{"id":54593,"date":"2024-12-21T17:26:54","date_gmt":"2024-12-21T09:26:54","guid":{"rendered":"https:\/\/swarma.org\/?p=54593"},"modified":"2024-12-21T17:26:54","modified_gmt":"2024-12-21T09:26:54","slug":"%e9%9c%8d%e6%99%ae%e5%a4%ab%e5%92%8c%e4%bb%96%e7%9a%84%e5%88%86%e5%b2%94%e7%90%86%e8%ae%ba%e4%b8%a8%e9%99%88%e5%85%b3%e8%8d%a3","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=54593","title":{"rendered":"\u970d\u666e\u592b\u548c\u4ed6\u7684\u5206\u5c94\u7406\u8bba\u4e28\u9648\u5173\u8363"},"content":{"rendered":"
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\u6211\u76ee\u524d\u4e3b\u7f16\u7684\u4e00\u4efd\u5b66\u672f\u671f\u520a\u540d\u53eb\u201c\u5206\u5c94\u4e0e\u6df7\u6c8c\u56fd\u9645\u6742\u5fd7\u201d<\/span>\uff08International Journal of Bifurcation and Chaos\uff09<\/span>\u3002\u4eca\u5929\uff0c\u5927\u5bb6\u6216\u591a\u6216\u5c11\u90fd\u77e5\u9053\u201c\u6df7\u6c8c\u201d<\/span>\uff08chaos\uff09<\/span>\uff0c\u56e0\u4e3a\u5b83\u6709\u9887\u957f\u7684\u5386\u53f2\u548c\u8bb8\u591a\u6709\u8da3\u7684\u6545\u4e8b[1,2]\u3002\u7279\u522b\u662f\uff0c\u5927\u5bb6\u90fd\u77e5\u9053\u6d1b\u4f26\u8328<\/span>\uff08Edward Norton Lorenz\uff0c1917-2008\uff09<\/span>\uff0c\u4ece\u4ed6\u90a3\u91cc\u4e86\u89e3\u5230\u8457\u540d\u7684\u201c\u8774\u8776\u6548\u5e94\u201d[3]\u3002\u4e0d\u8fc7\uff0c\u5927\u5bb6\u4e5f\u77e5\u9053\u201c\u5206\u5c94\u201d<\/span>\uff08bifurcation\uff09<\/span>\u5417\uff1f\u4e5f\u77e5\u9053\u5206\u5c94\u7406\u8bba\u7684\u5960\u57fa\u4eba\u5417\uff1f<\/span><\/section>\n

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\u4e8b\u5b9e\u4e0a\uff0c\u5206\u5c94\u7406\u8bba\u6bd4\u6df7\u6c8c\u7406\u8bba\u7684\u5386\u53f2\u66f4\u957f\u3002\u5206\u5c94\u7406\u8bba\u7684\u4e00\u4e2a\u91cd\u8981\u4eba\u7269\u662f\u827e\u4f2f\u54c8\u5fb7\u00b7\u970d\u666e\u592b<\/span>\uff08Eberhard Friedrich Ferdinand Hopf\uff0c1902\u5e744\u67084\u65e5-1983\u5e747\u670824\u65e5\uff09<\/span>\u3002\u4eca\u5929\uff0c\u4e16\u754c\u4e0a\u201c\u5206\u5c94\u4e0e\u6df7\u6c8c\u201d\u7406\u8bba\u53ca\u5176\u5e94\u7528\u7814\u7a76\u9886\u57df\u7684\u79d1\u7814\u4eba\u5458\uff0c\u5927\u6982\u6ca1\u6709\u8c01\u4e0d\u66fe\u5728\u8bba\u6587\u4e2d\u5206\u6790\u6216\u63d0\u53ca\u8fc7\u201c\u970d\u666e\u592b\u5206\u5c94\u201d<\/span>\uff08Hopf bifurcation\uff09<\/span>\u3002<\/span><\/section>\n

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\u56fe1 \u827e\u4f2f\u54c8\u5fb7\u00b7\u970d\u666e\u592b\uff081902-1983\uff09<\/span><\/section>\n

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1. \u751f\u5e73<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u970d\u666e\u592b\u4e8e1902\u5e74\u51fa\u751f\u5728\u5965\u5730\u5229\u8428\u5c14\u8328\u5821\uff0c\u5373\u97f3\u4e50\u5bb6\u83ab\u672d\u7279<\/span>\uff08Wolfgang Amadeus Mozart\uff0c1756-1791\uff09<\/span>\u7684\u6545\u4e61\u3002\u970d\u666e\u592b\u7684\u7236\u4eb2\u5f17\u91cc\u5fb7\u91cc\u5e0c<\/span>\uff08Friedrich Hopf\uff09<\/span>\u662f\u4e00\u4f4d\u5de7\u514b\u529b\u7cd6\u679c\u5236\u9020\u5546\u30021920\u5e74\uff0c\u970d\u666e\u592b\u5728\u5fb7\u56fd\u67cf\u6797\u4e2d\u5b66\u6bd5\u4e1a\u540e\u8fdb\u5165\u56fe\u5bbe\u6839<\/span>\uff08Tu\u0308bingen\uff09<\/span>\u5927\u5b66\uff0c\u5728\u90a3\u91cc\u8bfb\u4e86\u4e00\u4e2a\u5b66\u671f\uff0c\u7136\u540e\u8f6c\u5230\u67cf\u6797\u5927\u5b66\uff0c\u5728\u90a3\u91cc\u8bfb\u4e86\u4e03\u4e2a\u5b66\u671f\u30021925\u5e74-1926\u5e74\u95f4\uff0c\u4ed6\u5728\u5fb7\u56fd\u6570\u5b66\u5bb6\u57c3\u54c8\u5fb7\u00b7\u65bd\u5bc6\u7279<\/span>\uff08Erhard Schmidt\uff0c1876-1959\uff09<\/span>\u548c\u4fc4\u88d4\u5fb7\u56fd\u6570\u5b66\u5bb6\u4f0a\u8d5b\u00b7\u8212\u5c14<\/span>\uff08Issai Schur\uff0c1875-1941\uff09<\/span>\u8054\u5408\u6307\u5bfc\u4e0b\u5b8c\u6210\u4e86\u6570\u5b66\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u30021927\u5e74\uff0c\u4ed6\u5728\u67cf\u6797\u5927\u5b66\u5929\u6587\u7814\u7a76\u6240\u4efb\u804c\u79d1\u7814\u52a9\u7406\u30021929\u5e74\uff0c\u970d\u666e\u592b\u5728\u67cf\u6797\u5927\u5b66\u5b8c\u6210\u4e86\u79d1\u5b66\u535a\u58eb\u5b66\u4f4d\u5e76\u83b7\u5f97\u4e86\u6570\u5b66\u548c\u5929\u6587\u5b66\u7684\u6559\u5b66\u8d44\u683c\u3002\u540c\u5e74\uff0c\u4ed6\u548c\u5fb7\u56fd\u8457\u540d\u97f3\u4e50\u5bb6Johannes Wolf <\/span>\uff081869-1947\uff09<\/span>\u7684\u5973\u513f\u4f0a\u5c14\u65af\u00b7\u6c83\u5c14\u592b<\/span>\uff08Ilse Wolf\uff09<\/span>\u7ed3\u5a5a\u30022006\u5e74\uff0c\u4ed6\u4eec\u7684\u5973\u513f\u82ad\u82ad\u62c9\u00b7\u970d\u666e\u592b\u5728\u5199\u7ed9\u6570\u5b66\u5bb6Hans-Joachim Girlich<\/span>\uff081938-2018\uff09<\/span>\u7684\u4fe1\u4e2d\u56de\u5fc6\u9053\uff1a\u201c\u6211\u7684\u7236\u6bcd\uff0c\u5c24\u5176\u662f\u6211\u7684\u6bcd\u4eb2\uff0c\u975e\u5e38\u70ed\u60c5\u597d\u5ba2\uff0c\u5f88\u591a\u97f3\u4e50\u4e4b\u591c\u90fd\u6c89\u8ff7\u5728\u6f14\u51fa\u4e2d\u3002\u6211\u7684\u7236\u4eb2\u662f\u4e00\u4f4d\u51fa\u8272\u7684\u94a2\u7434\u6f14\u594f\u5bb6\uff0c\u6bcd\u4eb2\u662f\u4e00\u4f4d\u827a\u672f\u6b4c\u66f2\u6b4c\u624b\u3002\u201d<\/span><\/section>\n

