coupling, Nonlinearity, 18 (2005), 2035-2060.  doi: 10.1088/0951-7715/18/5/009.  Google Scholar

[15]

Y. Bakhtin, Noisy heteroclinic networks, Probability Theory and Related Fields, 150 (2011), 1–42, URL http://dx.doi.org/10.1007/s00440-010-0264-0. doi: 10.1007/s00440-010-0264-0.  Google Scholar

[16]

C. Bick and M. I. Rabinovich, On the occurrence of stable heteroclinic channels in lotkavolterra models, Dynamical Systems, 25 (2010), 97–110, URL http://dx.doi.org/10.1080/14689360903322227. doi: 10.1080/14689360903322227.  Google Scholar

[17]

W. Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity, 7 (1994), 1367–1384, URL http://stacks.iop.org/0951-7715/7/i=5/a=006. doi: 10.1088/0951-7715/7/5/006.  Google Scholar

[18]

F. H. Busse and R. M. Clever, Nonstationary convection in a rotating system, Springer Berlin Heidelberg, Berlin, Heidelberg, 1979,376–385, URL http://dx.doi.org/10.1007/978-3-642-67220-0_39. Google Scholar

[19]

S. B. S. D. Castro and A. Lohse, Switching in heteroclinic networks, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1085–1103, URL http://dx.doi.org/10.1137/15M1042176. doi: 10.1137/15M1042176.  Google Scholar

[20]

S. B. Castro and A. Lohse, Stability in simple heteroclinic networks in, Dynamical Systems, 29 (2014), 451–481, URL http://dx.doi.org/10.1080/14689367.2014.940853. doi: 10.1080/14689367.2014.940853.  Google Scholar

[21]

M. Dellnitz, M. Field, M. Golubitsky, J. Ma and A. Hohmann, Cycling chaos, International Journal of Bifurcation and Chaos, 5 (1995), 1243-1247.  doi: 10.1142/S0218127495000909.  Google Scholar

[22]

A. P. S. Dias and I. Stewart, Linear equivalence and ode-equivalence for coupled cell networks, Nonlinearity, 18 (2005), 1003–1020, URL http://stacks.iop.org/0951-7715/18/i=3/a=004. doi: 10.1088/0951-7715/18/3/004.  Google Scholar

[23]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[24]

G. L. dos Reis, Structural stability of equivariant vector fields on two-manifolds, Trans. Amer. Math. Soc., 283 (1984), 633–643, URL http://dx.doi.org/10.2307/1999151. doi: 10.1090/S0002-9947-1984-0737889-8.  Google Scholar

[25]

G. B. Ermentrout, A Guide to XPPAUT for Researchers and Students, SIAM, Pittsburgh, 2002. Google Scholar

[26]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Chapman & Hall/CRC Research Notes in Mathematics Series, Taylor & Francis, 1996, URL https://books.google.co.uk/books?id=4dqrLHmicR8C.  Google Scholar

[27]

M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001–1043, URL http://stacks.iop.org/0951-7715/4/i=4/a=001. doi: 10.1088/0951-7715/4/4/001.  Google Scholar

[28]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185–205, URL http://dx.doi.org/10.2307/1998153. doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[29]

M. J. Field, Heteroclinic networks in homogeneous and heterogeneous identical cell systems, Journal of Nonlinear Science, 25 (2015), 779–813, URL http://dx.doi.org/10.1007/s00332-015-9241-1. doi: 10.1007/s00332-015-9241-1.  Google Scholar

[30]

M. J. Field, Patterns of desynchronization and resynchronization in heteroclinic networks, Nonlinearity, 30 (2017), 516–557, URL http://stacks.iop.org/0951-7715/30/i=2/a=516. doi: 10.1088/1361-6544/aa4f48.  Google Scholar

[31]

M. Field, Combinatorial dynamics, Dynamical Systems, 19 (2004), 217-243.  doi: 10.1080/14689360410001729379.  Google Scholar

[32]

A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM Journal on Applied Mathematics, 52 (1992), 1476–1489, URL https://doi.org/10.1137/0152085. doi: 10.1137/0152085.  Google Scholar

[33]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate texts in mathematics, Springer-Verlag, New York-Heidelberg, 1973, URL https://books.google.co.uk/books?id=y9BFAQAAIAAJ.  Google Scholar

