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December  2020, 25(12): 4575-4583. doi: 10.3934/dcdsb.2020113

Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation

Yong-Guo Shi ,

Numerical Simulation Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China

* Corresponding author: Yong-Guo Shi

Received  May 2019 Revised  December 2019 Published  December 2020 Early access  March 2020

Fund Project: The author is supported by Scientific Research Fund of SiChuan Provincial Education Department (18ZA0274)

The Feigenbaum-Kadanoff-Shenker equation for universal scaling in circle maps characterizes the quasiperiodic route to chaos. In this paper, using two different iterative methods, we construct all strictly decreasing continuous solutions. Furthermore, we present respectively the corresponding conditions to guarantee $ C^1 $ smoothness of those continuous solutions.

Keywords: Feigenbaum-Kadanoff-Shenker equation, iterative construction method, continuously differentiable solution, infinite product.
Mathematics Subject Classification: Primary: 39B12; Secondary: 39B22.
Citation: Yong-Guo Shi. Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4575-4583. doi: 10.3934/dcdsb.2020113
References:
[1]

L. Berg, A piecewise linear solution of Feigenbaum' s equation, Aequationes Math., 76 (2008), 197-199.  doi: 10.1007/s00010-007-2925-3.  Google Scholar

[2]

K. M. Briggs, T. W. Dixon and G. Szekeres, Analytic solutions of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 347-357.  doi: 10.1142/S0218127498000206.  Google Scholar

[3]

K. M. Briggs, Feigenbaum Scaling in Discrete Dynamical Systems, Ph.D thesis, University of Melbourne, 1997. Google Scholar

[4]

M. Campanino and H. Epstein, On the existence of Feigenbaum's fixed point, Commun. Math. Phys., 79 (1981), 261-302.  doi: 10.1007/BF01942063.  Google Scholar

[5]

J.-P. Eckmann and H. Epstein, On the existence of fixed points of the composition operator for circle maps, Commun. Math. Phys., 107 (1986), 213-231.  doi: 10.1007/BF01209392.  Google Scholar

[6]

M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar

[7]

M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys., 21 (1979), 669-706.  doi: 10.1007/BF01107909.  Google Scholar

[8]

M. J. Feigenbaum, L. P. Kadanoff and S. J. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Phys. D, 5 (1982), 370-386.  doi: 10.1016/0167-2789(82)90030-6.  Google Scholar

[9]

J. Groeneveld, On constructing complete solution classes of the Cvitanović-Feigenbaum equation, Phys. A, 138 (1986), 137-166.  doi: 10.1016/0378-4371(86)90177-9.  Google Scholar

[10]

M. Kuczma, On the functional equation $\varphi ^{n}(x) = g(x)$, Ann. Polon. Math., 11 (1961), 161-175.  doi: 10.4064/ap-11-2-161-175.  Google Scholar

[11]

M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warszawa, 1968.  Google Scholar

[12]

M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, in Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639.  Google Scholar

[13]

O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N. S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[14]

O. E. Lanford III, Functional equations for circle homeomorphisms with golden ratio rotation number, J. Statist. Phys., 34 (1984), 57-73.  doi: 10.1007/BF01770349.  Google Scholar

[15]

B. D. Mestel, Computer Assisted Proof of Universality for Cubic Critical Maps of the Circle with Golden Mean Rotation Number, Ph.D thesis, University of Warwick, 1985. Google Scholar

[16]

M. Nauenberg, On the fixed points for circle maps, Phys. Lett. A, 92 (1982), 319-320.  doi: 10.1016/0375-9601(82)90898-2.  Google Scholar

[17]

J. Stephenson and Y. Wang, Relationships between the solutions of Feigenbaum's equation, Appl. Math. Lett., 4 (1991), 37-39.  doi: 10.1016/0893-9659(91)90031-P.  Google Scholar

[18]

J. Stephenson and Y. Wang, Relationships between eigenfunctions associated with solutions of Feigenbaum's equation, Appl. Math. Lett., 4 (1991), 53-56.  doi: 10.1016/0893-9659(91)90035-T.  Google Scholar