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1929\u5e74\uff0c\u5fb7\u56fd\u7684\u7ecf\u6d4e\u5f62\u52bf\u5f88\u5dee\uff0c\u970d\u666e\u592b\u627e\u4e0d\u5230\u5408\u9002\u7684\u5de5\u4f5c\u804c\u4f4d\u3002\u4e0d\u8fc7\uff0c\u4ed6\u5e78\u8fd0\u5730\u83b7\u5f97\u4e86\u7f8e\u56fd\u6d1b\u514b\u83f2\u52d2\u57fa\u91d1\u8d44\u52a9\uff0c\u524d\u5f80\u54c8\u4f5b\u56fd\u9645\u5929\u6587\u53f0\u5f53\u7814\u7a76\u5458\u30021930\u5e74\uff0c\u970d\u666e\u592b\u6765\u5230\u4e86\u6ce2\u58eb\u987f\uff0c\u5728\u7231\u56e0\u65af\u5766\u5854\u697c<\/span>\uff08Einstein Tower\uff09<\/span>\u4e0a\u73ed\u3002\u5728\u90a3\u91cc\uff0c\u4ed6\u4e3b\u8981\u5728\u6570\u5b66\u5bb6\u4e54\u6cbb\u00b7\u4f2f\u514b\u970d\u592b<\/span>\uff08George David Birkhoff\uff0c1884-1944\uff09<\/span>\u6307\u5bfc\u4e0b\u505a\u7814\u7a76\u5de5\u4f5c\u3002\u4ed6\u7814\u7a76\u62d3\u6251\u5b66\u548c\u904d\u5386\u7406\u8bba\uff0c\u4ee5\u53ca\u6052\u661f\u5927\u6c14\u6d41\u4f53\u52a8\u529b\u5b66\u30021931\u5e74\uff0c\u4ed6\u5728\u7f8e\u56fd\u56fd\u5bb6\u79d1\u5b66\u9662\u9662\u520a<\/span>\uff08PNAS<\/em>\uff09<\/span>\u53d1\u8868\u4e86\u4e00\u7bc7\u5173\u4e8e\u52a8\u529b\u7cfb\u7edf\u65f6\u95f4\u5e73\u5747\u5b9a\u7406\u7684\u8bba\u6587\u3002\u8be5\u8bba\u6587\u88ab\u8ba4\u4e3a\u662f\u73b0\u4ee3\u904d\u5386\u7406\u8bba\u7684\u521b\u59cb\u6587\u732e\u4e4b\u4e00\u3002\u5728\u6ce2\u58eb\u987f\uff0c\u970d\u666e\u592b\u8ba4\u8bc6\u4e86\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7684\u6570\u5b66\u5bb6\u8bfa\u4f2f\u7279\u00b7\u7ef4\u7eb3<\/span>\uff08Norbert Wiener\uff0c1894-1964\uff09<\/span>\u3002\u4e24\u4eba\u5408\u4f5c\u7814\u7a76\uff0c\u5bf9\u65e9\u671f\u8f90\u5c04\u4f20\u8f93\u7406\u8bba\u4e2d\u7684\u7c73\u5c14\u6069<\/span>\uff08Milne\uff09<\/span>\u65b9\u7a0b\u7ed9\u51fa\u4e86\u6c42\u89e3\u65b9\u6cd5\uff0c\u5e76\u4e8e1931\u5e74\u53d1\u8868\u4e86\u4e00\u7bc7\u8bba\u6587\uff0c\u53d1\u5c55\u4e86\u8457\u540d\u7684\u7ef4\u7eb3-\u970d\u666e\u592b\u65b9\u6cd5\u3002\u5176\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u79ef\u5206\u53d8\u6362\u5c06\u539f\u65b9\u7a0b\u5316\u4e3a\u4e00\u4e2a\u6cdb\u51fd\u65b9\u7a0b\uff0c\u540e\u4eba\u79f0\u4e4b\u4e3a\u7ef4\u7eb3-\u970d\u666e\u592b\u65b9\u7a0b\uff0c\u7136\u540e\u518d\u7528\u51fd\u6570\u56e0\u5b50\u5206\u89e3\u65b9\u6cd5\u6c42\u89e3\u3002\u8be5\u65b9\u6cd5\u7684\u79bb\u6563\u5f62\u5f0f\u672c\u8d28\u4e0a\u5c31\u662f\u540e\u6765\u7684\u7ef4\u7eb3\u6ee4\u6ce2<\/span>\uff08Wiener Filter\uff09<\/span>\u3002<\/span><\/section>\n