[34]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.  Google Scholar

[35]

M. Golubitsky, I. Stewart and A. Török, Patterns of synchrony in coupled cell networks with multiple arrows, SIAM Journal on Applied Dynamical Systems, 4 (2005), 78–100, URL http://dx.doi.org/10.1137/040612634. doi: 10.1137/040612634.  Google Scholar

[36]

J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. So, 103 (1988), 189-192.  doi: 10.1017/S0305004100064732.  Google Scholar

[37]

D. Hansel, G. Mato and C. Meunier, Clustering and slow switching in globally coupled phase oscillators, Physical Review E, 48 (1993), 3470-3477.  doi: 10.1103/PhysRevE.48.3470.  Google Scholar

[38]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, no. no. 583 in Lecture Notes in Mathematics, Springer-Verlag, 1977, URL https://books.google.co.uk/books?id=-rgZAQAAIAAJ.  Google Scholar

[39]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[40]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press Cambridge [England]; New York, 1988, URL http://www.loc.gov/catdir/toc/cam029/88017571.html. Google Scholar

[41]

C. Hou and M. Golubitsky, An example of symmetry breaking to heteroclinic cycles, Journal of Differential Equations, 133 (1997), 30–48, URL http://www.sciencedirect.com/science/article/pii/S0022039696932015. doi: 10.1006/jdeq.1996.3201.  Google Scholar

[42]

K. Kaneko and I. Tsuda, Focus issue on chaotic itinerancy, Chaos, 13 (2003), 926-936.  doi: 10.1063/1.1607783.  Google Scholar

[43]

O. Karabacak and P. Ashwin, Heteroclinic ratchets in networks of coupled oscillators, Journal of Nonlinear Science, 20 (2010), 105-129.  doi: 10.1007/s00332-009-9053-2.  Google Scholar

[44]

V. Kirk and M. Silber, A competition between heteroclinic cycles, Nonlinearity, 7 (1994), 1605–1621, URL http://stacks.iop.org/0951-7715/7/i=6/a=005. doi: 10.1088/0951-7715/7/6/005.  Google Scholar

[45]

C. Kirst and M. Timme, From networks of unstable attractors to heteroclinic switching Physical Review E, 78 (2008), 065201, 4pp. doi: 10.1103/PhysRevE.78.065201.  Google Scholar

[46]

W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos: An Interdisciplinary Journal of Nonlinear Science, 10 (2000), 427–469, URL http://dx.doi.org/10.1063/1.166509. doi: 10.1063/1.166509.  Google Scholar

[47]

H. Kori and Y. Kuramoto, Slow switching in globally coupled oscillators: Robustness and occurence through delayed coupling, Physical Review E, 63 (2001), 046214.  doi: 10.1103/PhysRevE.63.046214.  Google Scholar

[48]

M. Krupa, Robust heteroclinic cycles, Journal of Nonlinear Science, 7 (1997), 129-176.  doi: 10.1007/BF02677976.  Google Scholar

[49]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. Ⅰ, Ergodic Theory and Dynamical Systems, 15 (1995), 121-147.  doi: 10.1017/S0143385700008270.  Google Scholar

[50]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. Ⅱ, Proceedings of the Royal Socity of Edinburgh Sect. A, 134 (2004), 1177-1197.  doi: 10.1017/S0308210500003693.  Google Scholar

[51]

R. Lauterbach, S. Maier-Paape and E. Reissner, A systematic study of heteroclinic cycles in dynamical systems with broken symmetries, Proceedings/Royal Society of Edinburgh/Section A, Mathematics, 126 (1996), 885–909, URL http://publications.rwth-aachen.de/record/145691. doi: 10.1017/S030821050002312X.  Google Scholar

[52]

R. Lauterbach and M. Roberts, Heteroclinic cycles in dynamical systems with broken spherical symmetry, Journal of Differential Equations, 100 (1992), 22–48, URL http://www.sciencedirect.com/science/article/pii/0022039692901246. doi: 10.1016/0022-0396(92)90124-6.  Google Scholar

[53]

