Positive subharmonic solutions to superlinear ODEs with indefinite weight

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 $aabb$