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1932\u5e74\uff0c\u970d\u666e\u592b\u5728\u54c8\u4f5b\u56fd\u9645\u5929\u6587\u53f0\u7814\u7a76\u5458\u57fa\u91d1\u5373\u5c06\u7ed3\u675f\u65f6\u4e0e\u67cf\u6797\u5927\u5b66\u5929\u6587\u7814\u7a76\u6240\u8054\u7cfb\uff0c\u5e0c\u671b\u56de\u53bb\u590d\u804c\u3002\u53ef\u662f\uff0c\u67cf\u6797\u7684\u56de\u4fe1\u5374\u660e\u786e\u8868\u793a\uff0c\u7531\u4e8e\u5404\u79cd\u539f\u56e0\uff0c\u4e0d\u80fd\u4e3a\u4ed6\u63d0\u4f9b\u4efb\u4f55\u5de5\u4f5c\u3002\u5728\u6b64\u56f0\u96be\u65f6\u523b\uff0c\u7ef4\u7eb3\u5e2e\u52a9\u4ed6\u83b7\u5f97\u4e86\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u6570\u5b66\u7cfb\u52a9\u7406\u6559\u6388\u7684\u804c\u4f4d\u3002\u5728\u90a3\u91cc\uff0c\u970d\u666e\u592b\u7684\u6570\u5b66\u7814\u7a76\u4e3b\u8981\u5728\u904d\u5386\u7406\u8bba\u65b9\u9762\u3002<\/span><\/section>\n

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1933\u5e741\u6708\uff0c\u7eb3\u7cb9\u63a5\u7ba1\u4e86\u5fb7\u56fd\u653f\u6743\u3002\u65b0\u653f\u5e9c\u968f\u5373\u65bd\u884c\u7684\u4e00\u7cfb\u5217\u6cd5\u5f8b\u6781\u5927\u5730\u6539\u53d8\u4e86\u5927\u5b66\u53ca\u5404\u4e2a\u90e8\u95e8\u7684\u5de5\u4f5c\u72b6\u51b5\uff0c\u8ba9\u5927\u591a\u6570\u5177\u6709\u72b9\u592a\u8840\u7edf\u6216\u4e0d\u540c\u653f\u6cbb\u7acb\u573a\u7684\u4eba\u5931\u53bb\u4e86\u4ed6\u4eec\u7684\u5de5\u4f5c\u751a\u81f3\u517b\u8001\u91d1\u30021932 \u5e74\u5e95\uff0c\u970d\u666e\u592b\u4ece\u7f8e\u56fd\u5230\u5fb7\u56fd\u4f5c\u4e86\u4e00\u6b21\u77ed\u6682\u7684\u8bbf\u95ee\u3002\u8fd4\u56de\u7f8e\u56fd\u540e\uff0c\u4ed6\u5728\u7ed9\u82cf\u8054\u79fb\u6c11\u540c\u4e8bTamarkin\u7684\u4fe1\u4e2d\u5199\u8bf4\uff1a\u201c\u6211\u5f88\u60ca\u8bb6\u6709\u8fd9\u4e48\u591a\u5fb7\u56fd\u4eba\u6295\u7968\u652f\u6301\u5e0c\u7279\u52d2\u3002\u2026\u2026\u5927\u591a\u6570\u6295\u7968\u7ed9\u5e0c\u7279\u52d2\u7684\u4eba\u90fd\u662f\u5bf9\u653f\u5e9c\u548c\u81ea\u5df1\u7684\u5904\u5883\u4e0d\u6ee1\u3002\u4ed6\u4eec\u8ddf\u968f\u4efb\u4f55\u5411\u4ed6\u4eec\u63d0\u4f9b\u5404\u79cd\u4e0d\u53ef\u80fd\u5b9e\u73b0\u7684\u627f\u8bfa\u7684\u4eba\u3002\u201d<\/span><\/section>\n

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1934\u5e74\uff0c\u970d\u666e\u592b\u51fa\u7248\u4e86\u4e13\u8457\u300a\u8f90\u5c04\u5e73\u8861\u7684\u6570\u5b66\u95ee\u9898\u300b<\/span>\uff08Mathematical problems of radiative equilibrium<\/em>\uff09<\/span>\u300220\u4e16\u7eaa40\u5e74\u4ee3\u4ee5\u540e\uff0c\u8fd9\u4e00\u7406\u8bba\u5728\u89e3\u6790\u51fd\u6570\u8fb9\u503c\u95ee\u9898\u3001\u8c03\u548c\u5206\u6790\u53ca\u7b97\u5b50\u7406\u8bba\u7684\u57fa\u7840\u4e0a\u5f97\u5230\u4e86\u7cfb\u7edf\u7684\u53d1\u5c55\uff0c\u6210\u4e3a\u7814\u7a76\u5404\u79cd\u6570\u5b66\u7269\u7406\u95ee\u9898\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u6cd5\uff0c\u5176\u5e94\u7528\u4e5f\u4ece\u8f90\u5c04\u95ee\u9898\u6269\u5c55\u5230\u8bb8\u591a\u5176\u4ed6\u9886\u57df\u3002<\/span><\/section>\n

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1933\u5e74\u8d77\uff0c\u5fb7\u56fd\u83b1\u6bd4\u9521\u5927\u5b66\u6301\u7eed\u5730\u4e3a\u5df2\u6545\u6570\u5b66\u8bb2\u5ea7\u6559\u6388\u83b1\u6602\u00b7\u5229\u5e0c\u6ed5\u65af\u5766<\/span>\uff08Leon Lichtenstein\uff0c1878-1933\uff09<\/span>\u5bfb\u627e\u7ee7\u4efb\u8005\uff0c\u6700\u540e\u4e8e1936\u5e74\u9074\u9009\u51fa\u970d\u666e\u592b\u3002\u970d\u666e\u592b\u63a5\u53d7\u4e86\u8058\u8bf7\uff0c\u5e76\u4e8e\u540c\u5e7410\u6708\u62b5\u8fbe\u83b1\u6bd4\u9521\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6b64\u524d\u6ce2\u6069\u5927\u5b66\u3001\u54e5\u5ef7\u6839\u5927\u5b66\u548c\u67cf\u6797\u5927\u5b66\u4e5f\u66fe\u9080\u8bf7\u970d\u666e\u592b\u53bb\u5de5\u4f5c\uff0c\u4f46\u90fd\u6ca1\u6709\u6210\u529f\uff0c\u56e0\u4e3a\u6709\u4eba\u5411\u7eb3\u7cb9\u653f\u5e9c\u544a\u53d1\u4ed6\u201c\u5b66\u751f\u65f6\u671f\u6709\u4e00\u4e2a\u670b\u53cb\u662f\u5171\u4ea7\u515a\u4eba\u201d\u3002<\/span><\/section>\n

