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

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

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  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.

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.

[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.

[3]

M. AguiarP. AshwinA. 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.

[4]

M. A. D. AguiarA. P. S. DiasM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

M. DellnitzM. FieldM. GolubitskyJ. Ma and A. Hohmann, Cycling chaos, International Journal of Bifurcation and Chaos, 5 (1995), 1243-1247. doi: 10.1142/S0218127495000909.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[37]

D. HanselG. 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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[48]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[57]

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

[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.

[59]

G. OroszJ. 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.

[60]

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

[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.

[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.

[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.

[64]

M. RabinovichP. VaronaA. Selverston and H. Abarbanel, Dynamical principles in neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265. doi: 10.1103/RevModPhys.78.1213.

[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.

[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.

[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.

[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.

[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.

[70]

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

[71]

I. StewartM. 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.

[72]

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

[73]

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

[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.

[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.

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.

[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.

[3]

M. AguiarP. AshwinA. 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.

[4]

M. A. D. AguiarA. P. S. DiasM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

M. DellnitzM. FieldM. GolubitskyJ. Ma and A. Hohmann, Cycling chaos, International Journal of Bifurcation and Chaos, 5 (1995), 1243-1247. doi: 10.1142/S0218127495000909.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[37]

D. HanselG. 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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[48]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[57]

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

[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.

[59]

G. OroszJ. 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.

[60]

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

[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.

[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.

[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.

[64]

M. RabinovichP. VaronaA. Selverston and H. Abarbanel, Dynamical principles in neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265. doi: 10.1103/RevModPhys.78.1213.

[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.

[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.

[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.

[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.

[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.

[70]

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

[71]

I. StewartM. 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.

[72]

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

[73]

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

[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.

[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.

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.
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.
Figure 3.  The three cell network architecture of $\mathcal{N}_3$ from [3]: note the presence of two edge types.
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].
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 $0.05$.
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.
[1]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[2]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[3]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[4]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[5]

Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045

[6]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[7]

Reinhard Racke. Instability of coupled systems with delay. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753

[8]

Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187

[9]

Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078

[10]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449

[11]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[12]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

[13]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361

[14]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935

[15]

Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007

[16]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91

[17]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[18]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432

[19]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[20]

Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (59)
  • HTML views (155)
  • Cited by (0)

Other articles
by authors

[Back to Top]