{"id":32515,"date":"2022-02-18T23:17:58","date_gmt":"2022-02-18T15:17:58","guid":{"rendered":"https:\/\/swarma.org\/?p=32515"},"modified":"2022-02-18T23:17:58","modified_gmt":"2022-02-18T15:17:58","slug":"%e4%bb%80%e4%b9%88%e6%98%af%e5%90%b8%e5%bc%95%e5%ad%90%ef%bc%9a%e4%bb%8e%e6%95%b0%e5%ad%a6%e5%ae%9a%e4%b9%89%e5%88%b0%e5%8a%a8%e5%8a%9b%e5%ad%a6%e6%96%b9%e7%a8%8b%e8%bf%ad%e4%bb%a3-%e9%9b%86","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=32515","title":{"rendered":"\u4ec0\u4e48\u662f\u5438\u5f15\u5b50\uff1a\u4ece\u6570\u5b66\u5b9a\u4e49\u5230\u52a8\u529b\u5b66\u65b9\u7a0b\u8fed\u4ee3 | \u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
\n
<\/section>\n

<\/section>\n
\u201c\u96c6\u667a\u767e\u79d1\u7cbe\u9009\u201d\u662f\u4e00\u4e2a\u957f\u671f\u4e13\u680f\uff0c\u6301\u7eed\u4e3a\u5927\u5bb6\u63a8\u9001\u590d\u6742\u6027\u79d1\u5b66\u76f8\u5173\u7684\u57fa\u672c\u6982\u5ff5\u548c\u8d44\u6e90\u4fe1\u606f\u3002\u4f5c\u4e3a\u96c6\u667a\u4ff1\u4e50\u90e8\u7684\u5f00\u6e90\u79d1\u5b66\u9879\u76ee\uff0c\u96c6\u667a\u767e\u79d1\u5e0c\u671b\u6253\u9020\u590d\u6742\u6027\u79d1\u5b66\u9886\u57df\u6700\u5168\u9762\u7684\u767e\u79d1\u5168\u4e66\uff0c\u6b22\u8fce\u5bf9\u590d\u6742\u6027\u79d1\u5b66\u611f\u5174\u8da3\u3001\u70ed\u7231\u77e5\u8bc6\u6574\u7406\u548c\u5206\u4eab\u7684\u670b\u53cb\u52a0\u5165\uff01
<\/span><\/section>\n

<\/section>\n
\u672c\u6587\u662f\u5bf9\u96c6\u667a\u767e\u79d1\u4e2d\u201c\u5438\u5f15\u5b50<\/span>\u201d\u8bcd\u6761\u7684\u6458\u5f55\uff0c\u53c2\u8003\u8d44\u6599\u53ca\u76f8\u5173\u8bcd\u6761\u8bf7\u53c2\u9605\u767e\u79d1\u8bcd\u6761\u539f\u6587\u3002<\/span><\/section>\n

<\/section>\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n

\u672c\u8bcd\u6761\u7531\u96c6\u667a\u4ff1\u4e50\u90e8\u4f17\u5305\u751f\u4ea7\uff0c\u96be\u514d\u5b58\u5728\u7eb0\u6f0f\u548c\u95ee\u9898\uff0c\u6b22\u8fce\u5927\u5bb6\u7559\u8a00\u53cd\u9988\u6216\u8005\u524d\u5f80\u5bf9\u5e94\u7684\u767e\u79d1\u8bcd\u6761\u9875\u9762\u8fdb\u884c\u4fee\u6539\uff0c\u4e00\u7ecf\u4fee\u6539\uff0c\u53ef\u4ee5\u83b7\u5f97\u5bf9\u5e94\u7684\u79ef\u5206\u5956\u52b1\u5662\uff01<\/em><\/span><\/p>\n