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\u970d\u666e\u592b\u5728\u83b1\u6bd4\u9521\u5927\u5b66\u4efb\u804c\u81f31944\u5e74\u30021937\u5e74\uff0c\u4ed6\u7efc\u5408\u5728\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7684\u7814\u7a76\u6210\u679c\uff0c\u51fa\u7248\u4e86\u6570\u5b66\u4e13\u8457\u300a\u904d\u5386\u7406\u8bba\u300b<\/span>\uff08Ergodentheorie<\/em>\uff09<\/span>\u30021942\u5e74\uff0c\u67cf\u6797\u653f\u5e9c\u7684\u7eb3\u7cb9\u5b98\u5458\u6307\u6d3e\u970d\u666e\u592b\u5230\u4f4d\u4e8e\u5df4\u4f10\u5229\u4e9aAinring\u7684\u5fb7\u56fd\u79d1\u5b66\u7814\u7a76\u9662\u5de5\u4f5c\u3002\u5728\u90a3\u91cc\uff0c\u4ed6\u4e3b\u8981\u505a\u91cf\u5b50\u529b\u5b66\u3001\u6d41\u4f53\u52a8\u529b\u5b66\u548c\u6e4d\u6d41\u7406\u8bba\u7814\u7a76\u3002\u56e0\u4e3a\u8fd9\u6bb5\u5de5\u4f5c\u7ecf\u5386\uff0c\u540e\u6765\u6709\u4eba\u8bf4\u4ed6\u7684\u8fd9\u4e9b\u53ef\u88ab\u6218\u4e89\u5229\u7528\u7684\u7814\u7a76\u662f\u4e3a\u7eb3\u7cb9\u670d\u52a1\uff0c\u751a\u81f3\u8fd8\u6709\u4eba\u4f2a\u9020\u4ed6\u7684\u8bba\u6587\u52a0\u4ee5\u653b\u51fb\u30022017\u5e74\uff0c\u970d\u666e\u592b\u7684\u7269\u7406\u5316\u5b66\u5bb6\u5973\u513f\u82ad\u82ad\u62c9\u00b7\u970d\u666e\u592b\u00b7\u5965\u82ac\u54c8\u8328<\/span> (Barbara Hopf Offenhartz) <\/span>\u5728\u4e00\u7bc7\u6587\u7ae0\u4e2d\u56de\u5fc6\u9053\uff1a\u201c\u6211\u7684\u7236\u4eb2\u975e\u5e38\u53cd\u5bf9\u7eb3\u7cb9\uff0c\u6240\u4ee5\u4eb2\u7eb3\u7cb9\u5206\u5b50\u7684\u5404\u79cd\u6781\u7aef\u8a00\u884c\u8ba9\u4ed6\u6df1\u611f\u4e0d\u5b89\u3002\u6211\u7236\u4eb2\u4e3a\u4e86\u81ea\u5df1\u7684\u58f0\u8a89\u592a\u8fc7\u76f4\u8a00\u4e0d\u8bb3\u3002\u201d<\/span><\/section>\n

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1944\u5e74\uff0c\u6155\u5c3c\u9ed1\u5927\u5b66\u5728\u957f\u65f6\u95f4\u5bfb\u627e\u9000\u4f11\u6570\u5b66\u8bb2\u5ea7\u6559\u6388\u5eb7\u65af\u5766\u4e01\u00b7\u5361\u62c9\u897f\u5965\u591a\u91cc <\/span>\uff08Constantin Caratheodory\uff0c1873-1950\uff09<\/span>\u7684\u7ee7\u4efb\u8005\u4e4b\u540e\uff0c\u51b3\u5b9a\u628a\u8be5\u804c\u4f4d\u63d0\u4f9b\u7ed9\u970d\u666e\u592b\u3002\u970d\u666e\u592b\u63a5\u53d7\u4e86\u6155\u5c3c\u9ed1\u5927\u5b66\u7684\u8058\u4efb\uff0c\u5408\u7ea6\u52301948\u5e74\u7ec8\u6b62\u30021945\u5e746\u670823\u65e5\uff0c\u4ed6\u5199\u4e86\u4e00\u5c01\u4fe1\u7ed9\u7f8e\u56fd\u7ebd\u7ea6\u5927\u5b66\u6570\u5b66\u5bb6\u7406\u67e5\u5fb7\u00b7\u67ef\u6717<\/span>\uff08Richard Courant\uff0c1888-1972\uff09<\/span>\uff0c\u8bf4\u4ed6\u5728\u7eb3\u7cb9\u5fb7\u56fd\u5de5\u4f5c\u56e0\u7f3a\u4e4f\u653f\u6cbb\u652f\u6301\u800c\u4e00\u76f4\u611f\u53d7\u7740\u75db\u82e6\u3002\u67ef\u6717\u5f97\u77e5\u540e\uff0c\u4fbf\u9080\u8bf7\u4ed61947\u5e74\u5230\u7f8e\u56fd\u7ebd\u7ea6\u5927\u5b66\u67ef\u6717\u6570\u5b66\u79d1\u5b66\u7814\u7a76\u6240\u5f53\u5ba2\u5ea7\u6559\u6388\u3002\u970d\u666e\u592b\u4e8e\u662f\u53bb\u4e86\u7ebd\u7ea6\uff0c\u5e76\u57281949\u5e74\u6210\u4e3a\u7f8e\u56fd\u516c\u6c11\u3002\u968f\u540e\uff0c\u4ed6\u63a5\u53d7\u4e86\u5370\u7b2c\u5b89\u7eb3\u5927\u5b66\u6570\u5b66\u6559\u6388\u804c\u4f4d\uff0c\u5de5\u4f5c\u81f31972\u5e74\u9000\u4f11\u3002\u4ed6\u5728\u4f2f\u660e\u987f\u5e02\u751f\u6d3b\u52301983\u5e74\u75c5\u4e16\u30021951\u5e74-1981\u5e74\u8fd9\u4e09\u5341\u5e74\u95f4\uff0c\u970d\u666e\u592b\u4e00\u76f4\u62c5\u4efb\u5370\u7b2c\u5b89\u7eb3\u5927\u5b66\u6570\u5b66\u6742\u5fd7\u7f16\u8f91\u3002\u671f\u95f4\uff0c\u4ed6\u66fe\u591a\u6b21\u56de\u8bbf\u5fb7\u56fd\uff0c\u4e3b\u8981\u5230\u57c3\u5c14\u5170\u6839<\/span>\uff08Erlangen\uff09<\/span>\u5927\u5b66\u548c\u5965\u4f2f\u6c83\u5c14\u6cd5\u8d6b<\/span>\uff08Oberwolfach\uff09<\/span>\u6570\u5b66\u7814\u7a76\u6240\u4f5c\u5b66\u672f\u4ea4\u6d41\u3002<\/span><\/section>\n

