{"id":34215,"date":"2022-05-05T20:59:08","date_gmt":"2022-05-05T12:59:08","guid":{"rendered":"https:\/\/swarma.org\/?p=34215"},"modified":"2022-05-05T20:59:08","modified_gmt":"2022-05-05T12:59:08","slug":"%e6%88%91%e4%bb%ac%e5%a6%82%e4%bd%95%e7%90%86%e8%a7%a3%e6%b7%b7%e6%b2%8c%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=34215","title":{"rendered":"\u6211\u4eec\u5982\u4f55\u7406\u89e3\u201c\u6df7\u6c8c\u201d\uff1f"},"content":{"rendered":"
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David S. Richeson<\/span><\/span> <\/span>| \u4f5c\u8005<\/strong><\/span><\/strong><\/section>\n
\u51b6\u65e0\u60c5<\/span><\/span> | \u8bd1\u8005<\/strong><\/span><\/section>\n
\u66ae\u5927\u6cb3<\/span><\/span> | \u5ba1\u6821<\/strong><\/span><\/section>\n
\u5ed6\u6234\u4e3d<\/span><\/span> | \u7f16\u8f91<\/strong><\/span><\/section>\n

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1855\u5e74\uff0c\u745e\u5178\u56fd\u738b\u5965\u65af\u5361\u4e8c\u4e16\u5ba3\u5e03\u4e86\u4e00\u9879\u7531\u56db\u4e2a\u6570\u5b66\u95ee\u9898\u7ec4\u6210\u7684\u516c\u5f00\u6311\u6218\u3002\u6cd5\u56fd\u535a\u5b66\u5bb6<\/span>\u4ea8\u5229\u5e9e\u52a0\u83b1\uff08Henri Poincar\u00e9\uff09<\/strong><\/span><\/a>\u4e13\u6ce8\u4e8e\u5176\u4e2d\u4e00\u4e2a\u4e0e\u5929\u4f53\u8fd0\u52a8\u6709\u5173\u7684\u95ee\u9898\uff0c\u5373\u6240\u8c13\u7684n\u4f53\u95ee\u9898\uff1a<\/span>\u6211\u4eec\u7684\u592a\u9633\u7cfb\u4f1a\u65e0\u9650\u671f\u5730\u7ee7\u7eed\u5176\u65f6\u949f\u822c\u7684\u8fd0\u52a8\u5417\uff1f\u884c\u661f\u4f1a\u9041\u5165\u865a\u7a7a\u5417\uff1f\u8fd8\u662f\u884c\u661f\u4f1a\u574d\u7f29\u6210\u70bd\u70ed\u7684\u592a\u9633\u4ece\u800c\u6b7b\u4ea1\uff1f<\/span><\/span><\/section>\n

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Will our solar system continue its clocklike motion indefinitely, will the planets fly off into the void, or will they collapse into a fiery solar death?<\/span><\/span><\/strong><\/span><\/section>\n
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\u5e9e\u52a0\u83b1\u7684\u89e3\u51b3\u65b9\u6848\u8868\u660e\u81f3\u5c11\u6709\u4e00\u4e9b\u7cfb\u7edf\uff0c\u5982\u592a\u9633\u3001\u5730\u7403\u548c\u6708\u7403\uff0c\u662f\u7a33\u5b9a\u7684\uff0c\u800c\u8fd9\u4e00\u65b9\u6848\u8d62\u5f97\u4e86\u4eab\u6709\u76db\u8a89\u7684\u5956\u9879\uff0c\u5e76\u4e8e1889\u5e74\u5370\u5237\u548c\u53d1\u884c\u4e86\u4e00\u7bc7\u4e0e\u4e4b\u76f8\u5173\u7684\u6587\u7ae0\u3002\u4f46\u4ed6\u7684\u89e3\u51b3\u65b9\u6cd5\u662f\u4e0d\u6b63\u786e\u7684\u3002<\/span><\/section>\n