<\/p>\n

\n
\n
\n
\n
\n
\n
\n
\n
\n
\n

\u76ee\u5f55<\/strong><\/p>\n


<\/p>\n

\n
\u4e00\u3001\u5438\u5f15\u5b50\u7684\u63d0\u51fa\u52a8\u673a<\/span><\/strong><\/section>\n
\u4e8c\u3001\u6570\u5b66\u5b9a\u4e49<\/span><\/strong><\/section>\n
\u4e09\u3001\u5438\u5f15\u5b50\u7684\u7c7b\u578b<\/span><\/strong><\/section>\n
\u56db\u3001\u5438\u5f15\u5b50\u8868\u5f81\u7cfb\u7edf\u7684\u6f14\u5316<\/span><\/strong><\/section>\n
\u4e94\u3001\u5438\u5f15\u57df
<\/span><\/strong><\/section>\n
\u516d\u3001\u504f\u5fae\u5206\u65b9\u7a0b<\/span><\/strong><\/section>\n
<\/span><\/strong><\/section>\n
\u4e03\u3001\u7f16\u8005\u63a8\u8350<\/span><\/strong><\/section>\n
\u516b\u3001\u767e\u79d1\u9879\u76ee\u5fd7\u613f\u8005\u62db\u52df<\/span><\/strong><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
<\/section>\n
\u5728\u52a8\u529b\u7cfb\u7edf\u7684\u6570\u5b66\u9886\u57df\u4e2d\uff0c\u5438\u5f15\u5b50 attractor\u662f\u7cfb\u7edf\u5728\u4f17\u591a\u521d\u59cb\u6761\u4ef6\u4e0b\u6240\u8d8b\u5411\u7684\u4e00\u7ec4\u6570\u503c\u3002\u5373\u4f7f\u7a0d\u5fae\u53d7\u5230\u5e72\u6270\uff0c\u4e0e\u5438\u5f15\u5b50\u7684\u503c\u63a5\u8fd1\u7684\u7cfb\u7edf\u503c\u4ecd\u7136\u80fd\u591f\u4fdd\u8bc1\u8fd1\u4f3c\u6027\u3002<\/span>
<\/section>\n

\u5728\u6709\u9650\u7ef4\u7cfb\u7edf finite-dimensional systems\u4e2d\uff0c\u6f14\u5316\u53d8\u91cf\u53ef\u7528\u4ee3\u6570\u8868\u793a\u4e3a n \u7ef4\u5411\u91cf\u3002\u5438\u5f15\u5b50\u662fn\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u533a\u57df\u3002\u5728\u7269\u7406\u7cfb\u7edf\u4e2d\uff0cn\u7ef4\u53ef\u4ee5\u662f\u4e00\u4e2a\u6216\u591a\u4e2a\u7269\u7406\u5b9e\u4f53\u7684\u4e24\u4e2a\u6216\u4e09\u4e2a\u4f4d\u7f6e\u5750\u6807\uff1b\u5728\u7ecf\u6d4e\u7cfb\u7edf\u4e2d\uff0c\u5b83\u4eec\u53ef\u4ee5\u662f\u5355\u72ec\u7684\u53d8\u91cf\uff0c\u5982\u901a\u8d27\u81a8\u80c0\u7387 inflation rate\u548c\u5931\u4e1a\u7387 unemployment rate\u3002<\/span><\/section>\n

\u5982\u679c\u6f14\u5316\u53d8\u91cf\u662f\u4e8c\u7ef4\u6216\u4e09\u7ef4\u7684\uff0c\u5219\u52a8\u6001\u8fc7\u7a0b\u7684\u5438\u5f15\u5b50\u53ef\u4ee5\u7528\u51e0\u4f55\u65b9\u5f0f\u8868\u793a\u4e3a\u4e8c\u7ef4\u6216\u4e09\u7ef4\uff08\u4f8b\u5982\u53f3\u56fe\u6240\u793a\u7684\u4e09\u7ef4\u60c5\u51b5\uff09\u3002\u4e00\u4e2a\u5438\u5f15\u5b50\u53ef\u4ee5\u662f\u4e00\u4e2a\u70b9\uff0c\u4e00\u4e2a\u6709\u9650\u7684\u70b9\u96c6\uff0c\u4e00\u6761\u66f2\u7ebf\uff0c\u4e00\u4e2a\u6d41\u5f62\uff0c\u751a\u81f3\u662f\u4e00\u4e2a\u5177\u6709\u5206\u5f62\u7ed3\u6784\u7684\u590d\u6742\u96c6\u5408\u2014\u2014\u6211\u4eec\u79f0\u4e4b\u4e3a\u5947\u5f02\u5438\u5f15\u5b50 strange attractor\uff08\u89c1\u4e0b\u9762\u7684\u5947\u5f02\u5438\u5f15\u5b50\uff09\u3002\u5982\u679c\u53d8\u91cf\u662f\u6807\u91cf\uff0c\u90a3\u4e48\u5438\u5f15\u5b50\u5c31\u662f\u5b9e\u8f74\u7684\u5b50\u96c6\u3002\u63cf\u8ff0\u6df7\u6c8c\u52a8\u529b\u5b66\u7cfb\u7edf chaotic dynamical systems\u7684\u5438\u5f15\u5b50\u662f\u6df7\u6c8c\u7406\u8bba\u7684\u91cd\u8981\u6210\u679c\u4e4b\u4e00\u3002<\/span><\/section>\n

