{"id":22970,"date":"2020-12-26T12:51:57","date_gmt":"2020-12-26T04:51:57","guid":{"rendered":"https:\/\/swarma.org\/?p=22970"},"modified":"2020-12-26T12:51:57","modified_gmt":"2020-12-26T04:51:57","slug":"%e7%a6%bb%e6%95%a3%e6%b7%b7%e6%b2%8c%e4%bc%a0%e5%a5%87%e4%b8%a8%e9%99%88%e5%85%b3%e8%8d%a3","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=22970","title":{"rendered":"\u79bb\u6563\u6df7\u6c8c\u4f20\u5947\u4e28\u9648\u5173\u8363"},"content":{"rendered":"
<\/p>\n

\u5bfc\u8bed<\/p>\n

\u7f57\u4f2f\u7279\u00b7\u6885\u662f\u4e00\u4f4d\u6210\u7ee9\u5353\u8d8a\u3001\u4e94\u4f4d\u4e00\u4f53\u7684\u5b66\u8005\uff1a\u7406\u8bba\u7269\u7406\u5b66\u5bb6\u3001\u5e94\u7528\u6570\u5b66\u5bb6\u3001\u6570\u5b66\u751f\u6001\u5b66\u5bb6\u3001\u6570\u503c\u4f20\u67d3\u75c5\u5b66\u5bb6\u548c\u590d\u6742\u6027\u79d1\u5b66\u5bb6\u3002\u4ed6\u662f\u82f1\u56fd\u6700\u6709\u5f71\u54cd\u7684\u79d1\u5b66\u5bb6\u4e4b\u4e00\uff0c\u5728\u751f\u7269\u591a\u6837\u6027\u3001\u7fa4\u4f53\u52a8\u529b\u5b66\u548c\u6d41\u884c\u75c5\u5b66\u65b9\u9762\u90fd\u505a\u51fa\u4e86\u5960\u57fa\u6027\u7684\u8d21\u732e\uff0c\u6210\u5c31\u6590\u7136\u3002\u662f\u7684\uff0c\u6709\u4e9b\u4f20\u5947\u79d1\u5b66\u5bb6\u5c31\u662f\u8fd9\u4e48\u4f20\u5947\u3002
\u9648\u5173\u8363 | \u4f5c\u8005
\u9093\u4e00\u96ea | \u7f16\u8f91<\/p>\n

<\/div>\n

\u7f57\u4f2f\u7279\u00b7\u6885\uff08Robert McCredie May, 1936-2020\uff09\u65e0\u7591\u662f\u4f20\u5947\u5f0f\u4eba\u7269\u4e2d\u7684\u4f20\u5947\u3002<\/p>\n


Sir Robert M. May\uff081936-2020\uff09<\/p>\n

\u7f57\u4f2f\u7279\uff0c\u6635\u79f0 Bob\uff0c\u56e0\u8001\u5e74\u75f4\u5446\u75c7\u5e76\u53d1\u80ba\u708e\u4e8e2020\u5e744\u670828\u65e5\u5728\u82f1\u56fd\u725b\u6d25\u517b\u8001\u9662\u79bb\u4e16\uff0c\u4eab\u5e7484\u5c81\u3002<\/p>\n

9\u670824\u65e5\uff0c\u7f8e\u56fd\u751f\u6001\u5b66\u4f1a\u4f1a\u520a\uff08Bulletin of the Ecological Society of America\uff09\u5728\u7ebf\u53d1\u8868\u4e86\u666e\u6797\u65af\u987f\u5927\u5b66\u51e0\u4f4d\u751f\u6001\u5b66\u5bb6\u7684\u7eaa\u5ff5\u6587\u7ae0\uff0c\u5f00\u7bc7\u8d5e\u8bc4\u4fbf\u8bf4\uff1a\u201c\u5982\u679c\u80fd\u591f\u62e5\u6709\u4e00\u4e2a\u7cbe\u5f69\u7684\u804c\u4e1a\u4eba\u751f\uff0c\u6211\u4eec\u7edd\u5927\u591a\u6570\u4eba\u90fd\u4f1a\u611f\u5230\u65e0\u6bd4\u8363\u5e78\uff0c\u800cBob\u81f3\u5c11\u6709\u4e94\u4e2a\u3002\u201d<\/p>\n

\u867d\u7136\u8fd9\u53e5\u8bdd\u610f\u6307\u7f57\u4f2f\u7279\u00b7\u6885\u7684\u79d1\u5b66\u4eba\u751f\u7ecf\u5386\u4e86\u81f3\u5c11\u4e94\u4e2a\u8f89\u714c\u7684\u9636\u6bb5\uff0c\u4ed6\u786e\u5b9e\u4e5f\u662f\u4e00\u4f4d\u6210\u7ee9\u5353\u8d8a\u3001\u4e94\u4f4d\u4e00\u4f53\u7684\u5b66\u8005\uff1a\u7406\u8bba\u7269\u7406\u5b66\u5bb6\u3001\u5e94\u7528\u6570\u5b66\u5bb6\u3001\u6570\u5b66\u751f\u6001\u5b66\u5bb6\u3001\u6570\u503c\u4f20\u67d3\u75c5\u5b66\u5bb6\u548c\u590d\u6742\u6027\u79d1\u5b66\u5bb6\u3002\u4ed6\u662f\u82f1\u56fd\u6700\u6709\u5f71\u54cd\u7684\u79d1\u5b66\u5bb6\u4e4b\u4e00\uff0c\u5728\u751f\u7269\u591a\u6837\u6027\u3001\u7fa4\u4f53\u52a8\u529b\u5b66\u548c\u6d41\u884c\u75c5\u5b66\u65b9\u9762\u90fd\u505a\u51fa\u4e86\u5960\u57fa\u6027\u7684\u8d21\u732e\uff0c\u6210\u5c31\u6590\u7136\u3002<\/p>\n

\u8c08\u5230\u5b66\u672f\u6210\u5c31\uff0c\u4e0d\u77e5\u4ece\u4ec0\u4e48\u65f6\u5019\u5f00\u59cb\u5927\u5bb6\u4e60\u60ef\u4e86\u7528\u4e00\u628a\u53ef\u4ee5\u8ba1\u91cf\u7684\u5c3a\u5b50\u53bb\u91cf\u5ea6\u4e00\u4e0b\uff1a\u53d1\u51e0\u7bc7SNC\u4e86\uff1f\u6234\u51e0\u9876\u5e3d\u5b50\u4e86\uff1f\u5f97\u591a\u5c11\u4e2a\u5927\u5956\u4e86\uff1f\u5f53\u7136\uff0c\u5bf9\u4e8e\u8fd9\u4e9b\uff0c\u7f57\u4f2f\u7279\u00b7\u6885\u7684\u56de\u7b54\u5b8c\u5168\u4e0d\u662f\u4e00\u4e2a\u95ee\u9898\u3002<\/p>\n

\u8bb0\u5f55\u8868\u660e\uff0c\u5b66\u8005\u7f57\u4f2f\u7279\u00b7\u6885\u4e00\u751f\u53d1\u4e86224\u7bc7 Nature \u548c59\u7bc7 Science \uff0c\u5176\u4e2d\u6709\u8bb8\u591a\u79d1\u5b66\u8bba\u6587\u4e5f\u6709\u4e0d\u5c11\u5b66\u672f\u8bc4\u8bba\u3002\u4ed6\u7684h\u6307\u6570\u4e3a177\uff0c\u8fd8\u5728\u589e\u957f\u4e2d\u7684\u5f15\u7528\u603b\u6570\u8d85\u8fc7166,000\u3002<\/p>\n