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\u5e9e\u52a0\u83b1\u627f\u8ba4\u4e86\u4ed6\u7684\u9519\u8bef\uff0c\u5e76\u82b1\u94b1\u9500\u6bc1\u4e86\u4ed6\u89e3\u51b3\u65b9\u6848\u7684\u590d\u5370\u4ef6<\/span>\uff08\u82b1\u7684\u94b1\u6bd4\u5f97\u5230\u7684\u5956\u91d1\u8fd8\u591a\uff09<\/span>\u3002\u4e00\u4e2a\u6708\u540e\uff0c\u4ed6\u63d0\u4ea4\u4e86\u4e00\u4e2a\u66f4\u6b63\u7684\u7248\u672c\u3002\u4ed6\u6ce8\u610f\u5230\u5373\u4f7f\u662f\u4e00\u4e2a\u53ea\u6709\u4e09\u4e2a\u5bf9\u8c61\u7684\u7cfb\u7edf\u4e5f\u53ef\u80fd\u8868\u73b0\u5f97\u8fc7\u4e8e\u6df7\u4e71\u800c\u4e0d\u53ef\u9884\u6d4b\uff0c\u4ee5\u81f3\u4e8e\u65e0\u6cd5\u5efa\u6a21\u3002\u8fd9\u4e5f\u662f\u52a8\u6001\u7cfb\u7edf\u9886\u57df\u7684\u5f00\u7aef\u3002<\/span><\/span><\/section>\n

<\/span><\/em><\/span><\/strong><\/span><\/section>\n
He now saw that even a system with only three bodies could behave too unpredictably \u2014 too chaotically \u2014 to be modeled. So began the field of dynamical systems.<\/span><\/span><\/strong><\/span><\/section>\n

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\u4e3a\u4e86\u8fbe\u5230\u76ee\u7684\uff0c\u52a8\u6001\u7cfb\u7edf\u53ef\u4ee5\u7b80\u5316\u4e3a\u4e00\u4e2a\u51fd\u6570\uff0c\u5176\u53ef\u80fd\u7684\u8f93\u51fa\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u8f93\u5165\u3002\u8fd9\u4f7f\u6211\u4eec\u80fd\u591f\u5c06\u51fd\u6570\u7684\u8f93\u51fa\u4f5c\u4e3a\u8f93\u5165\u53cd\u590d\u4ee3\u5165\uff0c\u4ece\u800c\u4ea7\u751f\u4e0d\u65ad\u53d8\u5316\u7684\u884c\u4e3a\u3002\u6b63\u5982\u5e9e\u52a0\u83b1\u7684\u5de5\u4f5c\u6240\u8868\u660e\u7684\u90a3\u6837\uff0c\u8fd9\u4e2a\u7b80\u5355\u7684\u8bbe\u7f6e\u53ef\u4ee5\u4ea7\u751f\u975e\u5e38\u590d\u6742\u548c\u968f\u673a\u7684\u7ed3\u679c\uff0c\u6240\u4ee5\u5b83\u4eec\u5b9e\u9645\u4e0a\u4e5f\u88ab\u79f0\u4e3a<\/span><\/span>\u6df7\u6c8c\uff08chaotic\uff09<\/strong><\/span><\/a>\u3002<\/span><\/section>\n

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\u5927\u7ea6 70 \u5e74\u540e\uff0c\u4e00\u79cd\u4fbf\u4e8e\u7406\u89e3\u5e9e\u52a0\u83b1\u7ed3\u8bba\u5e76\u4e3a\u6df7\u4e71\u5e26\u6765\u79e9\u5e8f\u7684\u4f18\u96c5\u65b9\u5f0f\u51fa\u73b0\u4e86\u3002\u624d\u534e\u6a2a\u6ea2\u7684\u5e74\u8f7b\u62d3\u6251\u5b66\u5bb6<\/span>\uff08\u4ee5\u53ca\u672a\u6765\u7684\u83f2\u5c14\u5179\u5956\u7ae0\u83b7\u5f97\u8005\uff09<\/span>\u65af\u8482\u82ac\u00b7\u65af\u6885\u5c14(Stephen Smale)<\/strong>\u5728\u4ed6\u5199\u4e86\u5173\u4e8e\u52a8\u6001\u7cfb\u7edf\u7684\u7b2c\u4e00\u7bc7\u6587\u7ae0\u540e\u4e0d\u4e45\uff0c\u6536\u5230\u4e86\u4e00\u5c01\u4fe1\uff0c\u8fd9\u4f7f\u5f97\u8ba9\u4ed6\u53d1\u73b0\u4e86\u4e00\u4e2a\u76f8\u5bf9\u7b80\u5355\u4e14\u666e\u904d\u5b58\u5728\u7684\u51fd\u6570\uff0c\u5b83\u89e3\u91ca\u4e86\u5e9e\u52a0\u83b1\u5728\u4e09\u4f53\u95ee\u9898\u4e2d\u89c2\u5bdf\u5230\u7684\u6df7\u6c8c\u3002Smale\u79f0\u4e4b\u4e3a\u201c\u9a6c\u8e44\u94c1\u201d\uff08horseshoe\uff09<\/strong>\u3002<\/span><\/section>\n

