Decoding a time series for reconstruction

Question

The is based on digital communications. A continuous valued time series is generated from a chaotic Logistic map [https://en.wikipedia.org/wiki/Logistic_map ]. I then digitized it to binary which is the encoding part (transmitter end). After that I want to regenerate or reconstruct the continuous valued signal from the symbols (receiver end). This is the decoding part. I tried to apply this concept but I cannot get back the exact reconstructed signal as shown in the graph. Where did I go wrong? Will appreciate help.

Can somebody please help how to get the exact reconstruction from the symbols?

% Logistic Map Chaotic System
r = 3.8;
x = zeros(1, 50);
x(1) = 0.1;

for t = 2:length(x)
    x(t) = r * x(t-1) * (1 - x(t-1));
end

% Parameters for Binary Communication
threshold = 0.5; % Adjust the threshold based on your system characteristics
binarySymbols = zeros(1, length(x));

% Encode Chaotic Signal into Binary Symbols
binarySymbols(x >= threshold) = 1;

 

% Decode Binary Symbols (for demonstration)
decodedSignal = zeros(1, length(x));
decodedSignal(binarySymbols == 1) = x(binarySymbols == 1);
%the decoding process directly uses the binary symbols to index 
% the corresponding chaotic values, 
% providing a more accurate reconstruction of the original 
% chaotic signal. 

figure;
plot(x, 'b', 'LineWidth', 1.5);
hold on;
plot(decodedSignal, 'g--', 'LineWidth', 1.5);
legend('Original Chaotic Signal', 'Decoded Signal');
title('Decoding Binary Symbols back to Chaotic Signal');