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1956\u5e74\uff0c\u970d\u666e\u592b\u66fe\u6536\u5230\u5fb7\u56fd\u6d77\u5fb7\u5821\u5927\u5b66\u7684\u4e00\u4efd\u5de5\u4f5c\u9080\u8bf7\u3002\u4f46\u4ed6\u5bf9\u5f53\u5e74\u7684\u7eb3\u7cb9\u5fc3\u6709\u4f59\u60b8\uff0c\u201c\u4e0d\u60f3\u518d\u72af\u540c\u6837\u7684\u9519\u8bef\u201d\uff0c\u5a49\u8a00\u8c22\u7edd\u4e86\u3002<\/span><\/section>\n

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\u56fe2<\/strong> \u970d\u666e\u592b\u5728\u8bb2\u8bfe\uff081970\uff09<\/span><\/section>\n

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2. \u6210\u5c31<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u970d\u666e\u592b\u57281926\u5e74-1971\u5e74\u95f4\u51fa\u7248\u4e865\u90e8\u4e13\u8457\u5e76\u53d1\u8868\u4e8679\u7bc7\u8bba\u6587\uff0c\u5176\u4e2d1\u672c\u4e66\u548c41\u7bc7\u6587\u7ae0\u4e3a\u5fb7\u6587\uff0c4\u672c\u4e66\u548c38\u7bc7\u6587\u7ae0\u4e3a\u82f1\u6587\u3002<\/span><\/section>\n

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\u970d\u666e\u592b\u5728\u904d\u5386\u7406\u8bba\u3001\u504f\u5fae\u5206\u65b9\u7a0b\u3001\u79ef\u5206\u65b9\u7a0b\u3001\u6d41\u4f53\u52a8\u529b\u5b66\u548c\u5fae\u5206\u51e0\u4f55\u65b9\u9762\u90fd\u6709\u91cd\u8981\u8d21\u732e\u3002\u4ed6\u57281927\u5e74\u53d1\u8868\u7684\u6781\u5927\u503c\u539f\u7406\u662f\u692d\u5706\u504f\u5fae\u5206\u65b9\u7a0b\u7406\u8bba\u4e2d\u6700\u91cd\u8981\u7684\u6280\u672f\u4e4b\u4e00\u3002\u9664\u4e86\u4e0a\u9762\u63d0\u5230\u7684\u7ef4\u7eb3-\u970d\u666e\u592b\u65b9\u6cd5\u4e4b\u5916\uff0c\u4ed6\u5bf9\u62df\u7ebf\u6027\u53cc\u66f2\u65b9\u7a0b\u7684\u5904\u7406\u63d0\u4f9b\u4e86\u7814\u7a76\u5b88\u6052\u5b9a\u5f8b\u7684\u7b2c\u4e00\u4e2a\u73b0\u4ee3\u65b9\u6cd5\u3002\u4ed6\u8bc1\u660e\u4e86\u5177\u6709\u975e\u9f50\u6b21\u8fb9\u754c\u6761\u4ef6\u7684\u7eb3\u7ef4-\u65af\u6258\u514b\u65af<\/span>\uff08Navier-Stokes\uff09<\/span>\u65b9\u7a0b\u5f31\u89e3\u7684\u5b58\u5728\u6027\u5b9a\u7406\uff0c\u662f\u8be5\u9886\u57df\u4e00\u9879\u7a81\u7834\u6027\u7684\u8fdb\u5c55\u3002<\/span><\/section>\n

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\u5728\u8fd9\u91cc\u6211\u4eec\u5173\u5fc3\u7684\uff0c\u662f\u970d\u666e\u592b1942\u5e74\u5728\u4e00\u7bc7\u5960\u57fa\u6027\u8bba\u6587[4]\u4e2d\u63d0\u51fa\u7684\u52a8\u529b\u7cfb\u7edf\u5206\u5c94\u7406\u8bba\u3002\u4ed6\u5f15\u8fdb\u7684\u6982\u5ff5\u548c\u672f\u8bed\u4e3a\u540e\u4eba\u91c7\u7528\u3002\u4ed6\u8be6\u7ec6\u5730\u8ba8\u8bba\u4e86\u81ea\u5df1\u7684\u7ed3\u679c\u4e0e\u5e9e\u52a0\u83b1<\/span>\uff08Jules Henri Poincar\u00e9\uff0c1854-1912\uff09<\/span>\u5de5\u4f5c\u7684\u8054\u7cfb\uff0c\u5e76\u5c06\u8be5\u8bba\u6587\u732e\u7ed9\u57281907\u5e74\u89e3\u51b3\u4e86\u5e0c\u5c14\u4f2f\u7279\u7b2c22\u95ee\u9898\u7684\u5fb7\u56fd\u6570\u5b66\u5bb6\u4fdd\u7f57\u00b7\u514b\u4f2f<\/span>\uff08Paul Koebe\uff0c1882-1945\uff09<\/span>\u3002<\/span><\/section>\n

