April  2018, 11(2): 257-277. doi: 10.3934/dcdss.2018014

Positive subharmonic solutions to superlinear ODEs with indefinite weight

Département de Mathématique, Université de Mons, Place du Parc 20, B-7000 Mons, Belgium

Received  January 2016 Revised  April 2017 Published  January 2018

Fund Project: Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni". It is also partially supported by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14) of the Fonds de la Recherche Fondamentale Collective, Belgium.

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation
$\begin{equation*}u'' + q(t) g(u) = 0,\end{equation*}$
where
$g(u)$
has superlinear growth both at zero and at infinity, and
$q(t)$
is a
$T$
-periodic sign-changing weight. Under the sharp mean value condition
$\int_{0}^{T}{q\left( t \right)dt<0}$
, combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order
$k$
for any large integer
$k$
. Moreover, when the negative part of
$q(t)$
is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order
$k$
for any integer
$k≥2$
.
Citation: Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014
References:
[1]

N. Ackermann, Long-time dynamics in semilinear parabolic problems with autocatalysis, in Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, 2009, 1-30 doi: 10.1142/9789812834744_0001.

[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962.

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440.

[4]

M. Barnsley, Fractals Everywhere Academic Press, Inc. , Boston, MA, 1988.

[5]

V.L. BarutelloA. Boscaggin and G. Verzini, Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line, J. Differential Equations, 259 (2015), 3448-3489.  doi: 10.1016/j.jde.2015.04.026.

[6]

H. BerestyckiI. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear {L}iouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.

[7]

J. Berstel and D. Perrin, The origins of combinatorics on words, European J. Combin., 28 (2007), 996-1022.  doi: 10.1016/j.ejc.2005.07.019.

[8]

D. BonheureJ.M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.

[9]

A. Boscaggin, Positive periodic solutions to nonlinear ODEs with indefinite weight: an overview, Rend. Semin. Mat. Univ. Politec. Torino (to appear).

[10]

A. BoscagginW. Dambrosio and D. Papini, Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017.  doi: 10.1016/j.jde.2017.02.025.

[11]

A. Boscaggin and G. Feltrin, Positive subharmonic solutions to nonlinear ODEs with indefinite weight Commun. Contemp. Math. (to appear). doi: 10.1142/S0219199717500213.

[12]

A. BoscagginG. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 449-474.  doi: 10.1017/S0308210515000621.

[13]

A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree Trans. Amer. Math. Soc. (to appear). doi: 10.1090/tran/6992.

[14]

A. Boscaggin and F. Zanolin, Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics, J. Differential Equations, 252 (2012), 2922-2950.  doi: 10.1016/j.jde.2011.09.010.

[15]

A. Boscaggin and F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst., 33 (2013), 89-110.  doi: 10.3934/dcds.2013.33.89.

[16]

K.J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207. 

[17]

G.J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477.  doi: 10.1016/0022-0396(76)90041-3.

[18]

A. CapiettoM. HenrardJ. Mawhin and F. Zanolin, A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3 (1994), 81-100.  doi: 10.12775/TMNA.1994.005.

[19]

A. CapiettoW. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.  doi: 10.1006/jdeq.2001.4080.

[20]

B.S. Du, The minimal number of periodic orbits of periods guaranteed in Sharkovskiǐ's theorem, Bull. Austral. Math. Soc., 31 (1985), 89-103.  doi: 10.1017/S0004972700002306.

[21]

D. S. Dummit and R. M. Foote, Abstract Algebra 3rd edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2004.

[22]

G. Feltrin, Positive Solutions to Indefinite Problems: A Topological Approach Ph. D. thesis, SISSA (Trieste), 2016.

[23]

G. Feltrin and F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: The Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982. 

[24]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.

[25]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.  doi: 10.1016/j.jde.2017.01.009.

[26]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations Lecture Notes in Mathematics, 568, Springer-Verlag, Berlin-New York, 1977.

[27]

M. GaudenziP. Habets and F. Zanolin, An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272. 

[28]

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. 

[29]

P.M. Girão and J.M. Gomes, Multibump nodal solutions for an indefinite superlinear elliptic problem, J. Differential Equations, 247 (2009), 1001-1012.  doi: 10.1016/j.jde.2009.04.018.

[30]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.  doi: 10.1006/jdeq.2000.3772.

[31]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.  doi: 10.1080/03605308008820162.

[32]

M.R. JoglekarE. Sander and J.A. Yorke, Fixed points indices and period-doubling cascades, J. Fixed Point Theory Appl., 8 (2010), 151-176.  doi: 10.1007/s11784-010-0029-5.

[33]

T. Kociumaka, J. Radoszewski and W. Rytter, Computing k-th Lyndon word and decoding lexicographically minimal de Bruijn sequence, in Combinatorial Pattern Matching, vol. 8486 of Lecture Notes in Comput. Sci. , Springer, Cham, 2014,202-211. doi: 10.1007/978-3-319-07566-2_21.

