{"id":24396,"date":"2016-04-19T00:45:36","date_gmt":"2016-04-18T16:45:36","guid":{"rendered":"https:\/\/case.ntu.edu.tw\/blog\/?p=24396"},"modified":"2018-08-03T14:32:59","modified_gmt":"2018-08-03T06:32:59","slug":"%e3%80%90%e6%8e%a2%e7%b4%a215-2%e3%80%91%e6%b7%b7%e6%b2%8c%e8%88%87%e7%a7%a9%e5%ba%8f%ef%bc%9a%e7%94%b1-logistic-map-%e8%ab%87%e8%b5%b7","status":"publish","type":"post","link":"https:\/\/case.ntu.edu.tw\/blog\/?p=24396","title":{"rendered":"\u3010\u63a2\u7d2215-2\u3011\u6df7\u6c8c\u8207\u79e9\u5e8f\uff1a\u7531 Logistic map \u8ac7\u8d77"},"content":{"rendered":"<p><strong><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-24408\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/14-2-p03fy7fw.jpg\" alt=\"14-2 p03fy7fw\" width=\"624\" height=\"351\" srcset=\"https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/14-2-p03fy7fw.jpg 624w, https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/14-2-p03fy7fw-300x168.jpg 300w, https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/14-2-p03fy7fw-600x338.jpg 600w\" sizes=\"(max-width: 624px) 100vw, 624px\" \/><\/strong><\/p>\n<p><strong>\u8b1b\u5e2b\uff5c<\/strong>\u6771\u83ef\u5927\u5b78\u61c9\u7528\u6578\u5b78\u7cfb \u73ed\u69ae\u8d85\u6559\u6388<br \/>\n<strong>\u64b0\u6587\uff5c<\/strong>\u9673\u6587\u7fca<\/p>\n<p>\u6709\u4e9b\u81ea\u7136\u754c\u7684\u7269\u9ad4\uff0c\u5927\u81f3\u50cf\u96f2\u3001\u6d77\u5cb8\u7dda\uff0c\u5c0f\u81f3\u50cf\u5fae\u8840\u7ba1\u7db2\u3001\u82b1\u6930\u83dc\uff0c\u5b83\u5011\u7684\u5f62\u72c0\u4e0d\u80fd\u7528\u7c21\u55ae\u3001\u7d93\u5178\u7684\u5f62\u72c0\u53bb\u5206\u985e\uff0c\u4f8b\u5982\u65b9\u5f62\u8ddf\u5713\u5f62\uff0c\u4f46\u4ed4\u7d30\u53bb\u89c0\u5bdf\uff0c\u6703\u767c\u73fe\u5b83\u5011\u770b\u4f3c\u8907\u96dc\u7684\u5916\u89c0\uff0c\u5728\u4efb\u610f\u5c0f\u7684\u5c3a\u5ea6\u4e0b\u7684\u7d30\u5c0f\u7d50\u69cb\uff0c\u5176\u5be6\u662f\u7531\u7c21\u55ae\u7684\u5e7e\u4f55\u5f62\u72c0\u6240\u7d44\u6210\u7684\uff0c\u795e\u5947\u7684\u662f\uff0c\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\u4e0b\u6703\u6301\u7e8c\u51fa\u73fe\u985e\u4f3c\u7684\u5e7e\u4f55\u5f62\u72c0\uff0c\u9019\u6a23\u7684\u60c5\u6cc1\u70ba\u81ea\u76f8\u4f3c\u6027(self-similarity)\u3002\u6578\u5b78\u5bb6\u60f3\u8981\u4e86\u89e3\uff0c\u8907\u96dc\u7684\u8868\u8c61\u662f\u4e0d\u662f\u7531\u751a\u9ebc\u7c21\u55ae\u4e0d\u8b8a\u7684\u6a5f\u5236\uff0c\u5f9e\u9019\u4e9b\u7d30\u5fae\u7684\u7d50\u69cb\u6f14\u8b8a\u800c\u4f86\uff1f<\/p>\n<p><span style=\"color: #008080;\"><strong>\u25cfLogistic map<br \/>\n<\/strong><\/span>\u5f9eRobert