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

Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation

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.

Citation: Yong-Guo Shi. Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation. Discrete and 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.

[2]

K. M. BriggsT. 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.

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

M. J. FeigenbaumL. 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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

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.

[2]

K. M. BriggsT. 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.

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

M. J. FeigenbaumL. 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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

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