\u8bb0\u5f55\u4e5f\u8868\u660e\uff0c\u540d\u51a0\u7235\u58eb\u548c\u725b\u6d25\u7537\u7235\u7684\u7f57\u4f2f\u7279\u00b7\u6885\u662f\u524d\u82f1\u56fd\u653f\u5e9c\u9996\u5e2d\u79d1\u5b66\u987e\u95ee\uff081995-2000\uff09\u3001\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1a\u9662\u58eb\u548c\u524d\u4e3b\u5e2d\uff082000-2005\uff09\u3001\u82f1\u56fd\u7687\u5bb6\u5de5\u7a0b\u9662\u3001\u7f8e\u56fd\u79d1\u5b66\u9662\u3001\u6fb3\u5927\u5229\u4e9a\u79d1\u5b66\u9662\u3001\u6b27\u6d32\u79d1\u5b66\u9662\uff08Academia Europaea\uff09\u7b49\u591a\u4e2a\u56fd\u5bb6\u548c\u5730\u533a\u79d1\u5b66\u9662\u9662\u58eb\uff0c\u5e76\u4e14\u8363\u81ba\u666e\u6797\u65af\u987f\u3001\u8036\u9c81\u3001\u6089\u5c3c\u3001ETH\u3001\u725b\u6d25\u3001\u54c8\u4f5b\u7b49\u591a\u6240\u540d\u6821\u7684\u8363\u8a89\u535a\u58eb\u5b66\u4f4d\u3002\u4ed6\u8fd8\u66fe\u4efb1913\u5e74\u6210\u7acb\u7684\u82f1\u56fd\u751f\u6001\u5b66\u4f1a\u4e3b\u5e2d\uff081992-1993\uff09\u3001\u666e\u6797\u65af\u987f\u5927\u5b66\u5b66\u672f\u59d4\u5458\u4f1a\u4e3b\u5e2d\uff081977-1988\uff09\u4ee5\u53ca\u5723\u5854\u83f2\u7814\u7a76\u6240\u79d1\u5b66\u59d4\u5458\u4f1a\u4e3b\u5e2d\uff0c\u7b49\u7b49\u3002<\/p>\n

\u8bb0\u5f55\u8fd8\u8868\u660e\uff0c\u79d1\u5b66\u5bb6\u7f57\u4f2f\u7279\u00b7\u6885\u83b7\u5956\u65e0\u6570\u3002\u4ee3\u8868\u6027\u7684\u6709\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1aCopley Medal\uff082007\uff09\u3001\u65e5\u672cBlue Planet Prize\uff082001\uff09\u3001\u745e\u58eb-\u610f\u5927\u5229Balzan Prize\uff081998\uff09\u3001\u745e\u5178\u7687\u5bb6\u79d1\u5b66\u9662Crafoord Prize\uff081996\uff09\u3001\u7f8e\u56fd\u751f\u6001\u5b66\u4f1aMacArthur Prize\uff081984\uff09\u7b49\u91cd\u5927\u5956\u9879\u3002\u5176\u4e2d\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1aCopley Medal\u662f\u4e16\u754c\u4e0a\u6700\u53e4\u8001\u6700\u8457\u540d\u7684\u79d1\u5b66\u5956\uff0c\u59cb\u4e8e1731\u5e74\uff0c\u83b7\u5956\u8005\u5305\u62ec\u4f17\u6240\u5468\u77e5\u7684\u5bcc\u5170\u514b\u6797\u3001\u54c8\u5bc6\u987f\u3001\u9ad8\u65af\u3001\u6cd5\u62c9\u7b2c\u3001\u4ea5\u59c6\u970d\u5179\u3001\u5409\u5e03\u65af\u3001\u95e8\u6377\u5217\u592b\u3001\u5362\u745f\u798f\u3001\u7231\u56e0\u65af\u5766\u3001\u666e\u6717\u514b\u3001\u6ce2\u6069\u3001\u54c8\u4ee3\u3001\u72c4\u62c9\u514b\u3001\u970d\u91d1\u3001\u5e0c\u683c\u65af\u7b49\u7b49\u3002\u800c\u745e\u5178\u7687\u5bb6\u79d1\u5b66\u9662\u7684 Crafoord Prize \u57281983\u5e74\u6388\u4e88\u6df7\u6c8c\u5b66\u5148\u9a71\u7231\u5fb7\u534e\u00b7\u6d1b\u4f26\u8328\uff08Edward N. Lorenz\uff09\u3002<\/p>\n

\u5176\u5b9e\u7f57\u4f2f\u7279\u00b7\u6885\u5728\u5b66\u672f\u754c\u91cc\u66f4\u5e7f\u4e3a\u4eba\u77e5\u7684\u662f\u4ed6\u7684\u79d1\u5b66\u8d21\u732e\uff1a\u4ed6\u548cRoy M. Anderson\u5408\u8457\u3001\u57281992\u5e74\u7531\u725b\u6d25\u5927\u5b66\u51fa\u7248\u793e\u51fa\u7248\u7684\u8fd1800\u9875\u7684\u4e13\u8457\u300aInfectious Diseases of Humans: Dynamics and Control\u300b\u88ab\u79f0\u4e3a\u662f\u4f20\u67d3\u75c5\u6570\u5b66\u6a21\u578b\u548c\u5206\u6790\u7684\u5723\u7ecf\uff0c\u81f3\u4eca\u83b7\u5f9737,000\u591a\u6b21\u5f15\u7528\uff0c\u5176\u4e2d\u5f15\u8fdb\u5e76\u7814\u7a76\u4e86\u4eca\u5929\u719f\u77e5\u7684\u4f20\u64ad\u56e0\u5b50\u5373\u518d\u751f\u6570\uff0c\u7528\u4ee5\u754c\u5b9a\u75be\u75c5\u4f20\u64ad\u7684\u6536\u655b\u548c\u53d1\u6563\u7684\u901f\u5ea6\uff0c\u5e76\u4e14\u5efa\u7acb\u4e86\u65e9\u671f\u7684HIV\u4f20\u67d3\u75c5\u4f20\u64ad\u6570\u5b66\u6a21\u578b\u3002\u4ed6\u81ea\u5df1\u5199\u7684\u4e00\u672c\u8457\u4f5c\u300aStability and Complexity in Model Ecosystems\u300b\u57281973\u5e74\u7531\u666e\u6797\u65af\u987f\u5927\u5b66\u51fa\u7248\u793e\u51fa\u7248\u540e2001\u5e74\u518d\u7248\uff0c\u81f3\u4eca\u83b7\u5f979,000\u591a\u6b21\u5f15\u7528\u3002\u4ed6\u5173\u4e8e\u52a8\u7269\u6355\u98df\u6a21\u578b\u7684May-Wigner\u7a33\u5b9a\u6027\u5b9a\u7406\u5728\u8be5\u7814\u7a76\u9886\u57df\u4e2d\u7279\u522b\u6709\u540d\u3002\u800c\u4ed6\u6bd5\u751f\u5907\u53d7\u5173\u6ce8\u7684\u8bba\u6587\u5219\u662f1976\u5e74\u5728 Nature \u4e0a\u53d1\u8868\u7684\u9898\u4e3a\u201cSimple mathematical models with very complicated dynamics\u201d\u7684\u8bba\u6587\uff08Nature, 261: 459-467, 1976\uff09\uff0c\u81f3\u4eca\u88ab\u5f157,800\u591a\u6b21\u3002\u8fd9\u662f\u4e00\u7bc7\u91cc\u7a0b\u7891\u5f0f\u7684\u8bba\u6587\uff0c\u80cc\u540e\u6709\u8bb8\u591a\u6545\u4e8b\u3002<\/p>\n


\u56fe2\uff1a\u7f57\u4f2f\u7279\u00b7\u6885\u5f15\u8fdbLogistic\u6620\u5c04\u7684\u91cc\u7a0b\u7891\u8bba\u6587<\/p>\n

\u7f57\u4f2f\u7279\u00b7\u6885\u81ea\u79f0\u662f\u4e2a\u201cr-\u9009\u62e9\u578b\u79d1\u5b66\u5bb6\u201d\uff0c\u559c\u6b22\u505a\u201c\u7b80\u5355\u4f18\u96c5\u800c\u53c8\u91cd\u8981\u7684\u7814\u7a76\u201d\u3002\u8fd9\u91cc\u201cr-\u9009\u62e9\u578b\u201d\u662f\u751f\u6001\u5b66\u91cc\u7684\u884c\u8bdd\uff0c\u6307\u53d7\u81ea\u8eab\u751f\u7269\u6f5c\u80fd\uff08\u6700\u5927\u751f\u6b96\u80fd\u529b\uff0cr\uff09\u652f\u914d\u7684\u7269\u79cd\u3002<\/p>\n