May\u5206\u6790\u7684\u55ae\u5cf0\u6620\u5c04(Logistic map)\u8aaa\u8d77\uff0c\u5b83\u662f\u4e00\u7a2e\u4e8c\u6b21\u7684\u591a\u9805\u5f0f\u591a\u9805\u5f0f<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-24398\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/3-1.png\" alt=\"3-1\" width=\"156\" height=\"38\" srcset=\"https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/3-1.png 156w, https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/3-1-150x38.png 150w\" sizes=\"(max-width: 156px) 100vw, 156px\" \/><!--more--><\/p>\n<p>\u5728\u9019\u88e1\u9700\u8981\u5148\u8aaa\u660e\u6620\u5c04(map)\u7684\u6982\u5ff5\uff0c\u5b58\u5728\u4e00\u500b\u51fd\u6578\u8ddf\u521d\u59cb\u503c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_449b2b4dfb893846d176eaa51721d649.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \uff0c\u5c07\u9019\u500b\u521d\u59cb\u503c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_449b2b4dfb893846d176eaa51721d649.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u5e36\u5165\u51fd\u6578\u5f97\u5230 <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_d81d08e7061e1d570cd5d37cf4e3d109.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>  \uff0c\u518d\u5c07 <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_d81d08e7061e1d570cd5d37cf4e3d109.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u5e36\u5165\u51fd\u6578\u5f97\u5230 <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b2c324370551b9b53d56215b7bc43e25.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u2026\u2026\uff0c\u5982\u6b64\u4e0d\u65b7\u91cd\u8907\u4e0b\u53bb\u3002\u5728\u5206\u6790\u55ae\u5cf0\u6620\u5c04\u6642\uff0c\u6703\u5728\u539f\u672c\u7684\u5716\u4e0a\u518d\u52a0\u4e00\u689d\u7dda x=y \uff0c\u5982\u6b64\u8fed\u4ee3\u7684\u904e\u7a0b\u53ef\u4ee5\u4e00\u76ee\u4e86\u7136\u3002\u55ae\u5cf0\u6620\u5c04\u539f\u672c\u88ab\u7528\u4f5c\u4eba\u53e3\u5b78\u7684\u6a21\u578b\uff0c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u662f\u4ecb\u65bc0\u8ddf1\u4e4b\u9593\uff0c\u7528\u4f86\u4ee3\u8868\u7b2cn\u5e74\u7684\u4eba\u53e3\u6578\u76ee\uff1br\u662f\u4e00\u500b\u5f71\u97ff\u4eba\u53e3\u6210\u9577\u7684\u53c3\u6578\uff0c\u6839\u64da\u7e41\u6b96\u548c\u9913\u6b7b\u7387\u6240\u5f97\uff0c\u5728\u9019\u88e1\u6211\u5011\u8a0e\u8ad6 0\u2264 r \u22644 \u7684\u60c5\u6cc1\u3002<\/p>\n<p>\u7576 r&lt;1 \u6642\uff0c\u7d93\u904e\u591a\u6b21\u8fed\u4ee3\u5f8c\u767c\u73fe <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u21920\uff1b\u75761&lt;r&lt;3\u6642\uff0c\u767c\u73fe <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u6703\u8da8\u8fd1\u4e00\u500b\u56fa\u5b9a\u503c(attracting