\u52a8\u529b\u7cfb\u7edf\u5728\u5438\u5f15\u5b50\u4e2d\u7684\u8f68\u8ff9\uff0c\u82e5\u65f6\u95f4\u5411\u524d\uff0c\u4e0d\u5fc5\u6ee1\u8db3\u4efb\u4f55\u7279\u6b8a\u7684\u7ea6\u675f\u6761\u4ef6\u3002\u8f68\u8ff9\u53ef\u80fd\u662f\u5468\u671f\u6027\u7684\uff0c\u4e5f\u53ef\u80fd\u662f\u6df7\u6c8c\u7684\u3002\u5982\u679c\u4e00\u7ec4\u70b9\u662f\u5468\u671f\u6027\u7684\u6216\u6df7\u6c8c\u7684\uff0c\u4f46\u5176\u9644\u8fd1\u7684\u6d41\u8fdc\u79bb\u8be5\u96c6\u5408\uff0c\u5219\u8be5\u96c6\u5408\u4e0d\u662f\u5438\u5f15\u5b50\uff0c\u800c\u662f\u6392\u65a5\u70b9\uff08\u6216\u65a5\u70b9\uff09 repeller(or repellor<\/span><\/section>\n

<\/span><\/section>\n

<\/h2>\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n


<\/p>\n

\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

\u5438\u5f15\u5b50\u7684\u63d0\u51fa\u52a8\u673a<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

<\/span>
<\/h2>\n
\u6211\u4eec\u901a\u5e38\u7528\u4e00\u4e2a\u6216\u591a\u4e2a\u5fae\u5206\u65b9\u7a0b\u6216\u5dee\u5206\u65b9\u7a0b\u63cf\u8ff0\u52a8\u529b\u7cfb\u7edf\u3002\u4e00\u4e2a\u7ed9\u5b9a\u52a8\u529b\u7cfb\u7edf\u7684\u65b9\u7a0b\u53ef\u4ee5\u8868\u660e\u5b83\u5728\u4efb\u4f55\u7ed9\u5b9a\u77ed\u65f6\u95f4\u5185\u7684\u884c\u4e3a\u3002\u4e3a\u4e86\u786e\u5b9a\u7cfb\u7edf\u5728\u8f83\u957f\u65f6\u95f4\u5185\u7684\u884c\u4e3a\uff0c\u6211\u4eec\u5f80\u5f80\u9700\u8981\u901a\u8fc7\u5206\u6790\u624b\u6bb5\u6216\u8fed\u4ee3 Iteration\uff08\u901a\u5e38\u501f\u52a9\u4e8e\u8ba1\u7b97\u673a\uff09\u6765\u5bf9\u65b9\u7a0b\u8fdb\u884c\u79ef\u5206\u3002<\/span><\/section>\n