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\u4e3a\u4e86\u7406\u89e3\u5b83\uff0c\u8ba9\u6211\u4eec\u4ece\u4e00\u4e2a\u4e0d\u6df7\u6c8c\u52a8\u6001\u7cfb\u7edf\u7684\u7b80\u5355\u4f8b\u5b50\u5f00\u59cb\u3002<\/strong>\u5047\u8bbe\u4f60\u53ea\u60f3\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u8ba1\u7b97\u5668\u8ba1\u7b97 2 \u7684\u5e73\u65b9\u6839\u3002\u201c\u725b\u987f\u6cd5\u201d<\/span>\uff08Newton\u2019s method\uff09<\/span>\u8bf4\u4f60\u53ef\u4ee5\u4ece\u4efb\u610f\u503c\u5f00\u59cb\u2014\u2014\u6bd4\u5982\u8bf43\u2014\u2014\u7136\u540e\u628a\u5b83\u4ee3\u5165\u5230\u51fd\u6570 <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u3002\u51fd\u6570\u7684\u8f93\u51fa\u503cf(3) = 1.8333333\uff0c\u8f93\u51fa\u503c\u66f4\u63a5\u8fd1\u771f\u5b9e\u503c\u3002\u4e3a\u4e86\u8fdb\u4e00\u6b65\u63a5\u8fd1\u771f\u5b9e\u503c\uff0c\u5c06\u4e0a\u4e00\u6b65\u7684\u8f93\u51fa\u503c\u4f5c\u4e3a\u8f93\u5165\u5e26\u5165\u51fd\u6570\uff1af(1.8333333) = 1.4621212\u3002\u518d\u8fd9\u6837\u8fdb\u884c3 \u6b21\u5f97\u52301.4142136\uff0c\u8fd9\u53ef\u80fd\u662f\u8ba1\u7b97\u5668\u51c6\u786e\u5ea6\u7684\u6781\u9650\u3002<\/span><\/section>\n

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\u628a\u7b2c\u516d\u6b21\u8fed\u4ee3\u8868\u793a\u4e3a\u201cf(f(f(f(f(3))))\u201d\u6709\u4e9b\u5947\u602a\uff0c\u6240\u4ee5\u6211\u4eec\u9009\u62e9\u5199\u6210 <\/path><\/g><\/path><\/g><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u5e76\u4e14\u6211\u4eec\u628a\u8f93\u51fa\u7684\u65e0\u9650\u5e8f\u5217\u79f0\u4e3ax\u7684\u201c\u8f68\u9053\u201d\u3002\u628a\u6bcf\u6b21\u8fed\u4ee3\u5f53\u4f5c\u65f6\u949f\u53d1\u51fa\u7684\u6ef4\u7b54\u58f0\uff0c\u5e76\u4e14\u628a\u8f68\u9053\u89c6\u4e3a\u6cbf\u7740\u81ea\u7136\u6570\u8f74\u63a5\u8fd1 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u7684\u4e00\u79cd\u8df3\u8dc3\uff0c\u8fd9\u6837\u4f1a\u66f4\u6709\u5229\u4e8e\u7406\u89e3\u3002<\/span><\/section>\n<\/section>\n
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\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u79f0 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> <\/span>\u4e3a\u4e00\u4e2a\u56fa\u5b9a\u5438\u5f15<\/span>\u70b9\uff1a\u56fa\u5b9a\uff0c\u662f\u56e0\u4e3a\u5b83\u56f4\u7ed5 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u2026\u4ea7\u751f\u56fa\u5b9a\u7684\u8f68\u9053\uff1b\u5438\u5f15\uff0c\u662f\u56e0\u4e3a\u5b83\u50cf\u9ed1\u6d1e\u4e00\u6837\u5438\u6536\u5b83\u9644\u8fd1\u7684\u8f68\u9053\u3002<\/span><\/section>\n

