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Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(t) < 1$ for all $(n,t) \in \mathbb{N} \times \mathbb{R}_+$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Remark: The condition (ii) is a bit unclear, I should emphasize here that $a_n(t) < 1$ for all (fixed) $n \in \mathbb N$ and for all $t \geq 0$, this is the reason that I did not write condition (ii) as condition (ii') $a_n(t) \approx 1$ uniformly in $t$ for $n \gg 1$.

Remark: As discussed in the comments, I need to impose an initial datum ${a_n(0)}_{n \geq 0}$ such that $\sup_{n \in \mathbb N} a_n(0) = 1$ but there does not exists an $N \in \mathbb N$ for which $a_N(0) = 1$. This assumption is a mild assumption on the initial datum.

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(0) \leq 1$ for all $n\in \mathbb N$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Remark: The condition (ii) is a bit unclear, I should emphasize here that $a_n(t) < 1$ for all (fixed) $n \in \mathbb N$ and for all $t \geq 0$, this is the reason that I did not write condition (ii) as condition (ii') $a_n(t) \approx 1$ uniformly in $t$ for $n \gg 1$.

Remark: As discussed in the comments, I need to impose an initial datum ${a_n(0)}_{n \geq 0}$ such that $\sup_{n \in \mathbb N} a_n(0) = 1$ but there does not exists an $N \in \mathbb N$ for which $a_N(0) = 1$. Also, one can savely assume that $a_n(0) < a^2_{n+1}(0)$ for all $n \in \mathbb N$.

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(t) < 1$ for all $(n,t) \in \mathbb{N} \times \mathbb{R}_+$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Remark: The condition (ii) is a bit unclear, I should emphasize here that $a_n(t) < 1$ for all (fixed) $n \in \mathbb N$ and for all $t \geq 0$, this is the reason that I did not write condition (ii) as condition (ii') $a_n(t) \approx 1$ uniformly in $t$ for $n \gg 1$.

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(t) < 1$ for all $(n,t) \in \mathbb{N} \times \mathbb{R}_+$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Remark: The condition (ii) is a bit unclear, I should emphasize here that $a_n(t) < 1$ for all (fixed) $n \in \mathbb N$ and for all $t \geq 0$, this is the reason that I did not write condition (ii) as condition (ii') $a_n(t) \approx 1$ uniformly in $t$ for $n \gg 1$.

Remark: As discussed in the comments, I need to impose an initial datum ${a_n(0)}_{n \geq 0}$ such that $\sup_{n \in \mathbb N} a_n(0) = 1$ but there does not exists an $N \in \mathbb N$ for which $a_N(0) = 1$. This assumption is a mild assumption on the initial datum.

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(t) \leq 1$ for all $(n,t) \in \mathbb{N} \times \mathbb{R}_+$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(t) < 1$ for all $(n,t) \in \mathbb{N} \times \mathbb{R}_+$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'_n = a^2_{n+1} - a_n.$ My goal is to show that $$a_n(t) \xrightarrow{t \to \infty} \mathrm{e}^{-\mu 2^{-n}} \quad \text{for all $n \in \mathbb N$.}$$ Any help (or hints) will be greatly appreciated!

Remark: The condition (ii) is a bit unclear, I should emphasize here that $a_n(t) < 1$ for all (fixed) $n \in \mathbb N$ and for all $t \geq 0$, this is the reason that I did not write condition (ii) as condition (ii') $a_n(t) \approx 1$ uniformly in $t$ for $n \gg 1$.