1970\u5e74\u4ee3\uff0c\u7f57\u4f2f\u7279\u00b7\u6885\u5728\u666e\u6797\u65af\u987f\u5927\u5b66\u4efb\u804c\u751f\u6001\u548c\u52a8\u7269\u5b66\u6559\u6388\u3002\u4ed6\u5b5c\u5b5c\u4e0d\u5026\u5730\u7814\u7a76\u751f\u6001\u7cfb\u7edf\u4e2d\u7684\u52a8\u7269\u6355\u98df\u6a21\u578b\u4ee5\u53ca\u7269\u79cd\u751f\u5b58\u7ade\u4e89\u548c\u6f14\u5316\u95ee\u9898\u3002\u4ed6\u6ce8\u610f\u5230\u4e86\u6bd4\u5229\u65f6\u6570\u5b66\u5bb6Pierre Verhulst \u57281845-1847\u5e74\u671f\u95f4\u5efa\u7acb\u7684\u63cf\u8ff0\u4eba\u53e3\u6570\u76ee\u53d8\u5316\u7684\u8fde\u7eed\u65f6\u95f4Logistic\u65b9\u7a0b\u3002\u8fd9\u91cc\u7684\u5355\u8bcdLogistic\u6765\u81ea\u6cd5\u6587logistique\uff0c\u63cf\u8ff0\u90e8\u961f\u7684\u540e\u52e4\u4f9b\u9700\u53ca\u5bbf\u8425\u7ba1\u7406\u3002\u7f57\u4f2f\u7279\u00b7\u6885\u628a\u5b83\u79bb\u6563\u5316\uff0c\u83b7\u5f97\u4e86\u201cLogistic\u6620\u5c04\u201d\uff0c\u5373\u4ece\u7b2c k \u6b65\u5230\u7b2c k +1 \u6b65\u7684\u8fed\u4ee3\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n

x(k +1) = \u03bbx(k)[1-x(k)],   k =0, 1, 2, \u2026<\/p>\n

\u5f0f\u4e2d x(k) \u4e3a\u79bb\u6563\u5b9e\u6570\u53d8\u91cf\uff0c\u8868\u793a\u7b2c k \u5e74\u7684\u52a8\u7269\u4e2a\u4f53\u6570\u91cf\uff08\u6807\u51c6\u5316\u540e\u53d6\u503c\u57280\u548c1\u4e4b\u95f4\uff09\uff0c\u521d\u59cb\u503c x(0)\u2208(0, 1) \uff0c\u5b9e\u53c2\u6570 \u03bb\u2208(0, 4) \u4ee3\u8868\u751f\u6b7b\u53d8\u5316\u7387\u3002\u8fd9\u4e2a\u6570\u5b66\u516c\u5f0f\u7684\u610f\u601d\u4e0d\u96be\u7406\u89e3\uff1a\u5f53\u4e2a\u4f53\u6570\u91cf\u5c11\uff08\u5373x(k)\u5c0f\uff09\u7684\u65f6\u5019\uff0c\u4e0b\u4e00\u5e74\u7684\u6570\u91cf\u589e\u957f\u5927\u4f53\u4e0a\u662f\u4e2a\u5e38\u6570\uff1b\u5f53\u4e2a\u4f53\u6570\u91cf\u589e\u52a0\uff08\u5373 x(k) \u53d8\u5927\uff09\u65f6\uff0c\u5916\u754c\u8d44\u6e90\u6bd4\u5982\u98df\u7269\u4e0d\u591f\u4e86\uff0c\u4e2a\u4f53\u7684\u6570\u91cf\u4fbf\u4f1a\u51cf\u5c11\u3002<\/p>\n

\u7531\u4e8e\u8fd9\u4e2a\u51fd\u6570\u66f2\u7ebf\u5728\u5b9a\u4e49\u533a\u95f4\u4e0a\u662f\u4e00\u6761\u629b\u7269\u7ebf\uff0c\u53ea\u6709\u4e00\u4e2a\u5cf0\u503c\uff0c\u6545\u6b64\u4e5f\u79f0\u4e3a\u201c\u5355\u5cf0\u51fd\u6570\u201d\u3002\u7f57\u4f2f\u7279\u00b7\u6885\u7528\u5b83\u6765\u63cf\u8ff0\u4e00\u822c\u751f\u7269\u3001\u7ecf\u6d4e\u6216\u793e\u4f1a\u7684\u6f14\u5316\uff0c\u4f8b\u5982\u52a8\u7269\u6216\u6606\u866b\u7684\u6355\u98df\u548c\u7e41\u884d\u3002\u540e\u4eba\u5219\u628a\u5b83\u7c7b\u6bd4\u4e8e\u201c\u4eba\u53e3\u201d\u6570\u91cf\u7684\u6da8\u843d\uff0c\u79f0\u4e4b\u4e3a\u201c\u866b\u53e3\u201d\u6a21\u578b\u3002<\/p>\n

\u8fd9\u4e2a\u6570\u5b66\u6620\u5c04\u975e\u5e38\u795e\u5947\u6709\u8da3\u3002\u867d\u7136\u6570\u5b66\u516c\u5f0f\u770b\u4e0a\u53bb\u5f88\u7b80\u5355\uff0c\u4f46\u662f\u5b83\u63cf\u8ff0\u7684\u52a8\u529b\u5b66\u884c\u4e3a\u5374\u5f02\u5e38\u590d\u6742\u3002<\/p>\n

\u9996\u5148\uff0c\u8fd9\u4e2a\u8fed\u4ee3\u516c\u5f0f\u7684\u8fd0\u7b97\u8fc7\u7a0b\u53ef\u4ee5\u7406\u89e3\u5982\u4e0b\uff1a\u4ece\u4efb\u610f\u4e00\u4e2a\u521d\u59cb\u503c x(0)\u2208(0, 1) \u5f00\u59cb\uff0c\u4ee3\u5165\u53f3\u8fb9\u4fbf\u5f97\u5230\u5de6\u8fb9\u7684\u503c x(1) \u3002\u7136\u540e\u628a\u8fd9\u4e2a x(1) \u4ee3\u5165\u53f3\u8fb9\u4fbf\u5f97\u5230\u5de6\u8fb9\u7684\u503c x(2)\u3002\u5982\u6b64\u5468\u800c\u590d\u59cb\uff0c\u53ef\u4e0d\u65ad\u5730\u8ba1\u7b97\u4e0b\u53bb\u3002<\/p>\n

\u5bb9\u6613\u770b\u51fa\uff0c\u5f53 \u03bb=2 \u65f6\uff0c\u5982\u679c x(0)=0.5 \u5219\u6240\u6709\u540e\u9762\u7684 x(k)\u22610.5\uff0ck =0, 1, 2, \u2026 \u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u83b7\u5f97\u4e86\u4e00\u4e2a\u65e0\u7a77\u5e8f\u5217 {0.5, 0.5, 0.5, \u2026} \u7684\u89e3\uff0c\u79f0\u4e3a\u5468\u671f\u4e3a1\u7684\u5468\u671f\u89e3\u3002<\/p>\n

\u7136\u540e\uff0c\u4f5c\u4e3a\u7b80\u5355\u7c97\u7565\u7684\u89e3\u91ca\uff0c\u5f53 \u03bb=3.3 \u65f6\uff0c\u5982\u679c\u4ece x(0)=0.479 \u5f00\u59cb\uff0c\u6bcf\u4e00\u6b65\u8ba1\u7b97\u90fd\u4f5c\u56db\u820d\u4e94\u5165\u53ea\u4fdd\u7559\u4e09\u4f4d\u5c0f\u6570\uff0c\u5219\u6709<\/p>\n