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\u4f46\u540c\u6837\uff0c\u5e76\u975e\u6240\u6709\u52a8\u529b\u7cfb\u7edf\u90fd\u8868\u73b0\u51fa\u5982\u6b64\u7b80\u5355\u548c\u53ef\u9884\u6d4b\u7684\u884c\u4e3a\u3002\u52a8\u529b\u7cfb\u7edf\u7684\u8f68\u9053\u53ef\u4ee5\u6709\u5176\u4ed6\u884c\u4e3a\uff0c\u4f8b\u5982\uff1a\u5468\u671f\u6027\u5730\u5faa\u73af\u901a\u8fc7\u4e00\u7ec4\u6709\u9650\u7684\u70b9\uff1b\u8d8b\u4e8e\u65e0\u7a77\u6216\u8005\u6ca1\u6709\u660e\u663e\u7684\u987a\u5e8f\u3002<\/span><\/section>\n

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\u4e3a\u4e86\u7406\u89e3\u8fd9\u4e9b\u5bf9\u6df7\u6c8c\u7cfb\u7edf\u6765\u8bf4\u81f3\u5173\u91cd\u8981\u7684\u6982\u5ff5\uff0c\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u7279\u522b\u6709\u542f\u53d1\u6027\u7684\u4f8b\u5b50\uff0c\u79f0\u4e3a\u5e10\u7bf7\u56feT(the tent map)<\/strong>\uff0c\u5b83\u5b9a\u4e49\u4e860\u548c1 \u4e4b\u95f4\u7684\u503cx\u3002\u5c31\u50cf\u7cd6\u679c\u5236\u9020\u5546\u62c9\u201c\u592a\u5983\u7cd6\u201d<\/span>\u7684\u8fc7\u7a0b\u4e00\u6837\uff1a\u5b83\u5148\u5c06[0,1]\u533a\u95f4\u7684\u957f\u5ea6\u62c9\u4f38\u5230\u5176\u957f\u5ea6\u7684\u4e24\u500d\uff0c\u7136\u540e\u518d\u5c06\u5176\u5bf9\u6298\uff0c\u5373\u533a\u95f4\u957f\u5ea6\u91cd\u65b0\u53d8\u6210\u4e861\u3002\u6362\u53e5\u8bdd\u8bf4\uff1a0 \u548c 1 \u90fd\u6620\u5c04\u5230 0, <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u6620\u5c04\u5230 1\u3002\u56e0\u4e3a\u5e10\u7bf7\u56feT \u4ea7\u751f\u7684\u503c\u4e5f\u5728 0 \u548c 1 \u4e4b\u95f4\uff0c\u6240\u4ee5\u5b83\u53ef\u4ee5\u662f\u4e00\u4e2a\u52a8\u6001\u7cfb\u7edf\u3002\u4e0e\u725b\u987f\u6cd5\u4e00\u6837\uff0c\u8fed\u4ee3\u51fd\u6570\u610f\u5473\u7740\u91cd\u590d\u62c9\u4f38\u548c\u5bf9\u6298\u7684\u8fc7\u7a0b\u3002<\/span><\/section>\n

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\u5e10\u7bf7\u6620\u5c04\uff0c\u5373 <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container> \uff0c\u4e0d\u65ad\u62c9\u4f38\u548c\u6298\u53e0\u533a\u95f4 [0, 1]\u3002\u4e0d\u540c\u8fed\u4ee3\u51fd\u6570\u5bf9\u5e94\u4e8e\u4e0d\u540c\u7684\u62c9\u4f38\u548c\u6298\u53e0\u3002<\/span><\/section>\n

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\u6b63\u5982 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u7684\u4f8b\u5b50\u4e2d\uff0c\u5e10\u7bf7\u56fe\u6709\u56fa\u5b9a\u70b9\uff1a0\u548c <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u3002\u4f46\u5b83\u4e5f\u6709\u4e00\u4e2a\u5728\u4e24\u70b9\u4e4b\u95f4\u4ea4\u66ff\u7684\u8f68\u9053  <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u548c <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u2014\u2014\u6211\u4eec\u79f0\u4e4b\u4e3a\u5468\u671f\u4e3a2\u7684\u8f68\u9053\uff0c\u5468\u671f\u4e3a3\u7684\u8f68\u9053\u5faa\u73af\u901a\u8fc7 <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> , <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u548c <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span><\/span> \u3002<\/span>\u4ee4\u4eba\u60ca\u8bb6\u7684\u662f\uff0c\u56e0\u4e3a\u5e10\u7bf7\u6620\u5c04\u6709\u4e00\u4e2a\u70b9\u4ea7\u751f\u5468\u671f\u4e3a3\u7684\u8f68\u9053\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8bc1\u660e\u5b83\u6709\u4efb\u610f\u5468\u671f\u7684\u70b9\u2014\u2014\u65e0\u8bba\u4f60\u9009\u62e9\u4ec0\u4e48\u6b63\u6574\u6570\uff0c\u90fd\u4f1a\u6709\u4e00\u4e2a\u91cd\u590d\u7684\u8f68\u9053\uff0c\u8f68\u9053\u4e2d\u6709\u4e14\u4ec5\u6709\u90a3\u4e48\u591a\u70b9<\/span><\/span>\uff08Li-Yorke\u5b9a\u7406\uff09<\/span>\u3002<\/span><\/section>\n