\u7269\u7406\u4e16\u754c\u4e2d\u7684\u52a8\u529b\u7cfb\u7edf\u5f80\u5f80\u4ea7\u751f\u4e8e\u8017\u6563\u7cfb\u7edf dissipative system\uff1a\u5982\u679c\u6ca1\u6709\u67d0\u79cd\u9a71\u52a8\u529b\uff0c\u8fd0\u52a8\u5c31\u4f1a\u505c\u6b62\u3002\uff08\u8017\u6563\u53ef\u80fd\u6765\u81ea\u5185\u90e8\u6469\u64e6\uff0c\u70ed\u529b\u5b66\u635f\u5931\uff0c\u6750\u6599\u635f\u5931\u7b49\u8bb8\u591a\u539f\u56e0\u3002\uff09\u5f53\u8017\u6563\u548c\u9a71\u52a8\u529b\u8d8b\u4e8e\u5e73\u8861\u65f6\uff0c\u521d\u59cb\u77ac\u6001 Initial transients\u4f1a\u88ab\u6d88\u9664\uff0c\u4f7f\u7cfb\u7edf\u8fdb\u5165\u5176\u5178\u578b\u72b6\u6001\u3002\u4e0e\u5178\u578b\u72b6\u6001\u76f8\u5bf9\u5e94\u7684\u52a8\u529b\u7cfb\u7edf\u76f8\u7a7a\u95f4\u7684\u5b50\u96c6\u662f\u5438\u5f15\u5b50\u2014\u2014\u4e5f\u79f0\u4e3a\u5438\u5f15\u90e8\u5206\u3002<\/span><\/section>\n

\u4e0d\u53d8\u96c6 Invariant sets\u548c\u6781\u9650\u96c6 limit sets\u7684\u6982\u5ff5\u4e0e\u5438\u5f15\u5b50\u7c7b\u4f3c\u3002\u4e0d\u53d8\u96c6\u662f\u5728\u52a8\u529b\u5b66\u4f5c\u7528\u4e0b\u5411\u81ea\u8eab\u6f14\u5316\u7684\u96c6\u5408\u3002\u4e0d\u53d8\u96c6\u53ef\u80fd\u5305\u542b\u4e8e\u5438\u5f15\u5b50\u3002\u6781\u9650\u96c6\u662f\u4e00\u7ec4\u70b9\uff0c\u8fd9\u4e9b\u70b9\u5b58\u5728\u4e00\u5b9a\u7684\u521d\u59cb\u72b6\u6001\uff0c\u4f46\u662f\u968f\u7740\u6700\u7ec8\u65f6\u95f4\u8d8b\u8fd1\u65e0\u7a77\u8fdc\u65f6\u5c06\u4efb\u610f\u63a5\u8fd1\u6781\u9650\u96c6\uff08\u5373\u6536\u655b\u5230\u96c6\u5408\u7684\u6bcf\u4e2a\u70b9\uff09\u3002\u5438\u5f15\u5b50\u662f\u6781\u9650\u96c6\uff0c\u4f46\u4e0d\u662f\u6240\u6709\u7684\u6781\u9650\u96c6\u90fd\u662f\u5438\u5f15\u5b50: \u7cfb\u7edf\u7684\u67d0\u4e9b\u70b9\u53ef\u80fd\u4f1a\u6536\u655b\u5230\u6781\u9650\u96c6\uff0c\u4f46\u662f\u7a0d\u5fae\u504f\u79bb\u6781\u9650\u96c6\u7684\u70b9\u53ef\u80fd\u4f1a\u88ab\u6572\u6389\uff0c\u6c38\u8fdc\u4e0d\u4f1a\u56de\u5230\u6781\u9650\u96c6\u9644\u8fd1\u3002<\/span><\/section>\n

\u4f8b\u5982\uff0c\u963b\u5c3c\u6446 damped pendulum\u6709\u4e24\u4e2a\u4e0d\u53d8\u70b9: \u6700\u5c0f\u9ad8\u5ea6\u70b9x0\u548c\u6700\u5927\u9ad8\u5ea6\u70b9x1\u3002\u70b9x0\u4e5f\u662f\u4e00\u4e2a\u6781\u9650\u96c6,\u56e0\u4e3a\u8f68\u8ff9\u5411\u5b83\u6536\u655b\uff1b\u70b9 x1\u4e0d\u662f\u4e00\u4e2a\u6781\u9650\u96c6\u3002\u7531\u4e8e\u7a7a\u6c14\u963b\u529b\u7684\u8017\u6563\uff0c\u70b9x0\u4e5f\u662f\u5438\u5f15\u5b50\u3002\u5982\u679c\u6ca1\u6709\u8017\u6563\uff0cx0\u5c31\u4e0d\u4f1a\u51fa\u73b0\u5438\u5f15\u5b50\u3002\u4e9a\u91cc\u58eb\u591a\u5fb7 Aristotle\u8ba4\u4e3a\u7269\u4f53\u53ea\u6709\u5728\u88ab\u63a8\u52a8\u65f6\u624d\u4f1a\u79fb\u52a8\u2014\u2014\u8fd9\u662f\u8017\u6563\u5438\u5f15\u5b50 dissipative attractor\u7684\u65e9\u671f\u8868\u8ff0\u3002<\/span><\/section>\n