\u7531\u6b64\u6211\u4eec\u83b7\u5f97\u4e86\u4e00\u4e2a\u65e0\u7a77\u5e8f\u5217 \uff0c\u662f\u5468\u671f\u4e3a2\u7684\u5468\u671f\u89e3\u3002<\/p>\n

\u73b0\u5728\uff0c\u6211\u4eec\u4e00\u65b9\u9762\u53ef\u4ee5\u628a\u8fd9\u4e24\u4e2a\u5468\u671f1\u548c\u5468\u671f2\u7684\u89e3\u5728\u56fe\u7eb8\u4e0a\u5206\u522b\u6253\u4e0a\u4e00\u4e2a\u70b9\u548c\u4e24\u4e2a\u70b9\uff0c\u53e6\u4e00\u65b9\u9762\u53ef\u4ee5\u7ee7\u7eed\u628a\u6240\u6709\u4e0d\u540c\u53c2\u6570\u503c \u03bb\u2208(0, 4) \u548c\u6240\u6709\u5bf9\u5e94\u4e0d\u540c\u521d\u59cb\u503c x(0)\u2208(0, 1) \u90fd\u7b97\u4e00\u904d\uff0c\u5e76\u628a\u5f97\u5230\u7684\u89e3\u5e8f\u5217\u5206\u522b\u5728\u540c\u4e00\u5f20\u56fe\u4e0a\u6253\u4e0a\u5bf9\u5e94\u7684\u70b9\uff0c\u4fbf\u4f1a\u5f97\u5230\u56fe3\u6240\u793a\u7684\u66f2\u7ebf\u3002<\/p>\n


\u56fe3\uff1aLogistic \u6620\u5c04\u7684\u8ba1\u7b97\u7ed3\u679c\u66f2\u7ebf\uff08\u79f0\u4e3a\u5206\u53c9\u56fe\uff09<\/p>\n

\u4e0a\u9762\u7684Logistic\u6620\u5c04\u516c\u5f0f\u5f88\u7b80\u5355\uff0c\u8fde\u5fae\u79ef\u5206\u90fd\u7528\u4e0d\u4e0a\uff0c\u4e2d\u5b66\u751f\u90fd\u770b\u5f97\u660e\u767d\u3002\u4f46\u662f\uff0c\u7531\u5b83\u8ba1\u7b97\u51fa\u6765\u8fd9\u5e45\u56fe3\u5c31\u4e0d\u7b80\u5355\u4e86\uff0c\u7279\u522b\u662f\u5728\u53c2\u6570 \u03bb \u53d6\u503c\u63a5\u8fd14\u7684\u65f6\u5019\u3002\u5177\u4f53\u5730\u8bf4\uff1a<\/p>\n

\u5f53 \u03bb \u7531\u5f88\u5c0f\u7684\u6b63\u503c\uff08\u6bd4\u5982 0.1\uff09\u53d8\u52301\u65f6\uff0c\u66f2\u7ebf\u5f88\u5feb\u5730\u8d8b\u5411\u4e8e0\uff08\u662f\u4e00\u4e2a\u7a33\u5b9a\u503c\uff09\u3002
\u5f53 \u03bb \u7ee7\u7eed\u589e\u5927\uff0c\u66f2\u7ebf\u6162\u6162\u4e0a\u5347\uff0c\u9010\u6b21\u5230\u8fbe\u4e00\u4e2a\u63a5\u7740\u4e00\u4e2a\u7684\u7a33\u5b9a\u503c\uff08\u66f2\u7ebf\u4e0a\u4e00\u4e2a\u4e2a\u975e\u96f6\u70b9\uff0c\u5747\u4e3a\u5468\u671f1\u7684\u89e3\uff09\u3002
\u5f53 \u03bb \u7ee7\u7eed\u589e\u5927\uff0c\u66f2\u7ebf\u5f00\u59cb\u51fa\u73b0\u5206\u53c9\uff0c\u6bcf\u6b21\u67092\u4e2a\u7a33\u5b9a\u503c\uff08\u66f2\u7ebf\u4e0a\u4e24\u4e2a\u70b9\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5468\u671f2\u7684\u89e3\uff09\uff0c\u6bd4\u5982\u4e0a\u9762\u8ba1\u7b97 \u03bb=3.3 \u7684\u65f6\u5019\u3002
\u5f53 \u03bb \u7ee7\u7eed\u589e\u5927\uff0c\u66f2\u7ebf\u76f8\u7ee7\u51fa\u73b04\u4e2a\u30018\u4e2a\u300116\u4e2a\u300132\u4e2a … \u7a33\u5b9a\u503c\uff08\u4e0d\u540c\u5468\u671f\u7684\u5468\u671f\u89e3\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u79f0\u4e3a\u201c\u500d\u5468\u671f\u5206\u53c9\u201d\u3002
\u5f53 \u03bb \u7ee7\u7eed\u589e\u5927\u5230\u63a5\u8fd14\u65f6\uff0c\u7cfb\u7edf\u8fdb\u5165\u201c\u6df7\u6c8c\u72b6\u6001\u201d\uff0c\u8fd9\u65f6\u66f2\u7ebf\u4e0a\u5bc6\u5bc6\u9ebb\u9ebb\u7684\u70b9\u7684\u5168\u4f53\u7ec4\u6210\u4e86\u4e00\u4e2a\u5468\u671f\u5f88\u957f\u7684\u89e3\u3002
\u4e4b\u540e\uff0c\u5982\u679c \u03bb \u518d\u7ee7\u7eed\u589e\u5927\uff0c\u5230\u8d85\u8fc74\u65f6\uff0c\u590d\u6742\u7684\u66f2\u7ebf\u5c31\u7a81\u7136\u201c\u574d\u584c\u201d\u800c\u53d1\u6563\uff0c\u53d8\u5f97\u7b80\u5355\u65e0\u8da3\u4e86\u3002
\u8fd9\u4e2a\u8fc7\u7a0b\u4ee5\u53ca\u56fe3\u6240\u793a\u7684\u66f2\u7ebf\uff0c\u90fd\u51fa\u4e4e\u610f\u6599\u7684\u590d\u6742\u5427\uff1f
\u4e0d\u8fc7\uff0c\u806a\u660e\u7684\u4f60\u9a6c\u4e0a\u6ce8\u610f\u5230\u4e86\uff1a\u5468\u671f\u603b\u5728\u4e24\u500d\u6210\u5bf9\u5730\u51fa\u73b0\uff0c\u597d\u50cf\u8fd9Logistic\u6620\u5c04\u53ea\u6709\u5076\u6570\u5468\u671f\u7684\u5468\u671f\u89e3\u5427\uff1f
\u4e0d\u662f\u7684\u3002\u770b\u5230\u56fe3\u66f2\u7ebf\u53f3\u4fa7\u6709\u4e00\u4e9b\u201c\u7a97\u53e3\u201d\u4e86\u5417\uff1f\u4f8b\u5982 \u03bb\u22483.828 \u5bf9\u4e0a\u7684\u5730\u65b9\u6709\u4e00\u4e2a\u6bd4\u8f83\u5bbd\u7684\u7a97\u53e3\uff0c\u90a3\u91cc\u53ea\u67093\u4e2a\u70b9\uff0c\u5bf9\u5e94\u7740\u5468\u671f3\u7684\u89e3\u3002\u518d\u5f80\u5de6\u770b\uff0c \u03bb\u22483.738 \u5bf9\u5e94\u7740\u5468\u671f5\u7684\u89e3\uff0c \u03bb\u22483.702 \u5bf9\u5e94\u7740\u5468\u671f7\u7684\u89e3\uff0c\u7b49\u7b49\u3002\u8fd9Logistic\u6620\u5c04\u4e0d\u4f46\u6709\u5076\u6570\u7684\u5468\u671f\u89e3\uff0c\u8fd8\u6709\u591a\u79cd\u4e0d\u540c\u5947\u6570\u7684\u5468\u671f\u89e3\uff0c\u53ea\u4e0d\u8fc7\u5b83\u4eec\u4e0d\u50cf\u5076\u6570\u5206\u53c9\u8fc7\u7a0b\u90a3\u4e48\u6709\u5e8f\u800c\u5df2\u3002
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\u5982\u679c\u628a\u5bf9\u5e942\u30014\u30018\u300116 …\u7b49\u5206\u53c9\u70b9\u7684\u53c2\u6570 \u03bb \u503c\u8bb0\u4e3a \u03b11\u3001\u03b12\u3001\u03b13 …\u7b49\u7b49\uff0c\u90a3\u4e48 \u03b1n-\u03b1n-1 \u5c31\u662f\u5b83\u4eec\u4e24\u4e24\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002\u8003\u8651\u6bd4\u503c\t