<\/section>\n
\u7b2c\u4e00\u4e2a\u53d1\u73b0\u201c\u5b9e\u6570\u7ebf\u4e0a\u51fd\u6570\u201d\u8fd9\u4e00\u4e8b\u5b9e\u7684\u662f\u4e4c\u514b\u5170\u6570\u5b66\u5bb6\u4e9a\u5386\u5c71\u5927\u00b7\u6c99\u5c14\u79d1\u592b\u65af\u57fa<\/strong>(Alexander Sharkovsky)\u3002\u7136\u800c\uff0c\u4ed6 1964 \u5e74\u5173\u4e8e\u8be5\u4e3b\u9898\u7684\u8bba\u6587\u5728\u4e1c\u6b27\u4ee5\u5916\u4ecd\u7136\u4e0d\u4e3a\u4eba\u77e5\uff0c\u76f4\u5230 1975 \u5e74\u9a6c\u91cc\u5170\u5927\u5b66\u6570\u5b66\u5bb6\u674e\u5929\u4e00<\/span>(Tien-Yien Li)<\/span>\u548c\u8a79\u59c6\u65af\u00b7\u7ea6\u514b<\/span>(James Yorke)<\/span>\u91cd\u65b0\u53d1\u73b0\u5b83\u65f6\uff0c\u7ed3\u679c\u624d\u4e3a\u4eba\u6240\u77e5\u3002\u4ed6\u4eec\u8bc1\u660e\u4e86\u8fd9\u6837\u4e00\u4e2a\u52a8\u6001\u7cfb\u7edf\u4e5f\u6709\u4e0d\u53ef\u8fa8\u522b\u7684\u8f68\u9053\uff0c\u4f8b\u5982\u5e10\u7bf7\u56fe\u4e2d\u7684\u70b9 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u4ea7\u751f\u7684\u8f68\u9053\u3002\u4ed6\u4eec\u5199\u9053\uff1a \u201c\u5468\u671f\u4e3a3\u610f\u5473\u7740\u6df7\u6c8c\u201d\uff0c\u5728\u6b64\u8fc7\u7a0b\u4e2d\u521b\u9020\u4e86\u6570\u5b66\u672f\u8bed\u201c\u6df7\u6c8c\u201d-chaos\u3002<\/span><\/section>\n

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\u5e10\u7bf7\u56fe\u4e2d\u7684\u70b9 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container> \u4ea7\u751f\u7684\u8f68\u9053\u6ca1\u6709\u89c4\u5f8b\u3002<\/span><\/section>\n

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\u66f4\u6709\u8da3\u7684\u662f\uff0c\u5373\u4f7f\u70b9 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u548c\u70b9 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/path><\/path><\/path><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u9760\u5f97\u5f88\u8fd1\uff0c\u4f46\u5b83\u4eec\u7684\u8f68\u9053\u5f88\u5feb\u5c31\u5206\u5f00\u4e86\uff1a\u4f8b\u5982  <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u2014\u2014 <\/path><\/g><\/path><\/g><\/g><\/path><\/g><\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u800c <\/path><\/g><\/path><\/g><\/g><\/path><\/g><\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/path><\/path><\/path><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u201c\u5bf9\u521d\u59cb\u6761\u4ef6\u654f\u611f\u201d\uff0c\u6216\u8005\u66f4\u901a\u4fd7\u5730\u8bf4\u662f\u8774\u8776\u6548\u5e94\u3002\u5c0f\u7684\u521d\u503c\u53d8\u5316\u53ef\u80fd\u5bfc\u81f4\u5927\u7684\u7ed3\u679c\u53d8\u5316\u3002\u6b63\u5982\u6570\u5b66\u5bb6\u548c\u6c14\u8c61\u5b66\u5bb6\u7231\u5fb7\u534e\u6d1b\u4f26\u5179<\/span><\/span> (Edward Lorenz) <\/span>\u6240\u8bf4\uff0c\u201c\u8774\u8776\u5728\u5df4\u897f\u62cd\u6253\u7fc5\u8180\u4f1a\u4e0d\u4f1a\u5728\u5fb7\u514b\u8428\u65af\u5dde\u5f15\u53d1\u9f99\u5377\u98ce\uff1f\u201d\u867d\u7136\u6df7\u6c8c\u6ca1\u6709\u56fa\u5b9a\u7684\u5b9a\u4e49\uff0c\u4f46\u8fd9\u79cd\u654f\u611f\u4f9d\u8d56\u6027\u662f\u5b83\u7684\u6807\u5fd7\u4e4b\u4e00\u3002<\/span><\/section>\n

