
- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
On bounding exact models of epidemic spread on networks
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 |
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.
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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |





[1] |
Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 |
[2] |
Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248 |
[3] |
Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097 |
[5] |
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 and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785 |
[6] |
Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 |
[7] |
Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045 |
[8] |
Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185 |
[9] |
Reinhard Racke. Instability of coupled systems with delay. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753 |
[10] |
Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078 |
[11] |
Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187 |
[12] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[13] |
Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769 |
[14] |
Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569 |
[15] |
El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449 |
[16] |
Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 |
[17] |
Mauricio Achigar. Extensions of expansive dynamical systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399 |
[18] |
Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 |
[19] |
John Erik Fornæss. Sustainable dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361 |
[20] |
Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]