A. Lohse, Unstable attractors: Existence and stability indices, Dynamical Systems, 30 (2015), 324–332, URL http://dx.doi.org/10.1080/14689367.2015.1041879. doi: 10.1080/14689367.2015.1041879.  Google Scholar

[54]

A. Lohse and S. B. S. D. Castro, Construction of heteroclinic networks in $ R^4$, Nonlinearity, 29 (2016), 3677-3695.  doi: 10.1088/0951-7715/29/12/3677.  Google Scholar

[55]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29 (1975), 243-253.  doi: 10.1137/0129022.  Google Scholar

[56]

I. Melbourne, An example of a nonasymptotically stable attractor, Nonlinearity, 4 (1991), 835–844, URL http://stacks.iop.org/0951-7715/4/i=3/a=010. doi: 10.1088/0951-7715/4/3/010.  Google Scholar

[57]

F. S. Neves and M. Timme, Computation by switching in complex networks of states, Physical Review Letters, 109 (2012), 018701. Google Scholar

[58]

E. Nijholt, B. Rink and J. Sanders, Graph fibrations and symmetries of network dynamics, Journal of Differential Equations, 261 (2016), 4861–4896, URL http://www.sciencedirect.com/science/article/pii/S0022039616301784. doi: 10.1016/j.jde.2016.07.013.  Google Scholar

[59]

G. Orosz, J. Moehlis and P. Ashwin, Designing the dynamics of globally coupled oscillators, Progress of Theoretical Physics, 122 (2009), 611-630.  doi: 10.1143/PTP.122.611.  Google Scholar

[60]

J. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York, 1982.  Google Scholar

[61]

M. Peixoto, On an approximation theorem of kupka and smale, Journal of Differential Equations, 3 (1967), 214–227, URL http://www.sciencedirect.com/science/article/pii/0022039667900265. doi: 10.1016/0022-0396(67)90026-5.  Google Scholar

[62]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, no. 12 in Cambridge Nonlinear Science Series, Cambridge University Press, Cambridhe, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[63]

O. Podvigina and P. Ashwin, On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, 24 (2011), 887–929, URL http://stacks.iop.org/0951-7715/24/i=3/a=009. doi: 10.1088/0951-7715/24/3/009.  Google Scholar

[64]

M. Rabinovich, P. Varona, A. Selverston and H. Abarbanel, Dynamical principles in neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265.  doi: 10.1103/RevModPhys.78.1213.  Google Scholar

[65]

M. Rabinovich, P. Varona, I. Tristan and V. Afraimovich, Chunking dynamics: heteroclinics in mind, Frontiers in Computational Neuroscience, 8 (2014), 22, URL http://journal.frontiersin.org/article/10.3389/fncom.2014.00022. doi: 10.3389/fncom.2014.00022.  Google Scholar

[66]

M. Rabinovich, R. Huerta and G. Laurent, Transient dynamics for neural processing, Science, 321 (2008), 48–50, URL http://science.sciencemag.org/content/321/5885/48. Google Scholar

[67]

B. Rink and J. Sanders, Coupled cell networks and their hidden symmetries, SIAM Journal on Mathematical Analysis, 46 (2014), 1577-1609.  doi: 10.1137/130916242.  Google Scholar

[68]

B. Rink and J. Sanders, Coupled cell networks: semigroups, lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509–3548, Art no PII S0002-9947(2014)06221-1. doi: 10.1090/S0002-9947-2014-06221-1.  Google Scholar

[69]

B. Sandstede and A. Scheel, Forced symmetry breaking of homoclinic cycles, Nonlinearity, 8 (1995), 333–365, URL http://stacks.iop.org/0951-7715/8/i=3/a=003. doi: 10.1088/0951-7715/8/3/003.  Google Scholar

[70]

M. Schub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[71]

I. Stewart, M. Golubitsky and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM Journal on Applied Dynamical Systems, 2 (2003), 609-646.  doi: 10.1137/S1111111103419896.  Google Scholar

[72]

M. Timme, F. Wolf and T. Geisel, Unstable attractors induce perpetual synchronization and desynchronization, Chaos, 13 (2003), 377-387.  doi: 10.1063/1.1501274.  Google Scholar

[73]

I. Tsuda, Hypotheses on the functional roles of chaotic transitory dynamics, Chaos, 19 (2009), 015113, 10pp. doi: 10.1063/1.3076393.  Google Scholar