<\/span><\/section>\n
This phenomenon is known as \u201csensitive dependence on initial conditions,\u201d or more informally as the butterfly effect. Small initial changes can lead to big outcome changes. As the mathematician and meteorologist Edward Lorenz put it, \u201cDoes the flap of a butterfly\u2019s wings in Brazil set off a tornado in Texas?\u201d While there is no settled definition of chaos, this sensitive dependence is one of its hallmarks.<\/span><\/strong><\/span><\/strong><\/em><\/span><\/section>\n

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\u4e3a\u4e86\u5e2e\u52a9\u7406\u89e3\u8fd9\u4e9b\u6df7\u4e71\u7684\u7cfb\u7edf\u2014\u2014\u4ee5\u53ca Smale \u7684\u9a6c\u8e44\u94c1\u2014\u2014\u8ba9\u6211\u4eec\u5148\u4f7f\u7528\u53ef\u80fd\u7b2c\u4e00\u773c\u770b\u8d77\u6765\u5f88\u7c97\u7cd9\u7684\u6280\u672f\u3002\u9996\u5148\uff0c\u5c06\u533a\u95f4\u6807\u8bb0\u4e3a L \u548c R \u4e24\u90e8\u5206\u3002\u7136\u540e\uff0c\u968f\u7740\u8f68\u9053\u7684\u524d\u8fdb\uff0c\u53ea\u9700\u6ce8\u610f\u4e0b\u4e00\u6b21\u8fed\u4ee3\u843d\u5728\u54ea\u4e00\u534a\u4e0a\u3002\u8fd9\u4e2a\u5e8f\u5217\u662f\u8f68\u9053\u7684\u201c\u8def\u7ebf\u201d-itinerary\u3002\u4f8b\u5982\uff0c\u5468\u671f\u4e3a3\u7684\u8f68\u9053\u7684\u8def\u7ebf\uff0c\u4ece <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u5f00\u59cb\uff0c\u662fLLRLLRLLR… \u56e0\u4e3a <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u548c <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u5206\u522b\u5728L\u548cR\u4e2d\u3002\u800c\u4ece <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u5f00\u59cb\u7684\u8def\u7ebf\u5219\u662f\uff1aLRLRRRRRLL\u3002<\/span><\/section>\n

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\u5e10\u7bf7\u56fe\u4e0b\uff0c\u4ece\u70b9 <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container> \u5f00\u59cb\uff0c\u4ea7\u751f\u4e00\u4e2a\u5468\u671f\u4e3a3\u7684\u8f68\u9053\uff0c\u5176\u8def\u7ebf\u4e3a LLRLLRLLR\u2026\uff0c\u4ece\u70b9 <\/path><\/g><\/g><\/path><\/g><\/rect><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container> \u5f00\u59cb\uff0c\u4ea7\u751f\u7684\u8f68\u9053\u5f62\u6210\u7684\u8def\u7ebf\u4e3a LRLRRRRRLL\u2026<\/span><\/section>\n