fixed point)\u3002\u7576r\u8d8a\u4f86\u8d8a\u5927\uff0c\u50cf\u662f r=3.3 \u6642\uff0c\u6703\u767c\u73fe\u8fed\u4ee3\u591a\u6b21\u4ee5\u5f8c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u6703\u5728\u5169\u500b\u503c\u4e4b\u9593\u9707\u76ea(period-2 cycle)\uff1br=3.5 \u6642\uff0c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u5247\u6703\u5728\u56db\u500b\u503c\u4e4b\u9593\u9707\u76ea(period-4 cycle)\u3002\u70ba\u4e86\u5206\u6790\u4e0d\u540c r \u7684\u60c5\u6cc1\uff0c\u6211\u5011\u5c07\u4e0d\u540c\u7684 r \u8207\u5176\u5c0d\u61c9\u7684 <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> (n\u2192\u221e)\u4f5c\u5716\uff0c\u6b64\u5716\u7a31\u5206\u6b67\u5716(Bifurcation diagram)\u3002\u7576\u53c3\u6578 r \u8b8a\u5316\u6642\uff0c\u53ef\u4ee5\u767c\u73fe\u4e00\u958b\u59cb\u5e7e\u4e4e\u6240\u6709\u7684\u9ede\u90fd\u6703\u8da8\u8fd1\u4e8c\u5468\u671f\u9ede\uff0c\u518d\u4f86\u662f\u56db\u5468\u671f\u9ede\u3001\u516b\u9031\u671f\u9ede\u7b49\u7b49\uff0c\u800c\u5728\u6709\u4e9b\u53c3\u6578\u4e2d\u6211\u5011\u53ef\u4ee5\u770b\u51fa\u4efb\u4e00\u500b\u9ede\u51fa\u767c\u5f8c\u90fd\u4e0d\u6703\u505c\u5728\u67d0\u4e00\u500b\u9031\u671f\u9ede\uff0c\u5728\u9019\u4e9b\u53c3\u6578\u4e2d\u56e0\u70ba\u6211\u5011\u6c92\u6709\u8fa6\u6cd5\u8ffd\u8e64\u5b83\u7684\u7d42\u6975\u884c\u70ba\uff0c\u6240\u4ee5\u901a\u5e38\u7528\u6a5f\u7387\u6216\u8005\u662f\u6e2c\u5ea6\u7684\u8a9e\u8a00\u4f86\u63cf\u8ff0\u9019\u4e9b\u53c3\u6578\u7684\u884c\u70ba\uff0c <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u56de\u5230\u9019\u500b\u5340\u9593\u7684\u6a5f\u7387\u6709\u591a\u5927\u3002\u5728\u6b64\u5b9a\u7fa9\u5169\u500b\u65b0\u7684\u540d\u8a5e\uff0c\u4e00\u500b\u662f\u7a69\u5b9a\u53c3\u6578\uff0c\u4ee3\u8868\u5728\u6b64\u53c3\u6578\u4e0b\u7684 <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u6703\u88ab\u67d0\u500b\u9031\u671f\u9ede\u6240\u5438\u5f15\uff1b\u4e00\u500b\u662f\u96a8\u6a5f\u53c3\u6578\uff0c\u4ee3\u8868\u5728\u6b64\u53c3\u6578\u4e0b <span class='MathJax_Preview'><img src='https:\/\/case.ntu.edu.tw\/blog\/wp-content\/plugins\/latex\/cache\/tex_b4f72043bc4fcc7a87444ed73deea076.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> \u7684\u7d42\u6975\u884c\u70ba\u6700\u5f8c\u4e0d\u6703\u505c\u5728\u67d0\u4e9b\u9ede\u3002\u7e3d\u7684\u4f86\u8aaa\uff0c\u7576 r \u2248 3.57 \u6642\uff0c\u6620\u5c04\u6703\u8b8a\u5f97\u8d8a\u4f86\u8d8a\u6df7\u4e82\u3001\u8fed\u4ee3\u5f8c\u4e0d\u80fd\u8da8\u5411\u67d0\u4e9b\u9ede\uff0c\u4f46\u662f\u7576 r &gt; 