If I understand correctly what you want, you would need to use an AngularJS Service. For a good explanation, see this article from the Ionic docs:

http://learn.ionicframework.com/formulas/sharing-data-between-views/

In the section "Sharing Data with Services" the answer to your question is right there.

I had the opposite problem, where I wanted it to refresh if the user went back to the same page, but it wasn't. At some point during the last 10 months (between this question being asked and the point I'm writing this), I believe Ionic implemented a page cache to make things like their iOS slide-to-go-back effect work. So your problem may have been fixed by that, but just in case ...

My problem was fixed by adding "cache: false" to the options for each state in the module's .config() section. However, on some pages (or states, I guess) I didn't want them to reload, but I couldn't be sure the page/state would still be in the cache when the user came back to it. So I decided to set up a flag in $rootScope... or, rather, set up an object in $rootScope that contains a flag variable for each controller, and just check the flag to see if that page has already been loaded, and if so, don't run any of the controller code. So for the setup you described, you'd have something like this:

app.js (or whatever .js file defines your module):

angular.module('MyApp', ['ionic'])
.config(function($stateProvider, $urlRouterProvider) {
  $stateProvider
    .state('Search'  , { cache: true  , url:'/Search'  , templateUrl:'templates/Search.html'  , controller: 'SearchController'  })
    .state('All'     , { cache: true  , url:'/All'     , templateUrl:'templates/All.html'     , controller: 'AllController'     })
    .state('Dogs'    , { cache: true  , url:'/Dogs'    , templateUrl:'templates/Dogs.html'    , controller: 'CatsController'    })
    .state('Cows'    , { cache: true  , url:'/Cows'    , templateUrl:'templates/Cows.html'    , controller: 'DogsController'    })
    .state('Cats'    , { cache: true  , url:'/Cats'    , templateUrl:'templates/Cats.html'    , controller: 'CowsController'    })
    .state('Contact' , { cache: true  , url:'/Contact' , templateUrl:'templates/Contact.html' , controller: 'ContactController' });
  $urlRouterProvider.otherwise('/');
});

And then for the controllers, you could do:

controllers.js

.controller('SearchController', function($scope, $rootScope, $state) {
    // If $rootScope.flags does not already exist, create it as an empty object
    $rootScope.flags = $rootScope.flags || {};

    if($rootScope.flags.SearchPageHasBeenOpenedAlready) {
      console.log("Search page was already opened, not doing anything.");
      return;
    } else {
      // Assuming this is the first controller, I guess you could initialize the flags here...
      // We will initialize them all to false and not set them to true until the code for
      // each controller has run through successfully
      $rootScope.flags.SearchPageHasBeenOpenedAlready  = false;
      $rootScope.flags.AllPageHasBeenOpenedAlready     = false;
      $rootScope.flags.CatsPageHasBeenOpenedAlready    = false;
      $rootScope.flags.DogsPageHasBeenOpenedAlready    = false;
      $rootScope.flags.CowsPageHasBeenOpenedAlready    = false;
      $rootScope.flags.ContactPageHasBeenOpenedAlready = false;
    }

    /* Whatever you normally do in SearchController goes here */

    $rootScope.flags.SearchPageHasBeenOpenedAlready = true;
  }
)

.controller('AllController', function($scope, $rootScope, $state) {
    // If $rootScope.flags does not already exist, create it as an empty object
    $rootScope.flags = $rootScope.flags || {};

    if($rootScope.flags.AllPageHasBeenOpenedAlready) {
      console.log("All page was already opened, not doing anything.");
      return;
    }

    /* Whatever you normally do in AllController goes here */

    $rootScope.flags.AllPageHasBeenOpenedAlready = true;
  }
)

/* 
And so forth until the last one...
*/

.controller('ContactController', function($scope, $rootScope, $state) {
    // If $rootScope.flags does not already exist, create it as an empty object
    $rootScope.flags = $rootScope.flags || {};

    if($rootScope.flags.ContactPageHasBeenOpenedAlready) {
      console.log("Contact page was already opened, not doing anything.");
      return;
    }

    /* Whatever you normally do in ContactController goes here */

    $rootScope.flags.ContactPageHasBeenOpenedAlready = true;
  }
);