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\u7528\u8def\u7ebf\u6765\u8868\u793a\u8f68\u9053\u770b\u8d77\u6765\u4f1a\u635f\u5931\u5f88\u591a\u7684\u4fe1\u606f\uff0c\u4f46\u4e8b\u5b9e\u5e76\u975e\u5982\u6b64\u3002\u8fd9\u662f\u56e0\u4e3a\u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2aL\u548cR\u7684\u90fd\u5bf9\u5e94\u5e76\u4e14\u4ec5\u5bf9\u5e94\u4e00\u4e2a\u70b9\u3002\u4f8b\u5982\uff0c\u4ece\u70b9 <\/path><\/g><\/path><\/g><\/rect><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u5f00\u59cb\u7684\u8f68\u9053\u8def\u7ebf\u53ea\u80fd\u662f LLRLLRLLR\u2026\u8fd9\u6837\u7684\u7279\u70b9\u4e3a\u6709\u5229\u4e8e\u5206\u6790\u5e10\u7bf7\u6620\u5c04\u7684\u52a8\u6001\u7ed3\u6784\u3002\u8fd9\u8868\u660e\uff0c\u5f53\u4ed6\u4eec\u7684\u884c\u7a0b\u662f\u56fa\u5b9a\u7684\u65f6\uff0c\u8fd9\u4e9b\u70b9\u662f\u5468\u671f\u6027\u7684\u3002\u5b83\u8fd8\u5141\u8bb8\u6211\u4eec\u4ece\u4efb\u4f55\u7ed9\u5b9a\u7684\u8def\u7ebf\u4e2d\u786e\u5b9a\u4e00\u4e2a\u70b9\u7684\u7cbe\u786e\u4f4d\u7f6e\u3002<\/span><\/section>\n

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\u73b0\u5728\u8ba9\u6211\u4eec\u5c06\u5e10\u7bf7\u56fe\u5ef6\u4f38\u5230\u591a\u7ef4\u5ea6\uff0c\u6700\u7ec8\u6ee1\u8db3 Smale \u7684\u9a6c\u8e44\u5f62\u51fd\u6570h\u3002\u4ece\u4e00\u4e2a\u6b63\u65b9\u5f62\u5f00\u59cb\uff0c\u5c06\u5176\u62c9\u4f38\u6210\u4e00\u4e2a\u7ec6\u957f\u7684\u77e9\u5f62\uff0c\u5c06\u5176\u6298\u53e0\u6210\u9a6c\u8e44\u5f62\uff0c\u7136\u540e\u5c06\u5176\u653e\u5728\u539f\u6765\u7684\u6b63\u65b9\u5f62\u4e0a\u3002<\/span><\/section>\n

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Smale \u7684\u9a6c\u8e44\u51fd\u6570\u5c06\u6b63\u65b9\u5f62\u62c9\u4f38\u5e76\u5728\u5176\u4e0a\u653e\u7f6e\u521d\u59cb\u6b63\u65b9\u5f62\u3002<\/span><\/section>\n

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Smale\u2019s horseshoe map stretches and folds a square across itself.<\/span><\/section>\n

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\u4e0e\u6240\u6709\u52a8\u6001\u7cfb\u7edf\u4e00\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u91cd\u590d\u8fd9\u4e2a\u8fc7\u7a0b\u2014\u2014\u62c9\u4f38\u3001\u6298\u53e0\u3001\u62c9\u4f38\u3001\u6298\u53e0\u3001\u62c9\u4f38\u3001\u6298\u53e0\u2014\u2014\u5b9e\u73b0\u5728\u201c\u5927\u9a6c\u8e44\u94c1\u201d\u4e2d\u4ea7\u751f\u201c\u5c0f\u9a6c\u8e44\u94c1\u201d\u3002<\/span><\/section>\n

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\u8fed\u4ee3 Smale \u7684 \u201c\u9a6c\u8e44\u51fd\u6570\u201d\u4f1a\u4ea7\u751f\u9a6c\u8e44\u94c1\u5185\u7684\u9a6c\u8e44\u94c1\u3002<\/span><\/section>\n

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\u6211\u4eec\u53ea\u9700\u8981\u77e5\u9053\u70b9 <\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u7684\u8def\u7ebf\uff0c\u90a3\u4e48\u9a6c\u8e44\u5f62\u56fe\u5c31\u662f\u53ef\u9006\u7684\u3002\u6839\u636e <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u6211\u4eec\u53ef\u4ee5\u77e5\u9053 <\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u53bb\u5411\u4f55\u65b9\uff0c\u901a\u8fc7 <\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \uff0c\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u70b9 <\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u7684\u6765\u5904\u3002\u5c06 <\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/span> \u4f5c\u7528\u4e8e\u539f\u59cb\u6b63\u65b9\u5f62\uff0c\u5c06\u4f1a\u4ea7\u751f\u4e0e\u7b2c\u4e00\u4e2a\u6b63\u65b9\u5f62\u6210\u76f4\u89d2\u7684\u65b0\u9a6c\u8e44\u5f62\u3002\u5982\u679c\u7ee7\u7eed\u8fd9\u6837\u64cd\u4f5c\u4e0b\u53bb\uff0c\u5c06\u5728\u65b0\u9a6c\u8e44\u94c1\u5185\u83b7\u5f97\u66f4\u591a\u9a6c\u8e44\u94c1\u3002<\/span><\/section>\n