\u4f60\u4f1a\u53d1\u73b0\uff0c\u5f53 n\u2192\u221e  \u65f6\uff0c\ud835\udeffn\u21924.6692\u3002\u8fd9\u4e2a\u6781\u9650\u6570\u5b57\u53eb\u8d39\u6839\u9c8d\u59c6\u5e38\u6570\uff0c\u7531\u8d39\u6839\u9c8d\u59c6\uff08Mitchell Feigenbaum, 1944-2019\uff09\u57281975\u5e74\u7528\u7b80\u5355\u7684 HP-65 \u8ba1\u7b97\u5668\u7b97\u51fa\u3002\u8d39\u6839\u9c8d\u59c6\u8fd8\u901a\u8fc7\u9ad8\u9636\u5fae\u5206\u8fd0\u7b97\u53d1\u73b0\u4e86\u5173\u4e8e\u5206\u53c9\u9ad8\u5ea6\u5dee\u4e4b\u6bd4\u7684\u6781\u9650\uff0c\u5373\u8d39\u6839\u9c8d\u59c6\u7b2c\u4e8c\u5e38\u6570 2.5029\u3002\u4e0d\u8fc7\u66f4\u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0c\u8d39\u6839\u9c8d\u59c6\u5e38\u6570\u5bf9\u540c\u7c7b\u578b\u7684\u6620\u5c04\u5982  x(k+1) =\u03bc-x2(k) \u90fd\u662f\u6210\u7acb\u7684\uff0c\u4e5f\u5c31\u662f\u8bf4\u5b83\u5177\u6709\u4e00\u5b9a\u7684\u666e\u9002\u6027\u3002
\u56fe4\uff1aMitchell J. Feigenbaum\uff081944-2019\uff09
1976\u5e74, \u7f57\u4f2f\u7279\u00b7\u6885\u5728 Nature \u4e0a\u53d1\u8868\u7684\u9898\u4e3a”Simple mathematical models with very complicated dynamics”\u7684\u8bba\u6587\uff0c\u8be6\u5c3d\u5730\u4ecb\u7ecd\u548c\u5206\u6790\u4e86\u8fd9\u4e2a\u7b80\u5355\u795e\u5947\u7684Logistic\u6620\u5c04\u3002\u4eca\u5929\uff0c\u8fd9\u7bc7\u8bba\u6587\u5df2\u88ab\u89c6\u4e3a\u79bb\u6563\u6df7\u6c8c\u7406\u8bba\u7684\u5f00\u5c71\u4e4b\u4f5c\uff0c\u800cLogistic\u6620\u5c04\u5c31\u662f\u79bb\u6563\u6df7\u6c8c\u7cfb\u7edf\u7684\u7b2c\u4e00\u4e2a\u548c\u6700\u91cd\u8981\u7684\u4e00\u4e2a\u4ee3\u8868\u6027\u4f8b\u5b50\u3002
\u4e0d\u8fc7\uff0c\u79bb\u6563\u6df7\u6c8c\u7684\u7cbe\u5f69\u6545\u4e8b\u8fd8\u5f97\u4ece1973\u5e74\u7684\u9a6c\u91cc\u5170\u5927\u5b66\u8bb2\u8d77\u3002
\u9a6c\u91cc\u5170\u5927\u5b66\u57281949\u5e74\u6210\u7acb\u4e86\u4e00\u4e2a\u201c\u6d41\u4f53\u52a8\u529b\u5b66\u4e0e\u5e94\u7528\u6570\u5b66\u7814\u7a76\u6240\u201d\uff0c1976\u5e74\u540e\u6539\u540d\u4e3a\u201c\u7269\u7406\u79d1\u5b66\u4e0e\u6280\u672f\u7814\u7a76\u6240\u201d\u30021972\u5e74\uff0c\u8be5\u7814\u7a76\u6240\u6c14\u8c61\u7ec4\u7684A. Feller\u6559\u6388\u5c06\u7231\u5fb7\u534e\u00b7\u6d1b\u4f26\u8328\u5728\u6c14\u8c61\u5b66\u671f\u520a\u4e0a\u53d1\u8868\u7684\u5173\u4e8e\u6c14\u8c61\u9884\u6d4b\u6a21\u578b\u76844\u7bc7\u6587\u7ae0\u63a8\u8350\u7ed9\u6570\u5b66\u7cfb\u7684\u8a79\u59c6\u65af\u00b7\u7ea6\u514b\uff08James A. Yorke\uff09\uff0c\u8bf4\u8fd9\u4e9b\u6587\u7ae0\u592a\u7406\u8bba\u548c\u6570\u5b66\u5316\u4e86\uff0c\u4ed6\u4eec\u770b\u4e0d\u61c2\uff0c\u4f46\u4e5f\u8bb8\u6570\u5b66\u6559\u6388\u4eec\u4f1a\u611f\u5174\u8da3\u3002\u8fd9\u51e0\u7bc7\u6587\u7ae0\u4e2d\uff0c\u6700\u5173\u952e\u7684\u662f\u6d1b\u4f26\u8328\u53d1\u73b0\u7b2c\u4e00\u4e2a\u5177\u4f53\u6df7\u6c8c\u7cfb\u7edf\u7684\u8bba\u6587 “Deterministic nonperiodic flow”\uff08Journal of Atmospheric Sciences, 20: 130-141, 1963\uff09\uff0c\u8be5\u6587\u81f3\u4eca\u88ab\u5f15\u752823,000\u591a\u6b21\u3002
\u56fe5\uff1aEdward N. Lorenz\uff081917-2008\uff09
1973\u5e744\u6708\u7684\u4e00\u5929\u4e0b\u5348\uff0c\u7ea6\u514b\u4e00\u4f4d\u5f97\u610f\u95e8\u751f\u674e\u5929\u5ca9\uff08Tien-Yien Li\uff09\u6765\u5230\u4e86\u4ed6\u7684\u529e\u516c\u5ba4\u3002\u7ea6\u514b\u5174\u594b\u5730\u8bf4\uff1a\u201c\u6211\u6709\u4e2a\u597d\u60f3\u6cd5\u8981\u544a\u8bc9\u4f60\uff01\u201d\u674e\u5929\u5ca9\u5f53\u65f6\u662f\u4ed6\u7684\u535a\u58eb\u751f\uff0c1968\u5e74\u4ece\u53f0\u6e7e\u65b0\u7af9\u6e05\u534e\u5927\u5b66\u6570\u5b66\u7cfb\u6bd5\u4e1a\u540e\u670d\u4e86\u4e00\u5e74\u5175\u5f79\uff0c\u7136\u540e\u6765\u5230\u4e86\u7ea6\u514b\u95e8\u4e0b\uff0c\u505a\u5fae\u5206\u65b9\u7a0b\u7814\u7a76\u3002\u867d\u7136\u662f\u5b66\u751f\uff0c\u674e\u5929\u5ca9\u53ea\u6bd4\u5bfc\u5e08\u5c0f\u56db\u5c81\uff0c\u56e0\u6b64\u4e24\u4eba\u4ea6\u5e08\u4ea6\u53cb\u3002\u50cf\u5e73\u5e38\u90a3\u6837\uff0c\u674e\u5929\u5ca9\u534a\u5f00\u73a9\u7b11\u5730\u95ee\uff1a\u201c\u4f60\u8fd9\u4e2a\u65b0\u60f3\u6cd5\u8db3\u4ee5\u5f80\u300a\u7f8e\u56fd\u6570\u5b66\u6708\u520a\u300b\u6295\u7bc7\u6587\u7ae0\u4e48\uff1f\u201d\u5927\u5bb6\u90fd\u77e5\u9053\uff0c\u300a\u7f8e\u56fd\u6570\u5b66\u6708\u520a\u300b\uff08American Mathematical Monthly\uff09\u662f\u521b\u529e\u4e8e1894\u5e74\u7684\u8001\u6742\u5fd7\uff0c\u5f88\u6709\u540d\u6c14\uff0c\u4f46\u53ea\u662f\u6570\u5b66\u79d1\u666e\u7c7b\u578b\u7684\u6708\u520a\uff0c\u901a\u5e38\u4e0d\u53d1\u8868\u9ad8\u6df1\u6570\u5b66\u8bba\u6587\u3002\u7ea6\u514b\u7b11\u4e86\u7b11\uff0c\u8bf4\u4ed6\u771f\u6709\u4e2a\u65b0\u60f3\u6cd5\uff0c\u6e90\u4e8e\u6d1b\u4f26\u8328\u76844\u7bc7\u6587\u7ae0\u3002\u674e\u5929\u5ca9\u542c\u5b8c\u90a3\u4e2a\u60f3\u6cd5\u4e4b\u540e\uff0c\u611f\u6168\u5730\u8bf4\uff1a\u201c\u8fd9\u6837\u7684\u7ed3\u679c\u786e\u5b9e\u975e\u5e38\u9002\u5408\u90a3\u4e2a\u6708\u520a\uff01\u201d\u56e0\u4e3a\u4ed6\u9884\u611f\u5230\u4e86\uff0c\u5982\u679c\u80fd\u591f\u628a\u7ed3\u679c\u8bc1\u660e\u51fa\u6765\u7684\u8bdd\uff0c\u5176\u63cf\u8ff0\u5e76\u4e0d\u9700\u8981\u6d89\u53ca\u9ad8\u6df1\u7684\u6570\u5b66\u8bed\u8a00\uff0c\u5927\u5b66\u751f\u7814\u7a76\u751f\u5e94\u80fd\u770b\u5f97\u61c2\u3002