[34]

M. Lothaire, Combinatorics on Words Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511566097.

[35]

P.A. MacMahon, Applications of a theory of permutations in circular procession to the theory of numbers, Proc. London Math. Soc., 23 (1891/92), 305-313.  doi: 10.1112/plms/s1-23.1.305.

[36]

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, R. I. , 1979.

[37]

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, 1537, Springer, Berlin, 1993, 74-142 doi: 10.1007/BFb0085076.

[38]

J. MawhinC. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, Commun. Contemp. Math., 2 (2000), 87-126.  doi: 10.1142/S0219199700000074.

[39]

P. Morassi, A note on the construction of coincidence degree, Boll. Un. Mat. Ital. A (7), 10 (1996), 421-433. 

[40]

R.D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. Ⅱ, J. Differential Equations, 14 (1973), 360-394.  doi: 10.1016/0022-0396(73)90053-3.

[41]

R. D. Nussbaum, The Fixed Point Index and Some Applications Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 94, Presses de l'Université de Montréal, Montreal, QC, 1985.

[42]

R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Math. , 1537, Springer, Berlin, 1993,143-205. doi: 10.1007/BFb0085077.

[43]

D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233.  doi: 10.12775/TMNA.2000.017.

[44]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.  doi: 10.1515/ans-2004-0105.

[45]

N. J. A. Sloane, The on-line encyclopedia of integer sequences published electronically at [http://oeis.org, Sequence A001037.

[46]

E. Sovrano, How to get complex dynamics? A note on a topological approach, submitted.

[47]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics J. Math. Biol. (to appear). doi: 10.1007/s00285-017-1185-7.

[48]

S. Terracini and G. Verzini, Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities, Nonlinearity, 13 (2000), 1501-1514.  doi: 10.1088/0951-7715/13/5/305.

show all references

References:
[1]

N. Ackermann, Long-time dynamics in semilinear parabolic problems with autocatalysis, in Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, 2009, 1-30 doi: 10.1142/9789812834744_0001.

[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962.

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440.

[4]

M. Barnsley, Fractals Everywhere Academic Press, Inc. , Boston, MA, 1988.

[5]

V.L. BarutelloA. Boscaggin and G. Verzini, Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line, J. Differential Equations, 259 (2015), 3448-3489.  doi: 10.1016/j.jde.2015.04.026.

[6]

H. BerestyckiI. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear {L}iouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.

[7]

J. Berstel and D. Perrin, The origins of combinatorics on words, European J. Combin., 28 (2007), 996-1022.  doi: 10.1016/j.ejc.2005.07.019.

[8]

D. BonheureJ.M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.

[9]

A. Boscaggin, Positive periodic solutions to nonlinear ODEs with indefinite weight: an overview, Rend. Semin. Mat. Univ. Politec. Torino (to appear).

[10]

A. BoscagginW. Dambrosio and D. Papini, Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017.  doi: 10.1016/j.jde.2017.02.025.

[11]

A. Boscaggin and G. Feltrin, Positive subharmonic solutions to nonlinear ODEs with indefinite weight Commun. Contemp. Math. (to appear). doi: 10.1142/S0219199717500213.

[12]

A. BoscagginG. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 449-474.  doi: 10.1017/S0308210515000621.

[13]

A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree Trans. Amer. Math. Soc. (to appear). doi: 10.1090/tran/6992.

[14]

A. Boscaggin and F. Zanolin, Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics, J. Differential Equations, 252 (2012), 2922-2950.  doi: 10.1016/j.jde.2011.09.010.

[15]

A. Boscaggin and F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst., 33 (2013), 89-110.  doi: 10.3934/dcds.2013.33.89.

[16]

K.J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207. 

[17]

G.J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477.  doi: 10.1016/0022-0396(76)90041-3.

[18]

A. CapiettoM. HenrardJ. Mawhin and F. Zanolin, A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3 (1994), 81-100.  doi: 10.12775/TMNA.1994.005.

[19]

A. CapiettoW. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.  doi: 10.1006/jdeq.2001.4080.

[20]

B.S. Du, The minimal number of periodic orbits of periods guaranteed in Sharkovskiǐ's theorem, Bull. Austral. Math. Soc., 31 (1985), 89-103.  doi: 10.1017/S0004972700002306.

[21]

D. S. Dummit and R. M. Foote, Abstract Algebra 3rd edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2004.

[22]

G. Feltrin, Positive Solutions to Indefinite Problems: A Topological Approach Ph. D. thesis, SISSA (Trieste), 2016.

[23]

G. Feltrin and F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: The Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982. 

[24]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.

[25]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.  doi: 10.1016/j.jde.2017.01.009.

[26]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations Lecture Notes in Mathematics, 568, Springer-Verlag, Berlin-New York, 1977.

[27]

M. GaudenziP. Habets and F. Zanolin, An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272. 

[28]

E.N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. 