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\u5c06\u8fd9\u4e9b\u56fe\u50cf\u653e\u5728\u4e00\u8d77\u5c31\u53ef\u4ee5\u5f97\u5230\uff1a<\/span><\/section>\n

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\u8fd9\u4e2a\u9a6c\u8e44\u5f62\u56fe\u662f\u53ef\u9006\u7684\uff0c\u5e76\u4e14\u6709\u7740\u4e24\u7ec4\u5782\u76f4\u7684\u9a6c\u8e44\u5f62\u3002<\/span><\/section>\n

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\u6709\u4e00\u7ec4\u70b9\uff0c\u6211\u4eec\u79f0\u4e4bH<\/strong>\uff0c\u5b83\u7531\u6240\u6709\u6c34\u5e73\u9a6c\u8e44\u94c1\u548c\u5782\u76f4\u9a6c\u8e44\u94c1\u7684\u4ea4\u70b9\u7ec4\u6210\u3002\u8fd9\u4e9b\u70b9\u6240\u5728\u7684\u5730\u65b9\u4f1a\u53d1\u751f\u4e00\u4e9b\u6709\u8da3\u7684\u4e8b\u60c5\u3002<\/span><\/section>\n

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In his chaotic horseshoe, there must be a fixed point (let\u2019s call it p) with the itinerary \u2026 <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/svg><\/g><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span><\/span><\/strong> \u2026, because there exist points of every possible itinerary. That means there must also be a point \u2014 let\u2019s call this one q \u2014 with the itinerary \u2026 <\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/path><\/svg><\/g><\/g><\/path><\/g><\/path><\/g><\/path><\/g><\/g><\/g><\/svg><\/mjx-container><\/span><\/span><\/strong> \u2026. The forward orbit of this point approaches p (we say \u201cinto the future\u201d), as does its backward orbit (\u201cinto the past\u201d).<\/span><\/strong><\/span><\/strong><\/em><\/span><\/section>\n

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\u6df7\u6c8c\u7406\u8bba\u5c06\u7ee7\u7eed\u5f15\u8d77\u516c\u4f17\u7684\u6ce8\u610f\u3002\u5b83\u57281986\u5e74\u300a\u79d1\u5b66\u7f8e\u56fd\u4eba\u300b\u7684\u4e00\u7bc7\u6587\u7ae0\u4e2d\u88ab\u63cf\u8ff0\u4e3a\u201c\u79d1\u5b66\u5efa\u6a21\u7684\u65b0\u6a21\u8303\u201d \uff0c\u8a79\u59c6\u65af\u00b7\u683c\u83b1\u514b(James Gleick)1987\u5e74\u7684\u7545\u9500\u4e66\u300a\u6df7\u6c8c\u300b\u7684\u526f\u6807\u9898\u662f\u542f\u53d1\u6027\u7684\uff1a\u201c\u521b\u9020\u4e00\u95e8\u65b0\u79d1\u5b66\u201d\u3002\u6df7\u6c8c\u5f00\u59cb\u51fa\u73b0\u5728\u6d41\u884c\u6587\u5316\u4e2d\uff0c\u4f8b\u59821990\u5e74\u7684\u5c0f\u8bf4\u4f8f\u7f57\u7eaa\u516c\u56ed\u548c\u6c64\u59c6\u00b7\u65af\u6258\u5e15\u5fb7<\/span>(Tom Stoppard)<\/span>1993\u5e74\u7684\u620f\u5267\u300a\u963f\u5361\u8fea\u4e9a\u300b<\/span>\uff08Arcadia\uff09<\/span>\u3002<\/span><\/section>\n

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\u539f\u6587\u94fe\u63a5\uff1a<\/span><\/section>\n
https:\/\/www.quantamagazine.org\/how-mathematicians-make-sense-of-chaos-20220302\/<\/span><\/section>\n

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