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0343301308
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note
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Half of the bifurcations occur on each point if N is odd, while one more bifurcation happens on the initially stable point if N is even.
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7
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0343301309
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note
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In fact, only one attractor can emerge from the variety of Hopf bifurcations of a saddle-node pair of fixed points, but additional attractors may arise from secondary processes.
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8
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J. Farjas, J. I. Rosell, R. Herrero, R. Pons, F. Pi, and G. Orriols, Physica D 95, 107 (1996).
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Clarendon, Oxford
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(N-1)b] = N. See, e.g., S. Barnet and R. G. Cameron, Introduction to Mathematical Control Theory, 2nd ed. (Clarendon, Oxford, 1985), p. 85.
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Barnet, S.1
Cameron, R.G.2
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0343736841
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note
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N.
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19
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0008494528
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A. Wolf, B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16, 285 (1985).
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21
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0342431722
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note
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Complex sequences of bifurcations may occur near homoclinicity when the saddle set fulfills proper conditions [19], but mode mixing happens independently.
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22
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0002715889
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edited by V. I. Arnold Springer-Verlag, Berlin
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For a review see V. I. Arnold, V. S. Afrajmovich, Yyu. S. Il'yashenko, and L. P. Shil'nikov, in Dynamical Systems V, edited by V. I. Arnold (Springer-Verlag, Berlin, 1994), pp. 1-205.
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Arnold, V.I.1
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Shil'nikov, L.P.4
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25
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0342866230
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note
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Provided that a stable fixed point exists at a certain moment.
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26
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0003324839
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edited by T. Belford and J. Switf Cambridge University Press, Cambridge
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P. Glendinning, in New Directions in Dynamical Systems, edited by T. Belford and J. Switf (Cambridge University Press, Cambridge, 1988), p. 120.
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Glendinning, P.1
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R. Herrero, J. Farjas, R. Pons, F. Pi, and G. Orriols, Phys. Rev. E 57, 5366 (1998).
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Cambridge University Press, Cambridge
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A. Goldbeter, Biochemical Oscillations and Cellular Rhythms (Cambridge University Press, Cambridge, 1996); Dynamic Biological Networks, edited by R. M. Harris-Warrik, E. Marder, A. I. Selverton, and M. Moulins (MIT Press, Cambridge, 1992).
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Goldbeter, A.1
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MIT Press, Cambridge
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A. Goldbeter, Biochemical Oscillations and Cellular Rhythms (Cambridge University Press, Cambridge, 1996); Dynamic Biological Networks, edited by R. M. Harris-Warrik, E. Marder, A. I. Selverton, and M. Moulins (MIT Press, Cambridge, 1992).
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Selverton, A.I.3
Moulins, M.4
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0343301305
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note
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An interesting example to think about is the wake-sleep cycle of a brain.
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31
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0003851376
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Cambridge University Press, Cambridge
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P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, Cambridge, 1996).
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Holmes, P.1
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Berkooz, G.3
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