From 850e1f4668ac06fe0473f3110ce5edd105ad85d6 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Jon=C3=A1=C5=A1=20Koziorek?=
 <73384756+JonasKoziorek@users.noreply.github.com>
Date: Fri, 7 Jun 2024 14:29:14 +0200
Subject: [PATCH] docs fix (#333)

---
 docs/src/periodicity.md          |  2 +-
 src/periodicity/davidchacklai.jl | 15 +++++++++------
 2 files changed, 10 insertions(+), 7 deletions(-)

diff --git a/docs/src/periodicity.md b/docs/src/periodicity.md
index bd9416b8..f83b6eac 100644
--- a/docs/src/periodicity.md
+++ b/docs/src/periodicity.md
@@ -169,7 +169,7 @@ davidchacklai
 #### Logistic Map example
 
 The idea of periodic orbits can be illustrated easily on 1D maps. Finding all periodic orbits of period
-$n$ is equivalent to finding all points $x$ such that $f^{n}(x)=x$, where $f^{n}$ is $n$-th composition of $f$. Hence, solving $f^{n}(x)-x=0$ yields such points. However, this is impossible analytically. 
+$n$ is equivalent to finding all points $x$ such that $f^{n}(x)=x$, where $f^{n}$ is $n$-th composition of $f$. Hence, solving $f^{n}(x)-x=0$ yields such points. However, this is often impossible analytically. 
 Let's see how `davidchacklai` deals with it:
 
 First let's start with finding first $9$ periodic orbits of the logistic map for parameter $3.72$.
diff --git a/src/periodicity/davidchacklai.jl b/src/periodicity/davidchacklai.jl
index a69bc475..ac4cb059 100644
--- a/src/periodicity/davidchacklai.jl
+++ b/src/periodicity/davidchacklai.jl
@@ -25,7 +25,7 @@ periodic orbits will be used to detect periodic orbits of order `m+1` to `n`.
    is an `n`-periodic point.
 * `abstol = 1e-8`: A detected periodic point isn't stored if it is in `abstol` 
    neighborhood of some previously detected point. Distance is measured by 
-   euclidian norm. If you are getting duplicate periodic points, decrease this value.
+   euclidian norm. If you are getting duplicate periodic points, increase this value.
 
 ## Description
 
@@ -37,7 +37,7 @@ by turning fixed points of the original
 map `ds` to stable ones, through the transformation
 ```math
 \\mathbf{x}_{n+1} = \\mathbf{x}_{n} + 
-[\\beta |g(\\mathbf{x}_{n}| C^{T} - J(\\mathbf{x}_{n})]^{-1} g(\\mathbf{x}_{n})
+[\\beta |g(\\mathbf{x}_{n})| C^{T} - J(\\mathbf{x}_{n})]^{-1} g(\\mathbf{x}_{n})
 ```
 where
 ```math
@@ -46,7 +46,7 @@ g(\\mathbf{x}_{n}) = f^{n}(\\mathbf{x}_{n}) - \\mathbf{x}_{n}
 and
 ```math
 J(\\mathbf{x}_{n}) = \\frac{\\partial g(\\mathbf{x}_{n})}{\\partial \\mathbf{x}_{n}}
-````
+```
 
 The main difference between Schmelcher & Diakonos[Schmelcher1997](@cite) and 
 Davidchack & Lai[Davidchack1999](@cite) is that the latter uses periodic points of
@@ -58,9 +58,12 @@ while `davidchacklai` detects periodic points of all orders up to `n`.
 ## Important note
 
 For low periods `n` circa less than 6, you should select `m = n` otherwise the algorithm 
-detect periodic orbits correctly. For higher periods, you can select `m` as 6. 
-You can use initial grid of points for `ics`. Increase `m` in case the orbits are 
-not being detected correctly.
+won't detect periodic orbits correctly. For higher periods, you can select `m` as 6. 
+We recommend experimenting with `m` as it may depend on the specific problem. 
+Increase `m` in case the orbits are not being detected correctly.
+
+Initial conditions `ics` can be selected as a uniform grid of points in the state space or 
+subset of a chaotic trajectory.
 
 """
 function davidchacklai(