[74]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Applied Mathematical Sciences, Springer, 1994, URL https://books.google.co.uk/books?id= 0vHsHoOoPlQC. doi: 10.1007/978-1-4612-4312-0.  Google Scholar

[75]

J. Wordsworth and P. Ashwin, Spatiotemporal coding of inputs for a system of globally coupled phase oscillators, Phys. Rev. E, 78 (2008), 066203, 10pp, URL https://link.aps.org/doi/10.1103/PhysRevE.78.066203. doi: 10.1103/PhysRevE.78.066203.  Google Scholar

show all references

References:
[1]

N. Agarwal and M. J. Field, Dynamical equivalence of network architecture for coupled dynamical systems Ⅰ: asymmetric inputs, Nonlinearity, 23 (2010), 1245-1268.  doi: 10.1088/0951-7715/23/6/001.  Google Scholar

[2]

N. Agarwal and M. J. Field, Dynamical equivalence of network architecture for coupled dynamical systems Ⅱ: general case, Nonlinearity, 23 (2010), 1269-1289.  doi: 10.1088/0951-7715/23/6/002.  Google Scholar

[3]

M. Aguiar, P. Ashwin, A. Dias and M. Field, Dynamics of coupled cell networks: Synchrony, heteroclinic cycles and inflation, Journal of Nonlinear Science, 21 (2011), 271-323.  doi: 10.1007/s00332-010-9083-9.  Google Scholar

[4]

M. A. D. Aguiar, A. P. S. Dias, M. Golubitsky and M. da C A Leitee, Bifurcations from regular quotient networks: A first insight, Physica D, 238 (2009), 137-155.  doi: 10.1016/j.physd.2008.10.006.  Google Scholar

[5]

M. A. D. Aguiar, S. B. S. D. Castro and I. S. Labouriau, Dynamics near a heteroclinic network, Nonlinearity, 18 (2005), 391–414, URL http://stacks.iop.org/0951-7715/18/i=1/a=019. doi: 10.1088/0951-7715/18/1/019.  Google Scholar

[6]

P. Ashwin and P. Chossat, Attractors for robust heteroclinic cycles with continua of connections, Journal of Nonlinear Science, 8 (1998), 103–129, URL http://dx.doi.org/10.1007/s003329900045. doi: 10.1007/s003329900045.  Google Scholar

[7]

P. Ashwin and C. Postlethwaite, On designing heteroclinic networks from graphs, Physica D, 265 (2013), 26-39.  doi: 10.1016/j.physd.2013.09.006.  Google Scholar

[8]

P. Ashwin, S. Coombes and R. Nicks, Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience, The Journal of Mathematical Neuroscience, 6 (2016), Art. 2, 92 pp, URL https://doi.org/10.1186/s13408-015-0033-6. doi: 10.1186/s13408-015-0033-6.  Google Scholar

[9]

P. Ashwin and M. Field, Heteroclinic networks in coupled cell systems, Archive for Rational Mechanics and Analysis, 148 (1999), 107-143.  doi: 10.1007/s002050050158.  Google Scholar

[10]

P. Ashwin and Ö. Karabacak, Robust heteroclinic behaviour, synchronization, and ratcheting of coupled oscillators, Dynamics, Games and Science. II, Springer Proc. Math., Springer, Heidelberg, 2 (2011), 125–140, URL http://dx.doi.org/10.1007/978-3-642-14788-3_10. doi: 10.1007/978-3-642-14788-3_10.  Google Scholar

[11]

P. Ashwin, Ö. Karabacak and T. Nowotny, Criteria for robustness of heteroclinic cycles in neural microcircuits, The Journal of Mathematical Neuroscience, 1 (2011), Art. 13, 18 pp, URL http://dx.doi.org/10.1186/2190-8567-1-13. doi: 10.1186/2190-8567-1-13.  Google Scholar

[12]

P. Ashwin and C. Postlethwaite, Designing heteroclinic and excitable networks in phase space using two populations of coupled cells, Journal of Nonlinear Science, 26 (2016), 345–364, URL http://dx.doi.org/10.1007/s00332-015-9277-2. doi: 10.1007/s00332-015-9277-2.  Google Scholar

