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Complex Systems

  • Jakub SawickiEmail author
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

This Chapter is concerned with the general concepts of complex systems.

References

  1. 1.
    Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Atay FM (ed) (2010) Complex time-delay systems, understanding complex systems. Springer, BerlinzbMATHGoogle Scholar
  3. 3.
    Battiston F, Nicosia V, Latora V (2014) Structural measures for multiplex networks. Phys Rev E 89:032804Google Scholar
  4. 4.
    Benoit EE, Callot JL, Diener F, Diener MM (1981) Chasse au canard (première partie). Collect Math 32:37–119Google Scholar
  5. 5.
    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Boccaletti S, Bianconi G, Criado R, del Genio CI, Gómez-Gardeñes J, Romance M, Sendiña Nadal I, Wang Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544:1–122ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Choe CU, Dahms T, Hövel P, Schöll E (2010) Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states. Phys Rev E 81:025205(R)Google Scholar
  8. 8.
    Cozzo E, De Arruda GF, Rodrigues FA, Moreno Y (2018) Multiplex networks: basic formalism and structural properties. Springer, BerlinCrossRefGoogle Scholar
  9. 9.
    Criado R, Flores J, GarcÃa del Amo A, Gómez-Gardeñes J, Romance M (2012) A mathematical model for networks with structures in the mesoscale. Int J Comput Math 89:291MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dahms T, Lehnert J, Schöll E (2012) Cluster and group synchronization in delay-coupled networks. Phys Rev E 86:016202Google Scholar
  11. 11.
    Erneux T (2009) Applied delay differential equations. Springer, BerlinzbMATHGoogle Scholar
  12. 12.
    Euler L (1741) Solutio problematis ad geometriam situs pertinentis. Commentarii Acad Sci Petropolitanae 8:128–140Google Scholar
  13. 13.
    Farmer JD (1982) Chaotic attractors of an infinite-dimensional dynamical system. Phys D 4:366Google Scholar
  14. 14.
    FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445–466ADSCrossRefGoogle Scholar
  15. 15.
    Flunkert V (2011) Delay-coupled complex systems, Springer theses. Springer, HeidelbergCrossRefGoogle Scholar
  16. 16.
    Flunkert V, Fischer I, Schöll E (2013) Dynamics, control and information in delay-coupled systems. Theme Issue of Phil Trans R Soc A 371:20120465Google Scholar
  17. 17.
    Fridman E (2014) Introduction to time-delay systems: analysis and control. Springer, BerlinCrossRefGoogle Scholar
  18. 18.
    Hövel P (2010) Control of complex nonlinear systems with delay, Springer theses. Springer, HeidelbergCrossRefGoogle Scholar
  19. 19.
    Heinrich M, Dahms T, Flunkert V, Teitsworth SW, Schöll E (2010) Symmetry breaking transitions in networks of nonlinear circuit elements. New J Phys 12:113030ADSCrossRefGoogle Scholar
  20. 20.
    Hodgkin AL (1948) The local electric changes associated with repetitive action in a medullated axon. J Physiol 107:165Google Scholar
  21. 21.
    Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544CrossRefGoogle Scholar
  22. 22.
    Just W, Pelster A, Schanz M, Schöll E (2010) Delayed complex systems. Theme Issue of Phil Trans R Soc A 368:301–513Google Scholar
  23. 23.
    Keane A, Krauskopf B, Postlethwaite CM (2017) Climate models with delay differential equations. Chaos 27:114309MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kivelä M, Arenas A, Barthélemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. J Complex Netw 2:203–271CrossRefGoogle Scholar
  25. 25.
    Landau LD (1944) On the problem of turbulence. C R Acad Sci UESS 44:311Google Scholar
  26. 26.
    Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701Google Scholar
  27. 27.
    Lehnert J (2010) Dynamics of neural networks with delay. Master’s thesis, Technische Universität BerlinGoogle Scholar
  28. 28.
    Lehnert J (2016) Controlling synchronization patterns in complex networks, Springer theses. Springer, HeidelbergCrossRefGoogle Scholar
  29. 29.
    Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070CrossRefGoogle Scholar
  30. 30.
    Neves KW (1975) Automatic integration of functional differential equations: an approach. ACM Trans Math Softw 1:357MathSciNetCrossRefGoogle Scholar
  31. 31.
    Neves KW, Feldstein A (1976) Characterization of jump discontinuities for state dependent delay differential equations. J Math Anal Appl 5:689MathSciNetCrossRefGoogle Scholar
  32. 32.
    Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Newman MEJ, Barabási AL, Watts DJ (2006) The structure and dynamics of networks. Princeton University Press, Princeton, USAzbMATHGoogle Scholar
  34. 34.
    Newman MEJ (2010) Networks: an introduction. Oxford University Press Inc, New YorkCrossRefGoogle Scholar
  35. 35.
    Nicosia V, Latora V (2015) Measuring and modeling correlations in multiplex networks. Phys Rev E 92:032805Google Scholar
  36. 36.
    Ott E (2002) Chaos in dynamical systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  37. 37.
    Pecora LM, Carroll TL (1998) Master stability functions for synchronized coupled systems. Phys Rev Lett 80:2109–2112ADSCrossRefGoogle Scholar
  38. 38.
    Rosin DP, Callan KE, Gauthier DJ, Schöll E (2011) Pulse-train solutions and excitability in an optoelectronic oscillator. Europhys Lett 96:34001ADSCrossRefGoogle Scholar
  39. 39.
    Schöll E (2001) Nonlinear spatio-temporal dynamics and chaos in semiconductors. Nonlinear science series, vol 10. Cambridge University Press, CambridgeGoogle Scholar
  40. 40.
    Schöll E, Schuster HG (eds) Handbook of chaos control. Second completely revised and enlarged edition. Wiley-VCH, WeinheimGoogle Scholar
  41. 41.
    Schöll E (2013) Synchronization in delay-coupled complex networks. In: Sun J-Q, Ding Q (Eds) Advances in analysis and control of time-delayed dynamical systems, Chapter 4. World Scientific, Singapore, pp 57–83CrossRefGoogle Scholar
  42. 42.
    Schöll E, Klapp SHL, Hövel P (2016) Control of self-organizing nonlinear systems. Springer, BerlinCrossRefGoogle Scholar
  43. 43.
    Shima S, Kuramoto Y (2004) Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys Rev E 69:036213Google Scholar
  44. 44.
    Solá L, Romance M, Criado R, Flores J, Garcia del Amo A, Boccaletti S (2013) Eigenvector centrality of nodes in multiplex networks. Chaos 23:033131CrossRefGoogle Scholar
  45. 45.
    Sprott JC (2007) A simple chaotic delay differential equation. Phys Lett A 366:397ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Strogatz SH (1994) Nonlinear dynamics and chaos. Westview Press, Cambridge, MAGoogle Scholar
  47. 47.
    Stuart JT (1958) On the non-linear mechanics of hydrodynamic stability. J Fluid Mech 4:1ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Sun JQ, Ding G (2013) Advances in analysis and control of time-delayed dynamical systems. World Scientific, SingaporeCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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