192
Z. Zheng
and explore the order-parameter dynamics in terms of self-consistency approach,
the Ott-Antonsen ansatz or the ensemble order parameter technique. Symmetries
exist in many networks of coupled phase-oscillator systems, Watanabe and Strogatz
introduced the Mobius transformation (called the WS transformation) to reduce the
high-dimensional dynamics to a three-dimensional space, and the two-dimensional
invariant manifold obtained by using the OA ansatz is in fact a sub-manifold in the
WS space. We will not discuss these topics here. Interested readers may refer [6, 92,
93] for more information.
References
1. Needham, J.: Science and Civilisation in China, vol. 4. Science Press, Shanghai, Physics and
related technologies (2010)
2. Laplace, P. S.: A philosophical essay on probabilities. Truscott, F. W., Emory, F. L., trans.
New York: Wiley, (1902)
3. Anderson, P.W.: More is different: Broken symmetry and the nature of the hierarchical
structure of science. Science 177(4047), 393–396 (1972)
4. Palsson, B.O.: Systems Biology: Properties of Reconstructed Networks. Cambridge University Press, Cambridge (2006)
5. Meyers, R.A. (ed.): Encyclopedia of Complexity and Systems Science. Springer-Verlag,
Berlin (2009)
6. Zheng, Z. G.: Emergence Dynamics in Complex Systems: From Synchronization to Collective
Transport, vols. I and II, Science Press, Beijing (2019).
7. Prigogine, I., Nicolis, G.: Self-organization in Non-equilibrium Systems, from Dissipative
Structures to Order Through Fluctuations. Wiley, New Jersey (1977)
8. Kauffman, S.A.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford
University Press, USA (1993)
9. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)
10. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.
57, 617 (1985)
11. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics. Chemistry,
and Engineering, Westview Press;CRC Press, Biology (2018)
12. Fuchs, A.: Nonlinear Dynamics in Complex Systems: Theory and Applications for the Life-.
Springer, Neuro- and Natural Sciences (2012)
13. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72
(1952)
14. Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851
(1993)
15. Cross, M., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems.
Cambridge University Press, Cambridge (2009)
16. Gromov, M., Capasso, V., Gromov, M., Harel-Bellan, A., Morozova, N., Pritchard, L.L. (eds.):
Pattern Formation in Morphogenesis: Problems and Mathematical Issues. Springer-Verlag,
Berlin Heidelberg (2013)
17. Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)
18. Stanley, H.E. (ed.): Introduction to Phase Transitions and Critical Phenomena. Oxford
University Press, Oxford (1987)
19. Ma, S.K.: Modern theory of critical phenomena. Benjamin, New York (1976)
20. Haken, H.: Synergetics: An Introduction Nonequilibrium Phase Transitions and SelfOrganization in Physics. Springer-Verlag, Berlin Heidelberg, Chemistry and Biology (1978)
Z. Zheng
and explore the order-parameter dynamics in terms of self-consistency approach,
the Ott-Antonsen ansatz or the ensemble order parameter technique. Symmetries
exist in many networks of coupled phase-oscillator systems, Watanabe and Strogatz
introduced the Mobius transformation (called the WS transformation) to reduce the
high-dimensional dynamics to a three-dimensional space, and the two-dimensional
invariant manifold obtained by using the OA ansatz is in fact a sub-manifold in the
WS space. We will not discuss these topics here. Interested readers may refer [6, 92,
93] for more information.
References
1. Needham, J.: Science and Civilisation in China, vol. 4. Science Press, Shanghai, Physics and
related technologies (2010)
2. Laplace, P. S.: A philosophical essay on probabilities. Truscott, F. W., Emory, F. L., trans.
New York: Wiley, (1902)
3. Anderson, P.W.: More is different: Broken symmetry and the nature of the hierarchical
structure of science. Science 177(4047), 393–396 (1972)
4. Palsson, B.O.: Systems Biology: Properties of Reconstructed Networks. Cambridge University Press, Cambridge (2006)
5. Meyers, R.A. (ed.): Encyclopedia of Complexity and Systems Science. Springer-Verlag,
Berlin (2009)
6. Zheng, Z. G.: Emergence Dynamics in Complex Systems: From Synchronization to Collective
Transport, vols. I and II, Science Press, Beijing (2019).
7. Prigogine, I., Nicolis, G.: Self-organization in Non-equilibrium Systems, from Dissipative
Structures to Order Through Fluctuations. Wiley, New Jersey (1977)
8. Kauffman, S.A.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford
University Press, USA (1993)
9. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)
10. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.
57, 617 (1985)
11. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics. Chemistry,
and Engineering, Westview Press;CRC Press, Biology (2018)
12. Fuchs, A.: Nonlinear Dynamics in Complex Systems: Theory and Applications for the Life-.
Springer, Neuro- and Natural Sciences (2012)
13. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72
(1952)
14. Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851
(1993)
15. Cross, M., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems.
Cambridge University Press, Cambridge (2009)
16. Gromov, M., Capasso, V., Gromov, M., Harel-Bellan, A., Morozova, N., Pritchard, L.L. (eds.):
Pattern Formation in Morphogenesis: Problems and Mathematical Issues. Springer-Verlag,
Berlin Heidelberg (2013)
17. Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)
18. Stanley, H.E. (ed.): Introduction to Phase Transitions and Critical Phenomena. Oxford
University Press, Oxford (1987)
19. Ma, S.K.: Modern theory of critical phenomena. Benjamin, New York (1976)
20. Haken, H.: Synergetics: An Introduction Nonequilibrium Phase Transitions and SelfOrganization in Physics. Springer-Verlag, Berlin Heidelberg, Chemistry and Biology (1978)