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\u5e9e\u52a0\u83b1-\u5b89\u5fb7\u7f57\u8bfa\u592b-\u970d\u666e\u592b\u5206\u5c94\u7406\u8bba\uff0c\u901a\u79f0\u4e3a\u201c\u970d\u666e\u592b\u5206\u5c94\u201d\u7406\u8bba\uff0c\u662f\u9996\u5148\u7531\u5e9e\u52a0\u83b1\u57281892\u5e74\u5bf9\u5e73\u9762\u7cfb\u7edf\u8fdb\u884c\u7814\u7a76\u7684[5]\uff0c\u7136\u540e\u7531\u4e9a\u5386\u5c71\u5927\u00b7\u5b89\u5fb7\u7f57\u8bfa\u592b<\/span>\uff08Aleksandr Andronov\uff0c1901-1952\uff09<\/span>\u53ca\u5176\u5408\u4f5c\u8005\u57281930\u5e74\u8fdb\u884c\u4e86\u5b8c\u5584\u548c\u7ec6\u5316[6]\uff0c\u6700\u540e\u7531\u970d\u666e\u592b\u57281942\u5e74\u6269\u5c55\u5230\u4e00\u822c\u9ad8\u7ef4\u7cfb\u7edf\u3002\u7c97\u7565\u5730\u8bf4\uff0c\u970d\u666e\u592b\u5206\u5c94\u662f\u975e\u7ebf\u6027\u5e38\u5fae\u5206\u65b9\u7a0b\u81ea\u6cbb\u7cfb\u7edf\u7684\u4e00\u79cd\u52a8\u529b\u5b66\u73b0\u8c61\uff0c\u6307\u968f\u7740\u67d0\u4e2a\u5173\u952e\u53c2\u6570\u7684\u8fde\u7eed\u53d8\u5316\uff0c\u7cfb\u7edf\u5728\u5176\u5e73\u8861\u70b9\u9644\u8fd1\u76f8\u5e94\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u4e00\u5bf9\u5171\u8f6d\u590d\u6570\u7279\u5f81\u503c\u4ece\u5de6\u534a\u5e73\u9762\u5782\u76f4\u6a2a\u8de8\u865a\u8f74\u800c\u8f6c\u79fb\u5230\u53f3\u534a\u5e73\u9762\uff0c\u5728\u8de8\u8d8a\u7684\u77ac\u95f4\u6539\u53d8\u4e86\u5e73\u8861\u70b9\u7684\u7a33\u5b9a\u6027\uff0c\u5bfc\u81f4\u7cfb\u7edf\u4ea7\u751f\u51fa\u4e00\u4e2a\u65b0\u7684\u5468\u671f\u89e3\u3002<\/span><\/section>\n

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\u4eca\u5929\uff0c\u6570\u5b66\u6587\u732e\u4e2d\u4ee5\u970d\u666e\u592b\u547d\u540d\u7684\u672f\u8bed\u5305\u62ec\uff1aHopf bifurcation\u3001Hopf decomposition\u3001Hopf factorization\u3001Hopf lemma\u3001Hopf maximum principle\u3001Cole\u2013Hopf transformation\u3001Landau\u2013Hopf theory of turbulence\u3001Wiener\u2013Hopf method\u7b49\u3002<\/span><\/section>\n

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\u7531\u4e8e\u970d\u666e\u592b\u5bf9\u6570\u5b66\u7814\u7a76\u7684\u8bf8\u591a\u91cd\u8981\u6210\u679c\u548c\u5f00\u521b\u6027\u8d21\u732e\uff0c1950\u5e74\u4ed6\u88ab\u56fd\u9645\u6570\u5b66\u5bb6\u5927\u4f1a\u9080\u8bf7\u4f5c\u5927\u4f1a\u62a5\u544a\u30021971\u5e74\uff0c\u4ed6\u5728\u7f8e\u56fd\u6570\u5b66\u4f1a\u4f5c\u4e86\u8363\u8a89\u5f88\u9ad8\u7684Josiah Willard Gibbs\u8bb2\u5ea7\u30021981\u5e74\uff0c\u4ed6\u83b7\u5f97\u7f8e\u56fd\u6570\u5b66\u4f1a\u6700\u9ad8\u5956Leroy P. Steele Prize\u30022002\u5e74\uff0c\u7f8e\u56fd\u6570\u5b66\u5b66\u4f1a\u51fa\u7248\u4e86\u300a\u827e\u4f2f\u54c8\u5fb7\u00b7\u970d\u666e\u592b\u9009\u96c6\u300b<\/span>\uff08Selected Works of Eberhard Hopf<\/em>\uff09<\/span>\uff0c\u4f5c\u4e3a\u5bf9\u970d\u666e\u592b\u767e\u5c81\u8bde\u8fb0\u7684\u7eaa\u5ff5\u3002<\/span><\/section>\n

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\u5370\u7b2c\u5b89\u7eb3\u5927\u5b66\u6570\u5b66\u6742\u5fd7\u57281983\u5e74\u6700\u540e\u4e00\u671f\u520a\u767b\u7684\u60bc\u5ff5\u6587\u7ae0\u7ed3\u8bed\u4e2d\u8bf4\uff1a\u201c\u970d\u666e\u592b\u5728\u5176\u56db\u5341\u4e94\u5e74\u7684\u804c\u4e1a\u751f\u6daf\u4e2d\u4e0d\u4ec5\u4e3a\u6570\u5b66\u77e5\u8bc6\u7684\u589e\u957f\u505a\u51fa\u4e86\u91cd\u5927\u8d21\u732e\uff0c\u800c\u4e14\u5728\u786e\u7acb\u73b0\u4ee3\u6570\u5b66\u7814\u7a76\u7684\u5f62\u5f0f\u548c\u65b9\u5411\u65b9\u9762\u4e5f\u53d1\u6325\u4e86\u91cd\u8981\u4f5c\u7528\u3002\u867d\u7136\u4ed6\u4e00\u4e9b\u5de5\u4f5c\u7684\u5f71\u54cd\u662f\u7acb\u7aff\u89c1\u5f71\u7684\uff0c\u4f46\u5176\u5927\u90e8\u5206\u5de5\u4f5c\u7684\u6df1\u5ea6\u548c\u5e7f\u5ea6\u53ea\u6709\u5728\u7ecf\u8fc7\u591a\u5e74\u7684\u8fdb\u4e00\u6b65\u7814\u7a76\u540e\u624d\u4f1a\u5b8c\u5168\u663e\u73b0\u51fa\u6765\u3002\u4ed6\u6fc0\u52b1\u4e86\u6211\u4eec\u6240\u6709\u4eba\uff1b\u4ed6\u4f5c\u4e3a\u6570\u5b66\u9886\u5bfc\u8005\u5c06\u88ab\u6df1\u6df1\u6000\u5ff5\u3002\u201d<\/span><\/section>\n