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1974\u5e74\u662f\u9a6c\u91cc\u5170\u5927\u5b66\u6570\u5b66\u7cfb\u7684\u751f\u7269\u6570\u5b66\u201c\u7279\u6b8a\u5e74\u201d\uff0c\u671f\u95f4\u4ed6\u4eec\u6bcf\u5468\u90fd\u8bf7\u751f\u7269\u6570\u5b66\u8fd9\u4e2a\u9886\u57df\u4e2d\u4e00\u4f4d\u6700\u6770\u51fa\u7684\u5b66\u8005\u6765\u7cfb\u91cc\u6f14\u8bb2\u3002\u57285\u6708\u4efd\u7684\u7b2c\u4e00\u5468\uff0c\u4ed6\u4eec\u4ece\u666e\u6797\u65af\u987f\u5927\u5b66\u751f\u6001\u548c\u52a8\u7269\u7cfb\u8bf7\u6765\u4e86\u7f57\u4f2f\u7279\u00b7\u6885\uff0c\u8bf7\u4ed6\u6bcf\u5929\u7ed9\u4e00\u4e2a\u8bb2\u5ea7\u3002\u6700\u540e\u4e00\u5929\u662f\u5468\u4e94\uff0c\u4e0a\u5348\u6f14\u8bb2\u65f6\u7f57\u4f2f\u7279\u00b7\u6885\u4ecb\u7ecd\u4e86\u4ed6\u53d1\u73b0\u7684Logistic\u6620\u5c04\uff0c\u5373\u866b\u53e3\u6a21\u578b\uff0c\u4e5f\u5c31\u662f\u90a3\u5e45\u6709\u8da3\u7684\u56fe3\u3002\u5f53\u65f6\u7f57\u4f2f\u7279\u00b7\u6885\u6709\u628a\u63e1\u7684\u53ea\u662f\u56fe\u4e2d\u5de6\u8fb9\u5bf9\u5e94\u7740\u8f83\u5c0f\u6570\u503c \u03bb \u90a3\u90e8\u5206\u6bd4\u8f83\u89c4\u5219\u7684\u66f2\u7ebf\u6240\u8868\u8fbe\u7684\u751f\u7269\u542b\u4e49\u3002\u81f3\u4e8e\u5f53\u53c2\u6570 \u03bb \u63a5\u8fd14\u65f6\u56fe\u5f62\u8868\u73b0\u51fa\u6765\u201c\u4e71\u4e03\u516b\u7cdf\u201d\u7684\u884c\u4e3a\uff0c\u4ed6\u4e5f\u89e3\u91ca\u4e0d\u6e05\u695a\uff0c\u8bf4\u4e5f\u8bb8\u53ea\u662f\u8ba1\u7b97\u8bef\u5dee\u6240\u9020\u6210\u3002
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\u6587\u7ae0\u5728\u4e24\u5468\u5185\u5c31\u6539\u5199\u597d\u4e86\uff0c\u5f15\u7528\u4e86\u7f57\u4f2f\u7279\u00b7\u6885\u7684\u5de5\u4f5c\uff0c\u4eceLogistic\u6620\u5c04\u8c08\u8d77\uff0c\u5e76\u8865\u5145\u4e86\u4e00\u4e9b\u76f8\u5173\u6587\u732e\u3002\u4e24\u4f4d\u6570\u5b66\u5bb6\u4e0d\u6539\u521d\u8877\uff0c\u628a\u6587\u7ae0\u91cd\u6295\u300a\u7f8e\u56fd\u6570\u5b66\u6708\u520a\u300b\u3002
\u4e09\u4e2a\u6708\u540e\uff0c\u6708\u520a\u901a\u77e5\u4ed6\u4eec\uff0c\u6587\u7ae0\u73b0\u5728\u53ef\u4ee5\u63a5\u6536\u4e86\u3002\u8fd9\u7bc7\u201c\u6570\u5b66\u79d1\u666e\u201d\u6587\u7ae0\u4e0d\u4f46\u6709\u6d1b\u4f26\u8328\u7684\u6c14\u8c61\u7cfb\u7edf\uff0c\u8fd8\u6709\u7f57\u4f2f\u7279\u00b7\u6885\u7684Logistic\u6620\u5c04\uff0c\u66f4\u6709\u4e00\u6761\u6f02\u4eae\u7684\u6570\u5b66\u5b9a\u7406\uff0c\u6700\u540e\u4e8e1975\u5e7412\u6708\u9762\u4e16\uff08T.-Y. Li and J. A. Yorke,\u201cPeriod three implies chaos\u201d\uff0cAmerican Mathematical Monthly, 82: 985-992, 1975\uff09\u3002\u8be5\u6587\u81f3\u4eca\u88ab\u5f15\u75285,000\u6b21\u3002\u540e\u6765\u7f57\u4f2f\u7279\u00b7\u6885\u56de\u5fc6\u8bf4\uff0c\u5176\u5b9e\u4ed6\u5728\u770b\u5230\u674e-\u7ea6\u514b\u624b\u7a3f\u4e4b\u524d\uff0c\u5e76\u4e0d\u77e5\u9053\u6d1b\u4f26\u8328\u548c\u4ed6\u7684\u7cfb\u7edf\u3002
\u666e\u6797\u65af\u987f\u9ad8\u7b49\u7814\u7a76\u9662\u5df2\u6545\u7406\u8bba\u7269\u7406\u5b66\u5bb6\u5f17\u91cc\u66fc\u00b7\u6234\u68ee\uff08Freeman Dyson\uff0c1923-2020\uff09\u5728\u4ed6\u4e8e2009\u5e74\u521d\u7531\u300a\u7f8e\u56fd\u6570\u5b66\u4f1a\u4f1a\u520a\u300b\uff08Notices of the American Mathematical Society\uff09\u53d1\u8868\u7684\u3001\u5728\u201c\u7231\u56e0\u65af\u5766\u8bb2\u5ea7\u201d\u4e0a\u6f14\u8bb2\u8fc7\u7684\u6587\u7ae0\u201c\u9e1f\u4e0e\u86d9\u201d\uff08Birds and Frogs\uff09\u4e2d\u8bf4\uff1a\u201c\u5728\u6df7\u6c8c\u5b66\u7684\u9886\u57df\u4e2d\uff0c\u6211\u77e5\u9053\u7684\u53ea\u6709\u4e00\u6761\u4e25\u683c\u8bc1\u660e\u4e86\u7684\u5b9a\u7406\uff0c\u90a3\u662f\u7531\u674e\u5929\u5ca9\u548c\u8a79\u59c6\u65af\u00b7\u7ea6\u514b\u57281975\u5e74\u53d1\u8868\u7684\u4e00\u7bc7\u77ed\u6587\u2018\u5468\u671f\u4e09\u610f\u5473\u7740\u6df7\u6c8c\u2019\u4e2d\u6240\u5efa\u7acb\u7684\u3002\u201d\u4ed6\u5c06\u674e-\u7ea6\u514b\u7684\u8fd9\u7bc7\u8bba\u6587\u8a89\u4e3a\u201c\u6570\u5b66\u6587\u732e\u4e2d\u4e0d\u673d\u7684\u73cd\u54c1\u4e4b\u4e00\u201d\u3002
\u56fe6\uff1a\u7ea6\u514b\u4e3a\u5e86\u795d\u674e\u5929\u5ca9\uff081945-2020\uff0970\u5927\u5bff\u4e13\u95e8\u5b9a\u5236\u4e86\u4e00\u74f6\u6807\u6709\u201c\u6df7\u6c8c\u201d\u5546\u6807\u7684\u8461\u8404\u9152<\/p>\n