[19]

Y. Tang, $C^\infty$ even solutions of Feigenbaum functional equation, Acta Math. Sinica (Chin. Ser.), 40 (1997), 253-258.   Google Scholar

[20]

L. Yang and J. Z. Zhang, The second type of Feigenbaum's functional equations, Sci. Sinica Ser. A, 29 (1986), 1252-1262.   Google Scholar

show all references

References:
[1]

L. Berg, A piecewise linear solution of Feigenbaum' s equation, Aequationes Math., 76 (2008), 197-199.  doi: 10.1007/s00010-007-2925-3.  Google Scholar

[2]

K. M. Briggs, T. W. Dixon and G. Szekeres, Analytic solutions of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 347-357.  doi: 10.1142/S0218127498000206.  Google Scholar

[3]

K. M. Briggs, Feigenbaum Scaling in Discrete Dynamical Systems, Ph.D thesis, University of Melbourne, 1997. Google Scholar

[4]

M. Campanino and H. Epstein, On the existence of Feigenbaum's fixed point, Commun. Math. Phys., 79 (1981), 261-302.  doi: 10.1007/BF01942063.  Google Scholar

[5]

J.-P. Eckmann and H. Epstein, On the existence of fixed points of the composition operator for circle maps, Commun. Math. Phys., 107 (1986), 213-231.  doi: 10.1007/BF01209392.  Google Scholar

[6]

M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332.  Google Scholar

[7]

M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys., 21 (1979), 669-706.  doi: 10.1007/BF01107909.  Google Scholar

[8]

M. J. Feigenbaum, L. P. Kadanoff and S. J. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Phys. D, 5 (1982), 370-386.  doi: 10.1016/0167-2789(82)90030-6.  Google Scholar

[9]

J. Groeneveld, On constructing complete solution classes of the Cvitanović-Feigenbaum equation, Phys. A, 138 (1986), 137-166.  doi: 10.1016/0378-4371(86)90177-9.  Google Scholar

[10]

M. Kuczma, On the functional equation $\varphi ^{n}(x) = g(x)$, Ann. Polon. Math., 11 (1961), 161-175.  doi: 10.4064/ap-11-2-161-175.  Google Scholar

[11]

M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warszawa, 1968.  Google Scholar

[12]

M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, in Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639.  Google Scholar

[13]

O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N. S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[14]

O. E. Lanford III, Functional equations for circle homeomorphisms with golden ratio rotation number, J. Statist. Phys., 34 (1984), 57-73.  doi: 10.1007/BF01770349.  Google Scholar

[15]

B. D. Mestel, Computer Assisted Proof of Universality for Cubic Critical Maps of the Circle with Golden Mean Rotation Number, Ph.D thesis, University of Warwick, 1985. Google Scholar

[16]

M. Nauenberg, On the fixed points for circle maps, Phys. Lett. A, 92 (1982), 319-320.  doi: 10.1016/0375-9601(82)90898-2.  Google Scholar

[17]

J. Stephenson and Y. Wang, Relationships between the solutions of Feigenbaum's equation, Appl. Math. Lett., 4 (1991), 37-39.  doi: 10.1016/0893-9659(91)90031-P.  Google Scholar

[18]

J. Stephenson and Y. Wang, Relationships between eigenfunctions associated with solutions of Feigenbaum's equation, Appl. Math. Lett., 4 (1991), 53-56.  doi: 10.1016/0893-9659(91)90035-T.  Google Scholar

[19]

Y. Tang, $C^\infty$ even solutions of Feigenbaum functional equation, Acta Math. Sinica (Chin. Ser.), 40 (1997), 253-258.   Google Scholar

[20]

L. Yang and J. Z. Zhang, The second type of Feigenbaum's functional equations, Sci. Sinica Ser. A, 29 (1986), 1252-1262.   Google Scholar

Figure 1.  $ g(\alpha^2)=a=\alpha^2=1/4 $
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Figure 2.  $ g(\alpha^2)=a=1/2 $, $ \alpha^2=1/4 $
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