[13]

P. Ashwin and C. Postlethwaite, Quantifying noisy attractors: From heteroclinic to excitable networks, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1989–2016, URL http://dx.doi.org/10.1137/16M1061813. doi: 10.1137/16M1061813.  Google Scholar

[14]

P. Ashwin and M. Timme, Unstable attractors: Existence and robustness in networks of oscillators with delayed pulse coupling, Nonlinearity, 18 (2005), 2035-2060.  doi: 10.1088/0951-7715/18/5/009.  Google Scholar

[15]

Y. Bakhtin, Noisy heteroclinic networks, Probability Theory and Related Fields, 150 (2011), 1–42, URL http://dx.doi.org/10.1007/s00440-010-0264-0. doi: 10.1007/s00440-010-0264-0.  Google Scholar

[16]

C. Bick and M. I. Rabinovich, On the occurrence of stable heteroclinic channels in lotkavolterra models, Dynamical Systems, 25 (2010), 97–110, URL http://dx.doi.org/10.1080/14689360903322227. doi: 10.1080/14689360903322227.  Google Scholar

[17]

W. Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity, 7 (1994), 1367–1384, URL http://stacks.iop.org/0951-7715/7/i=5/a=006. doi: 10.1088/0951-7715/7/5/006.  Google Scholar

[18]

F. H. Busse and R. M. Clever, Nonstationary convection in a rotating system, Springer Berlin Heidelberg, Berlin, Heidelberg, 1979,376–385, URL http://dx.doi.org/10.1007/978-3-642-67220-0_39. Google Scholar

[19]

S. B. S. D. Castro and A. Lohse, Switching in heteroclinic networks, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1085–1103, URL http://dx.doi.org/10.1137/15M1042176. doi: 10.1137/15M1042176.  Google Scholar

[20]

S. B. Castro and A. Lohse, Stability in simple heteroclinic networks in, Dynamical Systems, 29 (2014), 451–481, URL http://dx.doi.org/10.1080/14689367.2014.940853. doi: 10.1080/14689367.2014.940853.  Google Scholar

[21]

M. Dellnitz, M. Field, M. Golubitsky, J. Ma and A. Hohmann, Cycling chaos, International Journal of Bifurcation and Chaos, 5 (1995), 1243-1247.  doi: 10.1142/S0218127495000909.  Google Scholar

[22]

A. P. S. Dias and I. Stewart, Linear equivalence and ode-equivalence for coupled cell networks, Nonlinearity, 18 (2005), 1003–1020, URL http://stacks.iop.org/0951-7715/18/i=3/a=004. doi: 10.1088/0951-7715/18/3/004.  Google Scholar

[23]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[24]

G. L. dos Reis, Structural stability of equivariant vector fields on two-manifolds, Trans. Amer. Math. Soc., 283 (1984), 633–643, URL http://dx.doi.org/10.2307/1999151. doi: 10.1090/S0002-9947-1984-0737889-8.  Google Scholar

[25]

G. B. Ermentrout, A Guide to XPPAUT for Researchers and Students, SIAM, Pittsburgh, 2002. Google Scholar

[26]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Chapman & Hall/CRC Research Notes in Mathematics Series, Taylor & Francis, 1996, URL https://books.google.co.uk/books?id=4dqrLHmicR8C.  Google Scholar

[27]

M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001–1043, URL http://stacks.iop.org/0951-7715/4/i=4/a=001. doi: 10.1088/0951-7715/4/4/001.  Google Scholar

[28]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185–205, URL http://dx.doi.org/10.2307/1998153. doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[29]

M. J. Field, Heteroclinic networks in homogeneous and heterogeneous identical cell systems, Journal of Nonlinear Science, 25 (2015), 779–813, URL http://dx.doi.org/10.1007/s00332-015-9241-1. doi: 10.1007/s00332-015-9241-1.  Google Scholar

[30]

M. J. Field, Patterns of desynchronization and resynchronization in heteroclinic networks, Nonlinearity, 30 (2017), 516–557, URL http://stacks.iop.org/0951-7715/30/i=2/a=516. doi: 10.1088/1361-6544/aa4f48.  Google Scholar

[31]

