Analytic solutions of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 2, 1998, Pages 347-357

  Briggs, K M ^a,b   Dixon, T W ^a,c   Szekeres, G ^d  

^a UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c UNIVERSITY OF OTAGO   (New Zealand)

^d NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032008442     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000206     Document Type: Article

References (31)

1. Briggs, K. M. [1991] "A precise computation of the Feigenbaum constants," Math. Comp. 57, 435-439.
2. Briggs, K. M., Quispel, G. R. W. & Thompson, C. [1991] "Feigenvalues for Mandelsets," J. Phys. A24, 3363-3368.
3. Briggs, K. M. [1998a] "A C++ package for automatic differentiation," http://www-epidem.plantsci.cam.ac.uk/̃kbriggs/math++1.9/a d.shar
4. Briggs, K. M. [1998b] "A C++ package for formal power series,"http://www-epidem.plantsci.cam.ac.uk/̃kbrigg s/math++1.9/ps.shar
5. Briggs, K. M. [1995a] "Formal solution of functional equations via algorithmic differentiation," Aust. Math. Soc. Gazette 22, 64-67.
6. Briggs, K. M. [1995b] "Algorithms for formal series solution of functional equations," MapleTech 2(2), 48-51.
7. Campanino, M., Epstein, H. & Ruelle, D. [1982] "On Feigenbaum's functional equation," Topology 21, 125-129.
8. Delbourgo, R. & Kenny, B. G. [1991] "Relations between universal scaling constants for the circle map near the golden mean," J. Math. Phys. 32, 1045-1051.
9. Dixon, T. W., Kenny, B. G. & Briggs, K. M. [1997] "On the universality of singular circle maps," Phys. Lett. 231A, 359-366.
10. Eckmann, J.-P. & Epstein, H. [1986] "On the existence of fixed points of the composition operator for circle maps," Comm. Math. Phys. 107, 213-231.
11. Eckmann, J.-P. & Epstein, H. [1990] "Bounds on the unstable eigenvalue for period doubling," Comm. Math. Phys. 128, 427-435.
12. Eckmann, J.-P. & Wittwer, P. [1985] Computer Methods and Borel Summability Applied to Feigenbaum's Equation, Lecture Notes in Physics 227 (Springer, Berlin).
13. Epstein, H. [1986] "New proofs of the existence of the Feigenbaum functions," Comm. Math. Phys. 106, 395-426.
14. Feigenbaum, M. J. [1978] "Quantitative universality for a class of nonlinear transformations," J. Stat. Phys. 19, 25-52.
15. Feigenbaum, M. J. [1979] "The universal metric properties of nonlinear transformations," J. Stat. Phys. 21, 669-702.
16. Feingold, M., Gonzalez, D. L., Magnasco, M. O. & Piro, O. [1991] "On the universality class dependence of period doubling indices," Phys. Lett. A156, 272-276.
17. Feigenbaum, M. J., Kadanoff, L. P. & Shenker, S. J. [1982] "Quasiperiodicity in dissipative systems: A renormalization group analysis," Physica 5D, 370-386.
18. Griewank, A. & Corliss, G. F. (eds.) [1991] Automatic Differentiation of Algorithms: Theory, Implementation, and Application, SIAM Proc. Applied Mathematics 53.
19. Hu, B., Valinia, A. & Piro, O. [1990] "Universal and asymptotic limits of the scaling exponents in circle maps," Phys. Lett. 144A, 7-10.
20. Kuczma, M. [1968] Functional Equations in a Single Variable (PWN - Polish Scientific Publishers, Warszawa).
21. Lanford, O. E. [1984] "Functional equations for circle homeomorphisms with golden ratio rotation number," J. Stat. Phys. 34, 57-73.
22. Loeffel, J.-J. [1976] "Transformation of an asymptotic series in [sic] a convergent one," in Workshop on Padé Approximants, eds. Bessis, D., Gilewicz, J. & Merry, P. Marseille 1975 (Commission de l'Energie Atomique).
23. McGuire, J. B. & Thompson, C. J. [1982] "Asymptotic properties of sequences of iterates of nonlinear transformations," J. Stat. Phys. 27, 183-200.
24. Mestel, B. D. [1985] "Computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number," PhD Thesis, Warwick University.
25. Ostlund, S., Rand, D., Sethna J. & Siggia, E. D. [1983] "Universal properties of the transition from quasiperiodicity to chaos in dissipative systems," Physica 8D, 303-342.
26. Rand, D., Ostlund, S., Sethna J. & Siggia, E. D. [1982] "Universal transition from quasiperiodicity to chaos in dissipative systems," Phys. Rev. Lett. 49(2), 132-135.
27. Sokal, A. D. [1980] "An improvement of Watson's theorem on Borel summability," J. Math. Phys. 21, 261-263.
28. Sokal, A. D. [1996] "Trangressing the boundaries: Towards a transformative hermeneutics of quantum gravity," Social Text #46/47, 217-252 (Spring/Summer 1996).
29. Szekeres, G. [1958] "Regular iteration of real and complex functions," Acta. Math. 100, 203-258.
30. Thompson, C. J. & McGuire, J. B. [1988] "Asymptotic and essentially singular solutions of the Feigenbaum equation," J. Stat. Phys. 51, 991-1007.
31. van der Weele, J. P., Capel, H. W. & Kluiving, R. [1987] "Period doubling in maps with a maximum of order z," Physica A145, 425-460.