\u6709\u4e9b\u5438\u5f15\u5b50\u662f\u6df7\u6c8c\u7684\uff08\u53c2\u89c1\u5947\u5f02\u5438\u5f15\u5b50\uff09\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5438\u5f15\u5b50\u7684\u4efb\u610f\u4e24\u4e2a\u4e0d\u540c\u70b9\u7684\u6f14\u5316\u90fd\u4f1a\u89e6\u53d1\u6307\u6570\u53d1\u6563\u8f68\u8ff9\u3002\u6b64\u65f6\u5373\u4f7f\u7cfb\u7edf\u4e2d\u6709\u4e00\u70b9\u566a\u58f0\uff0c\u9884\u6d4b\u4e5f\u4f1a\u56e0\u6b64\u800c\u53d8\u5f97\u590d\u6742\u3002<\/span><\/section>\n

<\/section>\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n


<\/p>\n

\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

\u6570\u5b66\u5b9a\u4e49<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

<\/span>
<\/h2>\n
\u8bbet\u8868\u793a\u65f6\u95f4\uff0c\u8bbef(t,\u2022)\u4e3a\u6307\u5b9a\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u51fd\u6570\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679ca\u662fn\u7ef4\u76f8\u7a7a\u95f4\u4e2d\u4e00\u4e2a\u8868\u793a\u7cfb\u7edf\u521d\u59cb\u72b6\u6001\u7684\u70b9\uff0c\u90a3\u4e48f(0,a)=a\u3002\u5bf9\u4e8et\u7684\u6b63\u503c\uff0cf(t,a)\u662f\u8be5\u72b6\u6001\u5728t\u4e2a\u65f6\u95f4\u5355\u4f4d\u4e4b\u540e\u6f14\u5316\u7684\u7ed3\u679c\u3002\u4f8b\u5982\uff0c\u5982\u679c\u7cfb\u7edf\u63cf\u8ff0\u4e86\u81ea\u7531\u7c92\u5b50\u5728\u4e00\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6f14\u5316\uff0c\u90a3\u4e48\u76f8\u7a7a\u95f4\u662f\u5750\u6807\u4e3a(x,v)\u7684\u5e73\u9762 R2 \uff0c\u5176\u4e2dx\u662f\u7c92\u5b50\u7684\u4f4d\u7f6e\uff0cv\u662f\u7c92\u5b50\u7684\u901f\u5ea6\uff0ca = (x,v),\u7531\u4ee5\u4e0b\u7ed9\u51fa<\/span><\/section>\n

<\/section>\n
<\/section>\n
\u5438\u5f15\u5468\u671f-3\u65cb\u56de\u53ca\u5176\u5bf9f(z) = z2 + c\u53c2\u6570\u5316\u7684\u76f4\u63a5\u5438\u5f15\u57df\u3002\u4e09\u4e2a\u6700\u6697\u7684\u70b9\u662f3\u5faa\u73af\u7684\u70b9\uff0c\u5b83\u4eec\u6309\u987a\u5e8f\u76f8\u4e92\u8fde\u63a5\uff0c\u4ece\u5438\u5f15\u57df\u4e2d\u7684\u4efb\u4f55\u70b9\u8fed\u4ee3\u90fd\u4f1a\uff08\u901a\u5e38\u662f\u6e10\u8fdb\u7684\uff09\u6536\u655b\u5230\u8fd9\u4e09\u4e2a\u70b9\u7684\u5e8f\u5217\u3002<\/span><\/section>\n

<\/section>\n
<\/section>\n

<\/section>\n
\u5438\u5f15\u5b50\u662f\u76f8\u7a7a\u95f4 phase space\u7684\u5b50\u96c6 subset A\u2014\u2014\u8fd9\u9700\u8981\u5177\u6709\u4ee5\u4e0b\u4e09\u4e2a\u6761\u4ef6\uff1a<\/span><\/section>\n