If you just want the title of #bottom to say 1 on every pageload, same as the #middle select is saying, you can use this:
$('#middle').change(function() {
  $('#output').text($(this).find(":selected") ? $(this).find(":selected").text() : $(this).find(":first").text());
}).change(); //trigger change on pageload
$('#middle').change(function() {
  $('#output').text($(this).find(":selected") ? $(this).find(":selected").text() : $(this).find(":first").text());
}).change();
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="form-group">
  <label class="col-md-4 control-label">Section:</label>
  <div class="col-md-8">
    <select id="middle" class="form-control">
      <option>1</option>
      <option>2</option>
      <option>3</option>
      <option>4</option>
      <option>5</option>
      <option>6</option>
      <option>7</option>
      <option>8</option>
      <option>9</option>
      <option>10</option>
    </select>
  </div>
</div>
<br /><br />
<div class="form-group">
  <label class="col-md-4 control-label" id="output"></label>
  <div class="col-md-8">
    <select id="bottom" class="form-control">
      <option>11</option>
      <option>12</option>
      <option>13</option>
    </select>
  </div>
</div>
 
 
jsfiddle: https://jsfiddle.net/hgep8hcg/
- This code checks if there is a selected option, and if so, uses its value. If there is no selectedoption, it uses the firstoption's value.
 This code, inside.text(), may look a little weird. It's called a ternary operator, click the link if you want to understand it.
IF YOU WANT TO STORE THE SELECTED VALUES, READ ON:
Your easiest option would be localStorage, provided that browser support is sufficient for your goal.
(I use sessionStorage here, just so everybody's localStorage doesn't get bloated with test values.)
This would do the trick:
$("#middle").change(function() {
  var value = $(this).find(":selected").text();
  $("#output").text(value);
  sessionStorage.setItem(this.id, value);
}).val(function(){return (sessionStorage.getItem(this.id) ? sessionStorage.getItem(this.id) : $(this).find("option:first").text());}).change(); //set values on pageload
jsfiddle: https://jsfiddle.net/k3htre4d/2/
codepen: https://codepen.io/anon/pen/OgOdLZ?editors=1010
- In your changehandler, I set thelocalStorage, using theselect's ID as the key, and theselect's text-value as value.
- After the changehandler, I set theselect's value, using thelocalStoragekey that matches theselect's ID, if that key exists. If it doesn't exist, I use theselect's firstoption's text.
 That's this line:return (... ? ... : ...);. It's called a ternary operator, click the link if you want to understand it.
MORE EFFICIENT MIGHT BE:
$("#middle").change(function() {
  $("#output").text($(this).find(":selected").text());
});
$(".form-control").change(function() {
  sessionStorage.setItem(this.id, $(this).find(":selected").text()); //store select's value on change
}).each(function() {
  if (sessionStorage.getItem(this.id)) {$(this).val(sessionStorage.getItem(this.id));} //set select's value on pageload
}).change(); //trigger change on pageload
jsfiddle: https://jsfiddle.net/k3htre4d/1/
codepen: https://codepen.io/anon/pen/dRZayW?editors=1010
- In this code the values of all the selects with classform-controlare stored intolocalStorageonchange. All theselects get their own key inlocalStoragebased on their ID. So now you don't have to duplicate the handler for every singleselect. Just make sure everyselectyou wish to store the value for, has classform-control(or whatever class you choose).
 Using this principle, you could also add an extra class to only thoseselects that you wish to store the values for, and than use that class for the handler instead of.form-control.
- The code that is only used by the #middleselect, is in a separate handler.