M. Field, Combinatorial dynamics, Dynamical Systems, 19 (2004), 217-243.  doi: 10.1080/14689360410001729379.  Google Scholar

[32]

A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM Journal on Applied Mathematics, 52 (1992), 1476–1489, URL https://doi.org/10.1137/0152085. doi: 10.1137/0152085.  Google Scholar

[33]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate texts in mathematics, Springer-Verlag, New York-Heidelberg, 1973, URL https://books.google.co.uk/books?id=y9BFAQAAIAAJ.  Google Scholar

[34]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.  Google Scholar

[35]

M. Golubitsky, I. Stewart and A. Török, Patterns of synchrony in coupled cell networks with multiple arrows, SIAM Journal on Applied Dynamical Systems, 4 (2005), 78–100, URL http://dx.doi.org/10.1137/040612634. doi: 10.1137/040612634.  Google Scholar

[36]

J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. So, 103 (1988), 189-192.  doi: 10.1017/S0305004100064732.  Google Scholar

[37]

D. Hansel, G. Mato and C. Meunier, Clustering and slow switching in globally coupled phase oscillators, Physical Review E, 48 (1993), 3470-3477.  doi: 10.1103/PhysRevE.48.3470.  Google Scholar

[38]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, no. no. 583 in Lecture Notes in Mathematics, Springer-Verlag, 1977, URL https://books.google.co.uk/books?id=-rgZAQAAIAAJ.  Google Scholar

[39]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[40]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press Cambridge [England]; New York, 1988, URL http://www.loc.gov/catdir/toc/cam029/88017571.html. Google Scholar

[41]

C. Hou and M. Golubitsky, An example of symmetry breaking to heteroclinic cycles, Journal of Differential Equations, 133 (1997), 30–48, URL http://www.sciencedirect.com/science/article/pii/S0022039696932015. doi: 10.1006/jdeq.1996.3201.  Google Scholar

[42]

K. Kaneko and I. Tsuda, Focus issue on chaotic itinerancy, Chaos, 13 (2003), 926-936.  doi: 10.1063/1.1607783.  Google Scholar

[43]

O. Karabacak and P. Ashwin, Heteroclinic ratchets in networks of coupled oscillators, Journal of Nonlinear Science, 20 (2010), 105-129.  doi: 10.1007/s00332-009-9053-2.  Google Scholar

[44]

V. Kirk and M. Silber, A competition between heteroclinic cycles, Nonlinearity, 7 (1994), 1605–1621, URL http://stacks.iop.org/0951-7715/7/i=6/a=005. doi: 10.1088/0951-7715/7/6/005.  Google Scholar

[45]

C. Kirst and M. Timme, From networks of unstable attractors to heteroclinic switching Physical Review E, 78 (2008), 065201, 4pp. doi: 10.1103/PhysRevE.78.065201.  Google Scholar

[46]

W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos: An Interdisciplinary Journal of Nonlinear Science, 10 (2000), 427–469, URL http://dx.doi.org/10.1063/1.166509. doi: 10.1063/1.166509.  Google Scholar

[47]

H. Kori and Y. Kuramoto, Slow switching in globally coupled oscillators: Robustness and occurence through delayed coupling, Physical Review E, 63 (2001), 046214.  doi: 10.1103/PhysRevE.63.046214.  Google Scholar

[48]

M. Krupa, Robust heteroclinic cycles, Journal of Nonlinear Science, 7 (1997), 129-176.  doi: 10.1007/BF02677976.  Google Scholar

[49]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. Ⅰ, Ergodic Theory and Dynamical Systems, 15 (1995), 121-147.  doi: 10.1017/S0143385700008270.  Google Scholar

[50]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. Ⅱ, Proceedings of the Royal Socity of Edinburgh Sect. A, 134 (2004), 1177-1197.  doi: 10.1017/S0308210500003693.  Google Scholar

[51]

R. Lauterbach, S. Maier-Paape and E. Reissner, A systematic study of heteroclinic cycles in dynamical systems with broken symmetries, Proceedings/Royal Society of Edinburgh/Section A, Mathematics, 126 (1996), 885–909, URL http://publications.rwth-aachen.de/record/145691. doi: 10.1017/S030821050002312X.  Google Scholar

[52]