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\u56fe3<\/strong>\u300a\u827e\u4f2f\u54c8\u5fb7\u00b7\u970d\u666e\u592b\u9009\u96c6\u300b\uff08\u7f8e\u56fd\u6570\u5b66\u5b66\u4f1a\uff0c2002\uff09<\/span><\/section>\n

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3. \u6ce8\u8bb0<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u548c\u827e\u4f2f\u54c8\u5fb7\u00b7\u970d\u666e\u592b\u540c\u65f6\u4ee3\u6709\u53e6\u4e00\u4f4d\u5fb7\u56fd\u6570\u5b66\u5bb6\u6d77\u56e0\u8328\u00b7\u970d\u666e\u592b<\/span>\uff08Heinz Hopf\uff0c1894\u5e7411\u670819\u65e5\u20131971\u5e746\u67083\u65e5\uff09<\/span>\uff0c\u662f20\u4e16\u7eaa\u6700\u91cd\u8981\u7684\u62d3\u6251\u5b66\u5bb6\u4e4b\u4e00\uff0c\u4ee5\u5176\u5728\u4ee3\u6570\u62d3\u6251\u5b66\u548c\u5fae\u5206\u51e0\u4f55\u5b66\u4e2d\u7684\u6770\u51fa\u8d21\u732e\u800c\u95fb\u540d\u4e8e\u4e16\u3002\u4ed6\u7684\u7814\u7a76\u5de5\u4f5c\u4e3b\u8981\u5728\u7ea4\u7ef4\u5316\u3001\u540c\u4f26\u8bba\u3001\u57fa\u672c\u7fa4\u4ee5\u53ca\u6d41\u5f62\u7684\u51e0\u4f55\u6027\u8d28\u65b9\u9762\uff0c\u5176\u6210\u679c\u4e3a\u73b0\u4ee3\u62d3\u6251\u5b66\u5960\u5b9a\u4e86\u57fa\u7840\uff0c\u5e76\u6df1\u523b\u5730\u5f71\u54cd\u4e86\u7269\u7406\u5b66\u91cf\u5b50\u573a\u8bba\u548c\u5f26\u7406\u8bba\u4ee5\u53ca\u8ba1\u7b97\u673a\u79d1\u5b66\u62d3\u6251\u6570\u636e\u5206\u6790\u7684\u8fdb\u5c55\u3002\u867d\u7136\u90fd\u59d3\u201c\u970d\u666e\u592b\u201d\uff0c\u6ca1\u6709\u8bb0\u5f55\u8868\u660e\u4ed6\u4fe9\u6709\u76f4\u63a5\u7684\u8840\u7f18\u5173\u7cfb\u3002\u4e0d\u8fc7\uff0c\u79d1\u666e\u6587\u732e\u4e2d\u4e0d\u4e4f\u5f20\u51a0\u674e\u6234\u7684\u4f5c\u54c1\uff0c\u628a\u4ed6\u4eec\u4e24\u4eba\u7684\u6570\u5b66\u8d21\u732e\u6df7\u4e3a\u4e00\u8c08\uff0c\u56e0\u6b64\u5927\u5bb6\u9605\u8bfb\u65f6\u9700\u8981\u5c0f\u5fc3\u9274\u522b\u3002<\/span><\/section>\n

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\u53c2\u8003\u6587\u732e\u53ca\u5ef6\u7533\u9605\u8bfb\uff1a<\/strong><\/span><\/section>\n
[1] \u9648\u5173\u8363\uff1a\u8fd9\u95e8\u590d\u6742\u6027\u79d1\u5b66\u6709\u4e2a\u522b\u81f4\u7684\u540d\u79f0\u2014\u2014\u6df7\u6c8c<\/a><\/span><\/section>\n
[2] \u9648\u5173\u8363\uff1a\u79bb\u6563\u6df7\u6c8c\u4f20\u5947<\/a><\/span><\/section>\n
[3] \u9648\u5173\u8363\uff1a\u8774\u8776\u6548\u5e94\u548c\u6df7\u6c8c\u6545\u4e8b<\/a><\/span><\/section>\n
[4] E. Hopf, Abzweigung einer periodischen L\u00f6sung von einer station\u00e4ren L\u00f6sung eines Differentialsystems, Berichte der Mathematisch-Physikalischen Klasse der S\u00e4chsischen Akademie der Wissenschaften zu Leipzig, Band XCIV, Sitzung vom 19, Januar 1942, pp. 3-22.<\/span><\/section>\n
[5] H. Poincar\u00e9, Les M\u00e9thodes Nouvelles de la M\u00e9canique C\u00e9leste, Vol. 1 (Gauthier-Villars, Paris, 1892).<\/span><\/section>\n
[6] A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators, translated from the Russian by F. Immirzi, edited and abridged by W. Fishwick (Pergamon Press, Oxford, 1966).<\/span><\/section>\n


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\u6df7\u6c8c\u79d1\u5b66\u7cfb\u5217\u79d1\u666e\u8bfe\u7a0b<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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21\u4e16\u7eaa\u662f\u590d\u6742\u6027\u7684\u4e16\u7eaa\uff0c\u7406\u89e3\u6df7\u6c8c\u662f\u63a2\u7d22\u590d\u6742\u6027\u7684\u5173\u952e\u73af\u8282\u3002\u6df7\u6c8c\u4e0e\u975e\u7ebf\u6027\u65b9\u6cd5\u662f\u7814\u7a76\u52a8\u529b\u7cfb\u7edf\u7684\u91cd\u8981\u624b\u6bb5\uff0c\u52a0\u6df1\u4e86\u5bf9\u6c14\u5019\u3001\u751f\u6001\u3001\u5927\u8111\u3001\u6d41\u884c\u75c5\u7b49\u8bf8\u591a\u590d\u6742\u7cfb\u7edf\u95ee\u9898\u7684\u7406\u89e3\uff0c\u5e76\u5728\u6e4d\u6d41\u3001\u52a0\u5bc6\u3001\u6570\u636e\u5206\u6790\u4ee5\u53ca\u751f\u547d\u79d1\u5b66\u4e2d\u6709\u5e7f\u6cdb\u5e94\u7528\uff0c\u5728\u901a\u8baf\u3001\u4ea4\u901a\u3001\u91d1\u878d\u5e02\u573a\u3001\u75be\u75c5\u4e0e\u4fe1\u606f\u4f20\u64ad\u7b49\u95ee\u9898\u4e2d\u4ea6\u6709\u8bf8\u591a\u542f\u53d1\u548c\u5e94\u7528\u3002\u8be5\u8bfe\u7a0b\u7531\u8457\u540d\u6df7\u6c8c\u7406\u8bba\u5b66\u8005\u3001\u9999\u6e2f\u57ce\u5e02\u5927\u5b66\u8bb2\u5e2d\u6559\u6388\u9648\u5173\u8363\u9886\u8854\uff0c9\u4f4d\u4ece\u4e8b\u6df7\u6c8c\u53ca\u76f8\u5173\u8de8\u5b66\u79d1\u7814\u7a76\u7684\u8d44\u6df1\u5b66\u8005\u62c5\u4efb\u5bfc\u5e08\uff0c\u5e26\u9886\u5927\u5bb6\u8d70\u8fd1\u6df7\u6c8c\u7406\u8bba\uff0c\u7406\u89e3\u5404\u7c7b\u6df7\u6c8c\u7cfb\u7edf\uff0c\u5e76\u5c1d\u8bd5\u8de8\u5b66\u79d1\u5e94\u7528\u3002\u53c2\u770b\uff1a<\/span>\u6df7\u6c8c\u79d1\u5b66\u7cfb\u5217\u79d1\u666e\u8bfe\u7a0b\uff1a\u786e\u5b9a\u6027\u7684\u7cfb\u7edf\u65e0\u6cd5\u9884\u6d4b\uff1f<\/a><\/span><\/span><\/span><\/section>\n