\u4f46\u662f\uff0c\u674e-\u7ea6\u514b\u5b9a\u7406\u7684\u6545\u4e8b\u5e76\u6ca1\u6709\u5c31\u6b64\u7ed3\u675f\u3002
\u7fcc\u5e74\uff0c\u7ea6\u514b\u5230\u524d\u82cf\u8054\u53c2\u52a0\u4e00\u4e2a\u56fd\u9645\u6570\u5b66\u4f1a\u8bae\uff0c\u62a5\u544a\u4e86\u674e-\u7ea6\u514b\u5b9a\u7406\u3002\u4f1a\u540e\u7ea6\u514b\u6e38\u901b\u65f6\uff0c\u4e00\u4f4d\u7565\u4e3a\u5e74\u957f\u7684\u6570\u5b66\u5bb6\u53eb\u4f4f\u4e86\u4ed6\uff0c\u53cb\u597d\u5730\u7b11\u7740\u8bf4\uff0c\u4f60\u8fd9\u201c\u5468\u671f3\u610f\u5473\u7740\u6240\u6709\u5468\u671f\u201d\u7684\u7ed3\u679c\u5440\uff0c\u6211\u5341\u5e74\u524d\u5c31\u53d1\u8868\u8fc7\u4e86\uff0c\u800c\u4e14\u5176\u4e2d\u7684\u5468\u671f\u9690\u542b\u89c4\u5f8b\u6211\u90fd\u8bf4\u6e05\u695a\u4e86\u3002
\u554a\uff1f\uff01\u8fd9\u7740\u5b9e\u8ba9\u7ea6\u514b\u5927\u5403\u4e86\u4e00\u60ca\u3002
\u8fd9\u4f4d\u8001\u5927\u54e5\u662f\u4e4c\u514b\u5170\u6570\u5b66\u5bb6\u6c99\u53ef\u592b\u65af\u57fa\uff08Oleksandr M. Sharkovsky\uff0c1936-\uff09\uff0c\u4ed6\u7684\u8bba\u6587\u7528\u4fc4\u6587\u53d1\u8868\u5728\u4e00\u4e2a\u4e0d\u751a\u77e5\u540d\u7684\u4e4c\u514b\u5170\u6570\u5b66\u6742\u5fd7\u4e0a\uff08”Co-existence of cycles of a continuous mapping of the line into itself”, Ukrainian Mathematical Journal, 16: 61-71, 1964\uff09\u3002
\u56fe7\uff1aOleksandr M. Sharkovsky (1936 – )
\u6c99\u53ef\u592b\u65af\u57fa\u5b9a\u7406\u8bf4\u7684\u662f\uff0c\u8ba9\u6211\u4eec\u628a\u6240\u6709\u7684\u6b63\u6574\u6570 n \u6309\u5982\u4e0b\u7684\u6b21\u5e8f\u6392\u5217\u8d77\u6765\uff1a\t\t