[29]

P.M. Girão and J.M. Gomes, Multibump nodal solutions for an indefinite superlinear elliptic problem, J. Differential Equations, 247 (2009), 1001-1012.  doi: 10.1016/j.jde.2009.04.018.

[30]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.  doi: 10.1006/jdeq.2000.3772.

[31]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.  doi: 10.1080/03605308008820162.

[32]

M.R. JoglekarE. Sander and J.A. Yorke, Fixed points indices and period-doubling cascades, J. Fixed Point Theory Appl., 8 (2010), 151-176.  doi: 10.1007/s11784-010-0029-5.

[33]

T. Kociumaka, J. Radoszewski and W. Rytter, Computing k-th Lyndon word and decoding lexicographically minimal de Bruijn sequence, in Combinatorial Pattern Matching, vol. 8486 of Lecture Notes in Comput. Sci. , Springer, Cham, 2014,202-211. doi: 10.1007/978-3-319-07566-2_21.

[34]

M. Lothaire, Combinatorics on Words Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511566097.

[35]

P.A. MacMahon, Applications of a theory of permutations in circular procession to the theory of numbers, Proc. London Math. Soc., 23 (1891/92), 305-313.  doi: 10.1112/plms/s1-23.1.305.

[36]

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, R. I. , 1979.

[37]

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, 1537, Springer, Berlin, 1993, 74-142 doi: 10.1007/BFb0085076.

[38]

J. MawhinC. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, Commun. Contemp. Math., 2 (2000), 87-126.  doi: 10.1142/S0219199700000074.

[39]

P. Morassi, A note on the construction of coincidence degree, Boll. Un. Mat. Ital. A (7), 10 (1996), 421-433. 

[40]

R.D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. Ⅱ, J. Differential Equations, 14 (1973), 360-394.  doi: 10.1016/0022-0396(73)90053-3.

[41]

R. D. Nussbaum, The Fixed Point Index and Some Applications Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 94, Presses de l'Université de Montréal, Montreal, QC, 1985.

[42]

R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Math. , 1537, Springer, Berlin, 1993,143-205. doi: 10.1007/BFb0085077.

[43]

D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233.  doi: 10.12775/TMNA.2000.017.

[44]

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.  doi: 10.1515/ans-2004-0105.

[45]

N. J. A. Sloane, The on-line encyclopedia of integer sequences published electronically at [http://oeis.org, Sequence A001037.

[46]

E. Sovrano, How to get complex dynamics? A note on a topological approach, submitted.

[47]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics J. Math. Biol. (to appear). doi: 10.1007/s00285-017-1185-7.

[48]

S. Terracini and G. Verzini, Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities, Nonlinearity, 13 (2000), 1501-1514.  doi: 10.1088/0951-7715/13/5/305.

Figure 1.  The figure shows an example of multiple positive solutions for the $T$-periodic boundary value problem associated with $(\mathscr{E}_{\mu})$. For this numerical simulation we have chosen $a(t) = \sin(6\pi t)$ for $t\in\mathopen{[}0, 1\mathclose{]}$, $\mu = 10$ and $g(s) = \max\{0,400\, s\arctan|s|\}$. Notice that the weight function $a(t)$ has $3$ positive humps. We show the graphs of the $7$ positive $T$-periodic solutions of $(\mathscr{E}_{\mu})$. We stress that $g(s)/s\not\to+\infty$ as $s\to+\infty$ (contrary to what is assumed in $(g_{s})$), indeed Theorem 2.2 is also valid assuming only that $g(s)/s$ is sufficiently large as $s\to+\infty$, as observed in Remark 2.1
Figure 2.  Using a numerical simulation we have studied the $2$-periodic solutions of equation $(\mathscr{E}_{\mu})$ where $a(t) = \sin(t)$, $T=2\pi$, $\mu = 6$ and $g(s) = 100(s^{2}+s^{3})$. In the upper part, the figure shows the graph of the weight $q(t)=a_{\mu}(t)$; while in the lower part, it shows the graphs of three $2T$-periodic positive solutions, in accordance with Theorem 4.1. The three solutions are in correspondence with the three strings $(1, 0)$, $(0, 1)$ and $(1, 1)$, respectively, as stated in Theorem 4.1. The first two solutions are subharmonic solutions of order $2$ and the third one is a $1$-periodic solution. As subharmonic solutions of order $2$, we consider only the first one, since the second one is a translation by $1$ of the first solution
Figure 3.  The figure shows the aperiodic necklaces made by arranging $k$ beads whose color is chosen from a list of 2 colors, when $k\in\{2, 3, 4\}$. The aperiodic necklaces depicted in the figure correspond to the following Lyndon words on the alphabet $\{a, b\}$: $ab$ for $k=2$, $abb$ and $aab$ for $k=3$, $abbb$, $aabb$ and $aaab$ for $k=4$ (which are the minimal elements in the class of equivalence in the lexicographic ordering). For instance, for $k = 4$, note that the string $abab$ is not acceptable as it represents a sequence of period $2$ and the string $bbaa$ is already counted as $aabb$
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