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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/p>\n

\u6df7\u6c8c\u79d1\u5b66\u7cfb\u5217\u79d1\u666e\u8bfe\u7a0b\uff1a\u786e\u5b9a\u6027\u7684\u7cfb\u7edf\u65e0\u6cd5\u9884\u6d4b\uff1f<\/a><\/span><\/strong><\/span><\/p>\n<\/section>\n


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Koopman\u5206\u6790\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684<\/strong><\/p>\n

\u5e94\u7528\u7cfb\u5217\u8bfe\u7a0b<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u662f\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u4e2d\u91cd\u8981\u7684\u7406\u8bba\u57fa\u7840\u548c\u7814\u7a76\u5de5\u5177\uff0c\u5b83\u6709\u52a9\u4e8e\u7406\u89e3\u7cfb\u7edf\u7684\u672c\u8d28\u7279\u6027\uff0c\u901a\u8fc7\u6784\u5efa\u548c\u5206\u6790\u7cfb\u7edf\u7684\u7c97\u7c92\u5316\u7a0b\u5ea6\u4e0d\u540c\u7684\u6570\u5b66\u6a21\u578b\u6765\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002\u5c3d\u7ba1\u4f4e\u7ef4\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u7406\u8bba\u5df2\u7ecf\u76f8\u5f53\u5b8c\u5907\uff0c\u4f46\u5982\u4f55\u5c06\u5176\u7528\u5230\u9ad8\u7ef4\u975e\u7ebf\u6027\u4f53\u7cfb\u4e2d\uff0c\u83b7\u53d6\u7cfb\u7edf\u7684\u5173\u952e\u52a8\u529b\u5b66\u4fe1\u606f\u662f\u4e00\u4e2a\u957f\u671f\u56f0\u6270\u7814\u7a76\u8005\u7684\u6311\u6218\u3002<\/span><\/section>\n

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\u597d\u7684\u7406\u8bba\u548c\u5de5\u5177\u4f1a\u5927\u5927\u52a0\u5feb\u79d1\u7814\u8fdb\u5c55\uff0cKoopman\u7b97\u7b26\u6709\u671b\u6210\u4e3a\u5206\u6790\u548c\u9884\u6d4b\u52a8\u6001\u7cfb\u7edf\u7684\u6807\u51c6\u7406\u8bba\u3002\u96c6\u667a\u5b66\u56ed\u9080\u8bf7\u5317\u4eac\u90ae\u7535\u5927\u5b66\u5170\u5cb3\u6052\u6559\u6388\u5f00\u8bbe\u300cKoopman\u5206\u6790\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684\u5e94\u7528\u300d\u7cfb\u5217\u8bfe\u7a0b<\/a>\uff0c\u9762\u5411\u81f4\u529b\u4e8e\u5b66\u4e60\u548c\u53d1\u5c55Koopman\u7b97\u7b26\u7406\u8bba\u7684\u793e\u533a\u670b\u53cb\uff0c\u57286\u8282\u8bfe\u7a0b\u65f6\u95f4\u4e2d\uff0c\u901a\u8fc7\u7ec6\u81f4\u7684\u8bb2\u89e3\uff0c\u6df1\u5165\u7684\u4e92\u52a8\uff0c\u5f62\u6210\u82cf\u683c\u62c9\u5e95\u5f0f\u8bfe\u5802\u6c1b\u56f4\uff0c\u5171\u540c\u8fdb\u6b65\u3002\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u52a0\u5165\u8bfe\u7a0b\uff01<\/span><\/section>\n

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\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/strong><\/section>\n
1. <\/strong>\u8fd9\u95e8\u590d\u6742\u6027\u79d1\u5b66\u6709\u4e2a\u522b\u81f4\u7684\u540d\u79f0\u2014\u2014\u6df7\u6c8c\u4e28\u9648\u5173\u8363<\/a><\/strong><\/span><\/section>\n
2. <\/strong>\u79bb\u6563\u6df7\u6c8c\u4f20\u5947\u4e28\u9648\u5173\u8363<\/a><\/strong><\/span><\/section>\n
3.<\/strong><\/strong> <\/strong><\/strong><\/a>\u8774\u8776\u6548\u5e94\u548c\u6df7\u6c8c\u6545\u4e8b | \u9648\u5173\u8363<\/a><\/strong><\/strong><\/span><\/section>\n
4. <\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
<\/strong>5.<\/strong> \u9f99\u5e74\u5927\u8fd0\u8d77\uff0c\u5b66\u4e60\u6b63\u5f53\u65f6\uff01\u89e3\u9501\u96c6\u667a\u5168\u7ad9\u5185\u5bb9\uff0c\u5f00\u542f\u65b0\u5e74\u5b66\u4e60\u8ba1\u5212<\/a><\/strong><\/strong><\/strong><\/strong><\/span><\/strong><\/strong><\/section>\n

6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/p>\n


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\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfe\u7a0b<\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n


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<\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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