3.57\uff0c\u6703\u767c\u73fe\u5716\u5448\u73fe\u4e0d\u80fd\u9810\u671f\u7684\u3001\u6709\u79e9\u5e8f(\u7a69\u5b9a\u53c3\u6578)\u53ca\u6df7\u4e82(\u96a8\u6a5f\u53c3\u6578)\u4ea4\u932f\u51fa\u73fe\u7684\u60c5\u6cc1\uff0c\u9019\u5c31\u662f\u4e00\u958b\u59cb\u6240\u63d0\uff0c\u4e00\u500b\u7c21\u55ae\u6a5f\u5236\u65e2\u57fa\u672c\u537b\u53c8\u80fd\u88fd\u9020\u51fa\u76f8\u7576\u8907\u96dc\u7684\u73fe\u8c61\u3002\u540c\u6642\u7576\u6211\u5011\u653e\u5927\u5206\u6b67\u5716\u4f86\u770b\uff0c\u6bd4\u5982\u50cf\u662fr \u2248 3.83 \u9078\u64c7\u6642\u4e0d\u65b7\u5730\u653e\u5927\uff0c\u6703\u767c\u73fe\u539f\u672c\u5206\u6b67\u5716\u7684\u6a21\u6a23\u6703\u4e0d\u65b7\u51fa\u73fe\uff0c\u9019\u60c5\u6cc1\u4e5f\u662f\u6211\u5011\u4e00\u958b\u59cb\u63d0\u7684\u81ea\u76f8\u4f3c\u6027\u3002<\/p>\n<p>\u5982\u6b64\u60f3\u554f\uff0c\u55ae\u5cf0\u6620\u5c04\u6709\u4e00\u822c\u6027\u55ce\uff1f\u4e5f\u5c31\u662f\u8aaa\u53ef\u4ee5\u7528\u63a2\u8a0e\u55ae\u5cf0\u6620\u5c04\u7684\u904e\u7a0b\u53bb\u4e86\u89e3\u5176\u4ed6\u7684\u6df7\u6c8c\u73fe\u8c61\u55ce\uff1f\u9996\u5148 John Guckenheimer \u8ddf Michal Misiurewicz \u8b49\u660e\u4efb\u4f55\u55ae\u5cf0\u6620\u5c04\u52a0\u4e0a\u4e00\u4e9b\u5f88\u5bec\u7684\u689d\u4ef6\uff0c\u5373\u6703\u548c\u67d0\u500b\u6210\u9577\u7684\u6620\u5c04\u62d3\u6a38\u5171\u8edb(topological conjugate)\uff1b\u63a5\u8457 John Milnor \u8ddf William Thurston \u4e5f\u63d0\u51fa\u5b9a\u7406\uff0c\u8aaa\u660e\u5e7e\u4e4e\u6240\u6709\u7684\u55ae\u5cf0\u6620\u5c04\u90fd\u548c\u6210\u9577\u6620\u5c04\u4e2d\u7684\u67d0\u53c3\u6578\u6709\u76f8\u540c\u7684\u52d5\u614b\u884c\u70ba\uff0c\u7c21\u55ae\u4f86\u8aaa\u662f\u7814\u7a76\u55ae\u5cf0\u6620\u5c04\u6703\u662f\u57fa\u672c\u3001\u7c21\u55ae\u4e14\u5177\u6709\u4ee3\u8868\u6027\u7684\u4e00\u985e\u51fd\u6578\u3002<\/p>\n<p>\u53e6\u5916\u9084\u6709\u5176\u4ed6\u554f\u984c\uff0c\u5176\u4e00\u662f\u96a8\u6a5f\u53c3\u6578\u662f\u5426\u5920\u591a\uff1f\u5728\u9019\u88e1\u5920\u591a\u7684\u610f\u601d\u662f\uff0c\u7576\u6211\u5011\u5c07\u96a8\u6a5f\u53c3\u6578\u6536\u96c6\u8d77\u4f86\uff0c\u7528\u5c3a\u53bb\u91cf\u662f\u5426\u53ef\u4ee5\u91cf\u5230\u5927\u65bc0\u7684\u9577\u5ea6\uff1f\u4e5f\u7a31\u5b83\u662f\u5426\u6709\u6b63\u7684\u52d2\u767d\u683c\u6e2c\u5ea6(Lebesgue measure)\u3002\u4e4b\u5f8c\u7531 Michael Benedicks\u3001Lennart Carleson \u8207 Michael Jacobson \u8b49\u660e\u96a8\u6a5f\u53c3\u6578\u7684\u78ba\u6709\u6b63\u7684\u52d2\u767d\u683c\u6e2c\u5ea6\u3002\u5176\u4e8c\u662f\u5728\u53c3\u6578\u96c6[0, 4]\u4e2d\uff0c\u7a69\u5b9a\u53c3\u6578\u662f\u4e0d\u662f\u7a20\u5bc6\u7684\uff1fMikhail Lyubich\u3001Jacek Graczyk \u7b49\u4eba\u5206\u5225\u8b49\u660e\u7a69\u5b9a\u53c3\u6578\u78ba\u5be6\u662f\u7a20\u5bc6\u7684\uff0c\u53e6\u5916\u4e5f\u5f97\u77e5\u7a69\u5b9a\u53c3\u6578\u7d93\u5f97\u8d77\u64fe\u52d5\uff0c\u9019\u500b\u610f\u601d\u662f\u5982\u679c\u78ba\u5b9a\u67d0\u500b\u53c3\u6578\u662f\u7a69\u5b9a\u53c3\u6578\uff0c\u5247\u9078\u53d6\u4e00\u500b\u7a0d\u5fae\u9060\u96e2\u8a72\u53c3\u6578\u4e00\u9ede\u9ede\u7684\u53c3\u6578\uff0c\u5b83\u4e5f\u6703\u662f\u7a69\u5b9a\u53c3\u6578\uff0c\u7136\u800c\u96a8\u6a5f\u53c3\u6578\u5247\u5426\uff0c\u4e0d\u80fd\u4fdd\u8b49\u9060\u96e2\u8a72\u53c3\u6578\u4e00\u9ede\u9ede\u662f\u5426\u80fd\u518d\u9078\u5230\u96a8\u6a5f\u53c3\u6578\u3002\u5176\u4e09\u662f\u82e5\u5c07\u9078\u53d6\u7a69\u5b9a\u53c3\u6578\u7684\u6a5f\u7387\u8207\u96a8\u6a5f\u53c3\u6578\u7684\u6a5f\u7387\u52a0\u8d77\u4f86\u6703\u4e0d\u6703\u662f\u4e00\uff1f Mikhail Lyubich \u8b49\u51fa\uff0c\u518d\u6263\u6389\u4e00\u500b\u6e2c\u5ea6\u96f6\u7684\u96c6\u5408\u4e4b\u5f8c\uff0c\u57280\u82071\u9593\u7684\u5340\u9593\u4e2d\u4efb\u610f\u9078\u53d6\u4e00\u500b\u53c3\u6578\uff0c\u5b83\u4e0d\u662f\u7a69\u5b9a\u53c3\u6578\u5c31\u662f\u96a8\u6a5f\u53c3\u6578\u3002\u4e0d\u904e\u5176\u5be6\u9664\u4e86\u9019\u5169\u8005\u5916\u9084\u6709\u5176\u4ed6\u7684\u53c3\u6578\uff0c\u53ea\u4e0d\u904e\u5b83\u5011\u6574\u9ad4\u7684\u52d2\u767d\u683c\u6e2c\u5ea6\u662f0\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong><strong>\u25cf<\/strong>\u63cf\u8ff0\u6df7\u6c8c<br \/>\n<\/strong><\/span>\u7136\u5f8c\u662f\u6211\u5011\u8981\u600e\u9ebc\u63cf\u8ff0\u6df7\u6c8c\uff1f\u9019\u88e1\u6709\u5169\u500b\u6982\u5ff5\u3002\u7b2c\u4e00\u500b\u662f\u5982James. A. Yorke\u8207Tien-Yien Li\u5408\u5beb\u7684\u8ad6\u6587Periodic 3 Implies Chaos\u63cf\u8ff0\uff0c\u82e5 f:R\u2192R \u662f\u4e00\u500b\u9023\u7e8c\u51fd\u6578\uff0c\u4e14f\u6709\u4e00\u500b\u9031\u671f\u4e09\u7684\u9ede\uff0c\u5247<\/p>\n<p>(1)\u00a0\u00a0 \u5c0d\u4efb\u4f55n\u90fd\u5b58\u5728\u9031\u671fn\u7684\u9031\u671f\u9ede<\/p>\n<p>(2)\u00a0\u00a0 \u5b58\u5728\u4e00\u500b\u4e0d\u53ef\u6578\u7684\u5b50\u96c6S\uff0c\u5c0d\u5176\u4e2d\u4efb\u5169\u9edex\u3001y\u6211\u5011\u6709<img decoding=\"async\" class=\"aligncenter size-full wp-image-24400\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/3-2.png\" alt=\"3-2\" width=\"257\" height=\"61\" \/><\/p>\n<p>(3)\u00a0\u00a0 \u5c0d\u4efb\u4f55\u4e00\u500b\u9031\u671f\u9edeP\u4ee5\u53caS\u5167\u7684\u9edex\uff0c\u6211\u5011\u6709<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-24401\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/3-3.png\" alt=\"3-3\" width=\"237\" height=\"41\" \/><\/p>\n<p>\u5f0f(1)\u7684\u610f\u601d\u662f\u5728\u4e0d\u65b7\u8fed\u4ee3\u7684\u904e\u7a0b\u4e2d\uff0c\u4efb\u5169\u9ede\u7684\u8ecc\u8de1\u7121\u7aae\u591a\u6b21\u6703\u7121\u9650\u63a5\u8fd1\uff1b\u5f0f(2)\u7684\u610f\u601d\u662f\u9019\u5169\u500b\u9ede\u7d42\u7a76\u6703\u5206\u958b\uff1b\u5f0f(3)\u7684\u610f\u601d\u662f\uff0c\u5728S\u88e1\u9762\u7684\u9ede\u7d55\u5c0d\u4e0d\u6703\u88ab\u67d0\u500b\u9031\u671f\u9ede\u5438\u5f15\uff0c\u4e5f\u5c31\u662f\u5b83\u4e0d\u6703\u505c\u5728\u67d0\u500b\u56fa\u5b9a\u7684\u4f4d\u7f6e\u3002\u800c\u6eff\u8db3\u9019\u4e9b\u689d\u4ef6\u7684\u96c6\u5408S\uff0c\u5b83\u7684\u884c\u70ba\u9ad4\u73fe\u7684\u6df7\u6c8c\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>\u7b2c\u4e8c\u500b\u662fDevaney Chaos\uff0c\u6240\u63cf\u8ff0\u7684\u4e09\u500b\u689d\u4ef6\uff1a<\/p>\n<p>(1)\u00a0\u00a0 \u5c0d\u521d\u59cb\u689d\u4ef6\u654f\u611f(sensitive dependence)\uff0c\u5728\u96c6\u5408U\u88e1\u9762\u7684\u4efb\u5169\u9ede\uff0c\u7d93\u904e\u5e7e\u6b21\u8fed\u4ee3\u5f8c\uff0c\u5b83\u5011\u4e4b\u9593\u7684\u8ddd\u96e2\u6703\u8d8a\u4f86\u8d8a\u5927\u3002<\/p>\n<p>(2)\u00a0\u00a0 \u7cfb\u7d71\u5177\u50b3\u905e\u6027(topological transitivity)\uff0c\u7d66\u5169\u500b\u96c6\u5408U\u8ddfV\uff0c\u5b58\u5728\u4e00\u500b\u6b21\u6578n\uff0c\u4f7f\u5f97U\u5728\u7cfb\u7d71n\u6b21\u8fed\u4ee3\u5f8c\u7e3d\u6703\u548cV\u6709\u4ea4\u96c6\u3002<\/p>\n<p>(3)\u00a0\u00a0 \u7a20\u5bc6\u7684\u9031\u671f\u9ede(Dense periodic orbits)\uff0c\u7a20\u5bc6\u7684\u9031\u671f\u9ede\u5176\u5be6\u662f\u8aaa\u660e\u9019\u7cfb\u7d71\u662f\u4e82\u4e2d\u6709\u5e8f\uff0c\u56e0\u70ba\u9031\u671f\u9ede\u662f\u500b\u6709\u5e8f\u3001\u6709\u898f\u5247\u7684\u72c0\u614b\uff0c\u5b83\u5e0c\u671b\u7cfb\u7d71\u88e1\u9762\u4e0d\u53ea\u662f\u4e82\uff0c\u800c\u662f\u4e82\u4e2d\u6709\u5e8f\u3002<\/p>\n<p>\u9806\u9053\u4e00\u63d0\uff0c\u9019\u500b\u6982\u5ff5\u5176\u4e2d\u5169\u500b\u5b9a\u7fa9\u53ef\u4ee5\u63a8\u51fa\u53e6\u5916\u4e00\u500b\u5b9a\u7fa9\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong><strong>\u25cf<\/strong>\u7d50\u8a9e<br \/>\n<\/strong><\/span>\u5176\u5be6\u81f3\u4eca\u9084\u672a\u6709\u4e00\u500b\u6240\u6709\u4eba\u90fd\u8a8d\u540c\u7684\u5b9a\u7fa9\uff0c\u53c8\u6df7\u6c8c\u53ea\u8aaa\u5b83\u5168\u90e8\u90fd\u662f\u6df7\u4e82\u7684\u5176\u5be6\u6709\u4e9b\u7c97\u7cd9\uff0c\u5b83\u4e8b\u5be6\u4e0a\u662f\u90e8\u5206\u6709\u5e8f\u7684\uff0c\u5982\u524d\u9762\u6240\u8ff0\u7684\u55ae\u5cf0\u6620\u5c04\uff0c\u6709\u6a5f\u7387\u4e00\u7684\u6a5f\u6703\u8981\u4e0d\u662f\u9078\u5230\u96a8\u6a5f\u53c3\u6578\u5c31\u662f\u7a69\u5b9a\u53