\u90a3\u4e48\uff0c\u5bf9\u4e8e\u8fde\u7eed\u533a\u95f4\u6620\u5c04 f: [0, 1]\u2192[0, 1] \uff0c\u5982\u679c f \u6709\u5468\u671f\u4e3a m \u7684\u89e3\uff0c\u5373 fm(x)=x \u4f46  fk(x) \u2260 x (0<k<m), x\u2208[0, 1] \uff0c\u5e76\u4e14\u5728\u4e0a\u9762\u7684\u6b21\u5e8f\u4e2d n \u6392\u5728 m \u540e\u9762\u7684\u8bdd\uff0c\u5219 f \u4e00\u5b9a\u6709\u5468\u671f\u4e3a n \u7684\u89e3\u3002
\u4e0d\u8fc7\uff0c\u6c99\u53ef\u592b\u65af\u57fa\u5b9a\u7406\u662f\u4e00\u4e2a\u62d3\u6251\u5b66\u800c\u4e0d\u662f\u52a8\u529b\u7cfb\u7edf\u65b9\u9762\u7684\u7ed3\u679c\u3002\u5b83\u57fa\u672c\u4e0a\u5305\u62ec\u4e86\u674e-\u7ea6\u514b\u5b9a\u7406\u4e2d\u7684\u7b2c\u4e00\u90e8\u5206\uff0c\u5373\u5982\u679c\u8be5\u6620\u5c04\u6709\u5468\u671f\u4e3a3\u7684\u89e3\uff0c\u5b83\u5c31\u6709\u6240\u6709\u6b63\u6574\u6570\u5468\u671f\u7684\u89e3\uff1b\u4f46\u5b83\u5b8c\u5168\u6ca1\u6709\u6d89\u53ca\u5230\u674e-\u7ea6\u514b\u5b9a\u7406\u4e2d\u7684\u7b2c\u4e8c\u90e8\u5206\uff0c\u5373\u8be5\u6620\u5c04\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u6781\u7aef\u654f\u611f\u6027\u3002\u800c\u4eca\u5929\u7684\u6df7\u6c8c\u6570\u5b66\u7406\u8bba\u5c31\u662f\u5efa\u7acb\u5728\u8fd9\u4e2a\u6700\u6839\u672c\u7684\u654f\u611f\u6027\u6761\u4ef6\u4e4b\u4e0a\uff0c\u4e0e\u201c\u5177\u6709\u6240\u6709\u5468\u671f\u201d\u8fd9\u4e00\u7279\u6027\u5173\u7cfb\u4e0d\u5927\u3002\u56e0\u6b64\uff0c\u4eca\u5929\u79d1\u5b66\u754c\u8bf4\u7684\u8457\u540d\u7684\u201c\u674e-\u7ea6\u514b\u5b9a\u7406\u201c\uff08Li-Yorke Theorem\uff09\uff0c\u6307\u7684\u662f\u5b83\u7684\u7b2c\u4e8c\u90e8\u5206\u3002\u4e0d\u8fc7\uff0c\u5c0a\u91cd\u539f\u6587\u7684\u5386\u53f2\u6027\u6807\u9898\uff0c\u4e5f\u4e3a\u4e86\u8ba9\u8bfb\u8005\u5bb9\u6613\u8bb0\u5fc6\uff0c\u4e60\u60ef\u4e0a\u5927\u5bb6\u8fd8\u662f\u4fdd\u7559\u539f\u6765\u7684\u8bf4\u6cd5\uff0c\u5373\u674e-\u7ea6\u514b\u5b9a\u7406\u662f\u4e00\u4e2a\u5173\u4e8e\u201c\u5468\u671f\u4e09\u610f\u5473\u7740\u6df7\u6c8c\u201d\u7684\u7ed3\u679c\u3002
\u674e\u5929\u5ca9\u548c\u7ea6\u514b\u8fd9\u4e24\u4f4d\u6570\u5b66\u5bb6\u59cb\u6599\u4e0d\u53ca\uff0c\u4ed6\u4eec\u51fa\u4e8e\u597d\u5947\u5fc3\u5199\u51fa\u6765\u7684\u8fd9\u7bc7\u79d1\u666e\u6742\u5fd7\u4e0a\u53d1\u8868\u7684\u201c\u5c0f\u6587\u7ae0\u201d\uff0c\u4ee5\u534a\u5f00\u73a9\u7b11\u7684\u65b9\u5f0f\u4f7f\u7528\u4e86\u201c\u6df7\u6c8c\u201d\uff08chaos\uff09\u4e00\u8bcd\uff0c\u5374\u4e3a\u6574\u4e2a\u79bb\u6563\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\u5f15\u8fdb\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u7814\u7a76\u65b9\u5411\uff0c\u5efa\u7acb\u4e86\u4e25\u683c\u7684\u79bb\u6563\u6df7\u6c8c\u7406\u8bba\u57fa\u7840\uff0c\u5e76\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5173\u4e8e\u5bf9\u521d\u59cb\u6761\u4ef6\u9ad8\u5ea6\u654f\u611f\u6027\u7684\u5173\u952e\u6570\u5b66\u5224\u636e\u3002
\u540e\u6765\uff0c\u7ea6\u514b\u548c\u201c\u5206\u5f62\u4e4b\u7236\u201d\u672c\u534e\u00b7\u66fc\u5fb7\u535a\uff08”Father of fractals”, Benoit B. Mandelbrot\uff0c1924-2010\uff09\u4e00\u9053\u5206\u4eab\u4e862003\u5e74\u7684\u5341\u5206\u8457\u540d\u7684\u65e5\u672c\u5956\uff08Japan Prize\uff09\u3002
\u56fe8\uff1aBenoit B. Mandelbrot\uff081924-2010\uff09
\u50cfLogistic\u6620\u5c04\u53cd\u590d\u8fed\u4ee3\u540e\u4f1a\u6709\u5468\u671f\u89e3\u90a3\u6837\uff0c\u6211\u4eec\u5173\u4e8e\u79bb\u6563\u6df7\u6c8c\u7684\u4f20\u5947\u6545\u4e8b\u4ece\u7f57\u4f2f\u7279\u00b7\u6885\u5f00\u59cb\uff0c\u56de\u987e\u4e86\u8bb8\u591a\u5386\u53f2\u548c\u4eba\u7269\u4e4b\u540e\uff0c\u6700\u7ec8\u8fd8\u8981\u56de\u5230\u8d77\u70b9\uff0c\u518d\u8bf4\u7f57\u4f2f\u7279\u00b7\u6885\u7684\u4eba\u751f\u3002
\u7f57\u4f2f\u7279\u00b7\u6885\u4e8e1936\u5e741\u67088\u65e5\u51fa\u751f\u4e8e\u6fb3\u5927\u5229\u4e9a\u6089\u5c3c\u5e02\uff0c\u7236\u4eb2\u662f\u5317\u7231\u5c14\u5170\u88d4\u7684\u4e00\u4f4d\u5f8b\u5e08\uff0c\u6bcd\u4eb2\u662f\u82cf\u683c\u5170\u4e00\u4f4d\u5de5\u7a0b\u5e08\u7684\u5973\u513f\u3002\u4ed6\u4e03\u5c81\u90a3\u5e74\uff0c\u7236\u6bcd\u79bb\u5f02\u3002\u4ed6\u7684\u672c\u79d1\u5728\u6089\u5c3c\u5927\u5b66\u4fee\u8bfb\u5316\u5b66\u5de5\u7a0b\u548c\u7406\u8bba\u7269\u7406\uff0c1956\u5e74\u83b7\u5f97\u7406\u5b66\u5b66\u58eb\u5b66\u4f4d\uff0c1959\u5e74\u4ee5\u201cInvestigations towards an understanding of superconductivity\u201d\u4e3a\u6bd5\u4e1a\u8bba\u6587\u83b7\u5f97\u7406\u8bba\u7269\u7406\u5b66\u535a\u58eb\u5b66\u4f4d\uff0c\u968f\u540e\u5230\u54c8\u4f5b\u5927\u5b66\u5f53\u4e86\u4e24\u5e74\u535a\u58eb\u540e\uff0c\u5176\u65f6\u62c5\u4efb\u51a0\u540dGordon MacKay\u5e94\u7528\u6570\u5b66\u8bb2\u5e08\u3002\u5728\u54c8\u4f5b\u671f\u95f4\uff0c\u4ed6\u548c\u5728\u7ebd\u7ea6\u66fc\u54c8\u987f\u4e00\u4e2a\u72b9\u592a\u5bb6\u5ead\u51fa\u751f\u957f\u5927\u7684Judith Feiner\u7ed3\u4e86\u5a5a\uff0c\u4e24\u4eba\u517b\u80b2\u6709\u4e00\u5973\u513fNaomi Felicity\u30021962\u5e74\uff0c\u4ed6\u56de\u5230\u6089\u5c3c\u5927\u5b66\u4efb\u6559\u7406\u8bba\u7269\u7406\u5b66\uff0c\u5148\u540e\u4efb\u804c\u9ad8\u7ea7\u8bb2\u5e08\u3001\u51c6\u6559\u6388\uff08Reader, 1964\uff09\u3001\u6559\u6388\uff08Personal Chair\uff0c1969\uff09\uff0c\u81f31972\u5e74\u30021973-1988\u5e74\u95f4\uff0c\u4ed6\u5230\u4e86\u666e\u6797\u65af\u987f\u5927\u5b66\uff0c\u63a5\u66ff\u53bb\u4e16\u7684\u56fd\u9645\u4e0a\u6700\u8457\u540d\u7684\u7406\u8bba\u751f\u6001\u5b66\u5bb6Robert MacArthur\u7684\u804c\u4f4d\uff0c\u6210\u4e3a\u51a0\u540d\u201cClass of 1877\u201d\u7684\u751f\u6001\u548c\u52a8\u7269\u5b66\u6559\u6388\u30021988-1995\u5e74\u95f4\u4ed6\u8f6c\u5230\u4e86\u82f1\u56fd\uff0c\u5728\u725b\u6d25\u5927\u5b66\u548c\u5e1d\u56fd\u7406\u5de5\u4efb\u7687\u5bb6\u5b66\u4f1a\u7814\u7a76\u6559\u6388\u30021995-2000\u5e74\u95f4\uff0c\u4ed6\u4efb\u804c\u82f1\u56fd\u653f\u5e9c\u9996\u5e2d\u79d1\u5b66\u987e\u95ee\u548c\u82f1\u56fd\u79d1\u5b66\u6280\u672f\u59d4\u5458\u4f1a\u4e3b\u5e2d\uff0c2000-2005\u5e74\u95f4\u51fa\u4efb\u82f1\u56fd\u7687\u5bb6\u5b66\u4f1a\u4e3b\u5e2d\uff0c2005\u5e74\u4e4b\u540e\u4e3a\u725b\u6d25\u5927\u5b66\u548c\u5e1d\u56fd\u7406\u5de5\u8363\u4f11\u6559\u6388\u3002
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