R. Lauterbach and M. Roberts, Heteroclinic cycles in dynamical systems with broken spherical symmetry, Journal of Differential Equations, 100 (1992), 22–48, URL http://www.sciencedirect.com/science/article/pii/0022039692901246. doi: 10.1016/0022-0396(92)90124-6.  Google Scholar

[53]

A. Lohse, Unstable attractors: Existence and stability indices, Dynamical Systems, 30 (2015), 324–332, URL http://dx.doi.org/10.1080/14689367.2015.1041879. doi: 10.1080/14689367.2015.1041879.  Google Scholar

[54]

A. Lohse and S. B. S. D. Castro, Construction of heteroclinic networks in $ R^4$, Nonlinearity, 29 (2016), 3677-3695.  doi: 10.1088/0951-7715/29/12/3677.  Google Scholar

[55]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29 (1975), 243-253.  doi: 10.1137/0129022.  Google Scholar

[56]

I. Melbourne, An example of a nonasymptotically stable attractor, Nonlinearity, 4 (1991), 835–844, URL http://stacks.iop.org/0951-7715/4/i=3/a=010. doi: 10.1088/0951-7715/4/3/010.  Google Scholar

[57]

F. S. Neves and M. Timme, Computation by switching in complex networks of states, Physical Review Letters, 109 (2012), 018701. Google Scholar

[58]

E. Nijholt, B. Rink and J. Sanders, Graph fibrations and symmetries of network dynamics, Journal of Differential Equations, 261 (2016), 4861–4896, URL http://www.sciencedirect.com/science/article/pii/S0022039616301784. doi: 10.1016/j.jde.2016.07.013.  Google Scholar

[59]

G. Orosz, J. Moehlis and P. Ashwin, Designing the dynamics of globally coupled oscillators, Progress of Theoretical Physics, 122 (2009), 611-630.  doi: 10.1143/PTP.122.611.  Google Scholar

[60]

J. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York, 1982.  Google Scholar

[61]

M. Peixoto, On an approximation theorem of kupka and smale, Journal of Differential Equations, 3 (1967), 214–227, URL http://www.sciencedirect.com/science/article/pii/0022039667900265. doi: 10.1016/0022-0396(67)90026-5.  Google Scholar

[62]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, no. 12 in Cambridge Nonlinear Science Series, Cambridge University Press, Cambridhe, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[63]

O. Podvigina and P. Ashwin, On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, 24 (2011), 887–929, URL http://stacks.iop.org/0951-7715/24/i=3/a=009. doi: 10.1088/0951-7715/24/3/009.  Google Scholar

[64]

M. Rabinovich, P. Varona, A. Selverston and H. Abarbanel, Dynamical principles in neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265.  doi: 10.1103/RevModPhys.78.1213.  Google Scholar

[65]

M. Rabinovich, P. Varona, I. Tristan and V. Afraimovich, Chunking dynamics: heteroclinics in mind, Frontiers in Computational Neuroscience, 8 (2014), 22, URL http://journal.frontiersin.org/article/10.3389/fncom.2014.00022. doi: 10.3389/fncom.2014.00022.  Google Scholar

[66]

M. Rabinovich, R. Huerta and G. Laurent, Transient dynamics for neural processing, Science, 321 (2008), 48–50, URL http://science.sciencemag.org/content/321/5885/48. Google Scholar

[67]

B. Rink and J. Sanders, Coupled cell networks and their hidden symmetries, SIAM Journal on Mathematical Analysis, 46 (2014), 1577-1609.  doi: 10.1137/130916242.  Google Scholar

[68]

B. Rink and J. Sanders, Coupled cell networks: semigroups, lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509–3548, Art no PII S0002-9947(2014)06221-1. doi: 10.1090/S0002-9947-2014-06221-1.  Google Scholar

[69]

B. Sandstede and A. Scheel, Forced symmetry breaking of homoclinic cycles, Nonlinearity, 8 (1995), 333–365, URL http://stacks.iop.org/0951-7715/8/i=3/a=003. doi: 10.1088/0951-7715/8/3/003.  Google Scholar

[70]

M. Schub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[71]

I. Stewart, M. Golubitsky and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM Journal on Applied Dynamical Systems, 2 (2003), 609-646.  doi: 10.1137/S1111111103419896.  Google Scholar

