Add your Twitter timeline to your CakePHP website. I also included The MVC for your website.

index.ctp

<?php if(!isset($tweets['error'])) : ?>
    <h2><?php echo __("Twitter"); ?></h2>
    <?php foreach($tweets as $k => $v) :?>
        <div class="sidebar-tweet">
            <p>
                <?php echo $v['text']; ?>
                <br/>
                <em><?php echo $v['created_at']; ?></em>
            </p>
        </div> <!-- .tweet -->
    <?php endforeach; ?>
<?php endif; ?>

PBTwitter.php

<?php

/**
 *
 * Twitter class to display your tweets in a CakePHP application
 *
 *
 * @author      Pierre Baron <[email protected]>
 *
 * @copyright   2012-2012 Pierre Baron
 * @link        http://www.pierrebaron.fr
 * @package     app.Lib
 * @since       October 2012
 * @version     1.0
 */
class PBTwitter {

    /**
     * Twitter API base URL
     *
     * @var integer
     * @access private
     */
    private $baseURL = 'https://api.twitter.com/1.1/statuses/user_timeline.json';

    /**
     * Default options for twitterapi array
     *
     * @var integer
     * @access private
     */
    private $TWITTERAPI = array(
        'count'            => 10,
        'exclude_replies'  => true,
        'include_entities' => true,
        'include_rts'      => true,
        'trim_user'        => true
    );

    /**
     * Default options for pbtwitter array
     *
     * @var integer
     * @access private
     */
    private $PBTWITTER = array(
        'cache_duration' => '+30mins',
        'hashtag_class'  => 'pbtwitter-hashtag',
        'url_class'      => 'pbtwitter-url',
        'user_class'     => 'pbtwitter-user'
    );

    /**
     * twitterapi array
     *
     * @var string
     * @access private
     */
    private $twitterapi;

    /**
     * pbtwitter array
     *
     * @var integer
     * @access private
     */
    private $pbtwitter;


     /**
      * Find tweets
      *
      * @param  string  $name       The name of your Twitter account
      * @param  array   $twitterapi the options for the Twitter api, see $TWITTERAPI for options
      * @param  array   $pbtwitter  the options for the PBTwitter class, see $PBTWITTER for options
      * @access public
      * @return array
      */
    public function find($name, $twitterapi = array(), $pbtwitter = array()) {
        // Merge the arrays with the default ones
        $this->twitterapi = array_merge($this->TWITTERAPI, $twitterapi);
        $this->pbtwitter    = array_merge($this->PBTWITTER, $pbtwitter);
        
        if(($twits = Cache::read('Twitter.lines')) == false) {
            $twits = $this->_getTwits($name);
            
            Cache::set(array('duration' => $this->pbtwitter['cache_duration'])); 
            Cache::write('Twitter.lines',$twits); 
        }
        
        return $twits;
    }


    /**
     * Get all tweets
     *
     * @param   string $name    The name of your Twitter account
     * @access private
     * @return array
     */
    private function _getTwits($name){
        // Create the full url with the options
        $twitterURL  = $this->baseURL.'?screen_name='.$name;
        $twitterURL .= '&include_entities='.$this->twitterapi['include_entities'];
        $twitterURL .= '&include_rts='.$this->twitterapi['include_rts'];
        $twitterURL .= '&count='.$this->twitterapi['count'];
        $twitterURL .= '&trim_user='.$this->twitterapi['trim_user'];
        $twitterURL .= '&exclude_replies='.$this->twitterapi['exclude_replies'];
        // send the Get request
        App::uses('HttpSocket', 'Network/Http');
        $httpSocket = new HttpSocket();
        $tweets = $httpSocket->get($twitterURL);
        // parse the JSON string
        $tweets = json_decode($tweets);
        $out = array();
        // Add the tweet to a new array
        foreach($tweets as $k => $tweet) {
            $out[$k] = $this->_createTweetArray($tweet);
            
            // Add links for Hashtags, urls, Users
            if(!isset($out['errors']) && $this->twitterapi['include_entities'])
                $out[$k] = $this->_linkEntities($out[$k]);
        }
        return $out;
    }


    /**
     * Create an array from an object
     *
     * @param   string $name    The name of your Twitter account
     * @access private
     * @return array
     */
    private function _createTweetArray($tweetObject) {
        // create new empty array
        $tweetArray = array();
        // Transform Object to Array
        foreach($tweetObject as $key => $value) {
            $tweetArray[$key] = $value;
        }
        
        return $tweetArray;
    }


    /**
     * Add links for hastags, urls, users
     *
     * @param   array $tweetArray   array of one tweet
     * @access private
     * @return array
     */
    private function _linkEntities($tweetArray) {
        // Add link for each hashtag
        foreach($tweetArray['entities']->hashtags as $k => $v) {
            $tweetArray['text'] = preg_replace(
                '~#'.$v->text.'~', 
                '<a href="http://www.twitter.com/search/%23'.$v->text.'" class="'.$this->pbtwitter['hashtag_class'].'"'.'>#'.$v->text.'</a>', 
                $tweetArray['text']
            );
        }

        // Add link for each url
        foreach($tweetArray['entities']->urls as $k => $v) {
            $tweetArray['text'] = preg_replace(
                '~'.$v->url.'~', 
                '<a href="'.$v->url.'" class="'.$this->pbtwitter['url_class'].'"'.'>'.$v->url.'</a>', 
                $tweetArray['text']
            );
        }
        
        // Add link for each user
        foreach($tweetArray['entities']->user_mentions as $k => $v) {
            $tweetArray['text'] = preg_replace(
                '~@'.$v->screen_name.'~', 
                '<a href="http://www.twitter.com/'.$v->screen_name.'" class="'.$this->pbtwitter['user_class'].'"'.'>@'.$v->screen_name.'</a>', 
                $tweetArray['text']
            );
        }

        return $tweetArray;
    }
}

Twitter.php

<?php
class Twitter extends AppModel{
    public $useTable = false;
}

TwitterController.php

<?php
class TwitterController extends AppController{

    public $uses = array('Twitter');

    public function index(){
        App::import('Lib', 'PBTwitter');
        
        $Twitter = new PBTwitter();
        return $Twitter->find("YourTwitterName", array('count' => 3));
    }
}