c3\u6578\uff0c\u7cfb\u7d71\u4e2d\u4ea4\u7e54\u7a7f\u63d2\u7c21\u55ae\u7d50\u69cb\u8207\u8907\u96dc\u7d50\u69cb\u3002\u7e3d\u62ec\u4f86\u8aaa\u7c21\u55ae\u7684\u6a5f\u5236\u53ef\u7522\u751f\u51fa\u8907\u96dc\u7684\u73fe\u8c61\uff0c\u53c8\u6df7\u6c8c\u548c\u79e9\u5e8f\u662f\u5bc6\u4e0d\u53ef\u5206\u7684\u3002<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u53c3\u8003\u8cc7\u6599\uff1a<\/strong><br \/>\n1. \u73ed\u69ae\u8d85\uff0c\u6dfa\u8ac7\u52d5\u529b\u7cfb\u7d71\u4e2d\u7684\u798d\u798f\u76f8\u501a\uff0c\u6578\u7406\u4eba\u6587\u7b2c\u516d\u671f<br \/>\n2. Steven H. Strogatz, Nonlinear Dynamics and Chaos<\/p>\n<p><strong style=\"color: #000000;\">--<br \/>\n\u672c\u6587\u6574\u7406\u81ea\uff1a105\/4\/9\u00a0<\/strong><strong style=\"color: #000000;\">\u7531\u73ed\u69ae\u8d85\u6559\u6388\u5728\u81fa\u5927\u601d\u4eae\u9928\u570b\u969b\u6703\u8b70\u5ef3\u6240\u4e3b\u8b1b\u4e4b\u300c\u7c21\u55ae\u7684\u958b\u59cb\uff0c\u537b\u901a\u5f80\u8907\u96dc\uff1f\uff01\u300d\u6f14\u8b1b\u5167\u5bb9\u3002<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/line.me\/R\/ti\/p\/%40fcx6182w\" target=\"_blank\" rel=\"noopener noreferrer\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/scdn.line-apps.com\/n\/line_add_friends\/btn\/zh-Hant.png\" alt=\"\u52a0\u5165\u597d\u53cb\" width=\"212\" height=\"36\" border=\"0\"><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8b1b\u5e2b\uff5c\u6771\u83ef\u5927\u5b78\u61c9\u7528\u6578\u5b78\u7cfb \u73ed\u69ae\u8d85\u6559\u6388 \u64b0<\/p>\n","protected":false},"author":20,"featured_media":24408,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3293,2413],"tags":[2475,2512,2509,2510,2511,149,2417],"aioseo_notices":[],"jetpack_featured_media_url":"https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2016\/04\/14-2-p03fy7fw.jpg","_links":{"self":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/24396"}],"collection":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/users\/20"}],"replies":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=24396"}],"version-history":[{"count":11,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/24396\/revisions"}],"predecessor-version":[{"id":32523,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/24396\/revisions\/32523"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/media\/24408"}],"wp:attachment":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=24396"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=24396"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=24396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}