[72]

M. Timme, F. Wolf and T. Geisel, Unstable attractors induce perpetual synchronization and desynchronization, Chaos, 13 (2003), 377-387.  doi: 10.1063/1.1501274.  Google Scholar

[73]

I. Tsuda, Hypotheses on the functional roles of chaotic transitory dynamics, Chaos, 19 (2009), 015113, 10pp. doi: 10.1063/1.3076393.  Google Scholar

[74]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Applied Mathematical Sciences, Springer, 1994, URL https://books.google.co.uk/books?id= 0vHsHoOoPlQC. doi: 10.1007/978-1-4612-4312-0.  Google Scholar

[75]

J. Wordsworth and P. Ashwin, Spatiotemporal coding of inputs for a system of globally coupled phase oscillators, Phys. Rev. E, 78 (2008), 066203, 10pp, URL https://link.aps.org/doi/10.1103/PhysRevE.78.066203. doi: 10.1103/PhysRevE.78.066203.  Google Scholar

Figure 1.  (a) Coupled cell structure of the Guckenheimer-Holmes system, see Example 2. (b) The Guckenheimer-Holmes heteroclinic cycle in phase space as the limiting set of a trajectory between the saddle equilibria $(x,y,z) = (\xi,0,0)$, $(0,\xi,0)$ and $(0,0,\xi)$ for some $\xi>0$, with active cells along the cycle indicated. Note that this is part of a larger network of twelve connections between the six equilibria $(\pm \xi,0,0)$, $(0,\pm \xi,0)$ and $(0,0,\pm \xi)$; typical initial conditions limit to one of eight possible cycles. (c) Corresponding time-series showing asymptotic slowing down as the trajectory approaches the heteroclinic cycle.
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Figure 2.  A coupled cell network with two cell types and three edge types, graphically indicated by different box and arrow styles. As in $1.$ of Def. 3.1, edges of the same type have equivalent sources and targets. Note also that the input sets $I(c_1)$ and $I(c_2)$, and $I(c_3)$ and $I(c_4)$ respectively, consist of the same number of edges of each type, which is the content of $2.$ in Def. 3.1.
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3]: note the presence of two edge types.">Figure 3.  The three cell network architecture of $\mathcal{N}_3$ from [3]: note the presence of two edge types.
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Fig. 5: calculations using xppaut [25].">Figure 4.  (a) The heteroclinic cycle of Example 4 seen in phase space, with two nodes and four connections. (b) Time-series for a trajectory along the cycle in (a) with added independent noise of amplitude $10^{-7}$ in all components and initial condition $(x,y,z) = (1,1.001,0.999)$. Observe heteroclinic switching between two synchronized states $x = y = z = \pm 1$ but two types of switching owing to the two branches of the unstable manifold shown in Fig. 5: calculations using xppaut [25].
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25] and a Runge-Kutta integrator with time step $0.05$.">Figure 5.  The system (2, 5) in the synchrony subspace ${\bf{P}}_2$. Green lines show the nullcline for the $y$-component and red lines show the nullcline for the $x$-component, with parameters as in text. Blue curves are trajectories that approximate of the unstable manifold of $(x,y,z) = (-1,-1,-1)$: note that both branches of the unstable manifold are asymptotic to the sink at $(1,1,1)$. The numerical integration of (2, 5) is performed using xppaut [25] and a Runge-Kutta integrator with time step $0.05$.
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Figure 6.  Time-series showing the $p_i$ components of trajectories exploring a realization of the Kirk-Silber network using (6, 7), with low amplitude noise. (a) shows a typical time-series for a heteroclinic network realization - note that the system moves around between four equilibria $P_i$, where $p_i = 1$ and $p_j = 0$ for $j\neq i$. The network is the union of the two cycles $P_1\rightarrow P_2\rightarrow P_4\rightarrow P_1$ and $P_1\rightarrow P_3\rightarrow P_4\rightarrow P_1$. (b) shows the case where one connection from $P_1$ has been made more unstable and in addition the connection from $P_3$ has been made excitable rather than heteroclinic. In particular, the noise-induced escape times for the heteroclinic transitions are fairly uniform while those for excitable transition are more widely distributed, consistent with exponential distribution.
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