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$$m_1 \\dfrac{d^2 q_{1i}}{dt^2} = k m_1 m_2 \\dfrac{q_{2i}-q_{1i}}{r^{3}{12}}+km_1 m_3 \\dfrac{q{3i}-q_{1i}}{r^{3}_{13}}$$<\/p>\n\n\n\n
$$m_2 \\dfrac{d^2 q_{2i}}{dt^2}= k m_2 m_1 \\dfrac{q_{1i}-q_{2i}}{r^{3}{12}}+km_2 m_3 \\dfrac{q{3i}-q_{2i}}{r^{3}_{23}}$$<\/p>\n\n\n\n
$$m_3 \\dfrac{d^2 q_{3i}}{dt^2}= k m_3 m_1 \\dfrac{q_{1i}-q_{3i}}{r^{3}{13}}+km_3 m_2 \\dfrac{q{32i}-q_{3i}}{r^{3}_{23}}$$<\/p>\n\n\n\n
$$ (i=1,2,3)$$<\/p>\n\n\n\n
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<\/p>\n\n\n\n
1885\u5e74\uff0c\u745e\u5178\u73b0\u4ee3\u6570\u5b66\u4e4b\u7236\u7c73\u5854-\u5217\u592b\u52d2\uff08G. Mittag-Leffler\uff09\uff0c\u662f\u4e00\u4e2a\u6781\u5176\u6709\u7406\u60f3\u62b1\u8d1f\u7684\u6570\u5b66\u5bb6\uff0c\u5f53\u65f6\u7684\u6b27\u6d32\u5927\u9646\u5b66\u672f\u6c1b\u56f4\u6d53\u539a\uff0c\u745e\u5178\u5374\u6ca1\u4ec0\u4e48\u540d\u6c14\uff0c\u4e8e\u662f\u521b\u529e\u4e86\u6742\u5fd7\u745e\u5178\u7684Acta Mathematica\u79d1\u5b66\u6742\u5fd7\uff0c\u81f3\u4eca\u8fd9\u4e2a\u6742\u5fd7\u4f9d\u7136\u662f\u4e16\u754c\u77e5\u540d\u5b66\u672f\u6742\u5fd7\u3002<\/p>\n\n\n\n
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\u53c2\u52a0\u6bd4\u8d5b\u7684\u5404\u56fd\u6570\u5b66\u5bb6\u5fc5\u987b\u57281888\u5e74\u76846\u67081\u65e5\u524d\u628a\u4ed6\u4eec\u7684\u53c2\u8d5b\u8bba\u6587\u5bc4\u7ed9\u6742\u5fd7\u7684\u521b\u529e\u4eba\u548c\u4e3b\u7f16\uff0c\u6240\u6709\u8bba\u6587\u5c06\u88ab\u533f\u540d\u5730\u88ab\u4e00\u4e2a\u56fd\u9645\u59d4\u5458\u4f1a\u8bc4\u5224\u4ee5\u51b3\u51fa\u4f18\u80dc\u8005\uff0c\u7136\u540e\u4f18\u80dc\u8005\u7684\u8bba\u6587\u5c06\u53d1\u8868\u5728Acta Mathematica\u4e0a\uff0c\u4e3a\u4e86\u4f53\u73b0\u6bd4\u8d5b\u7684\u6743\u5a01\u6027\uff0c\u7c73\u5854-\u5217\u592b\u52d2\u627e\u6765\u4e86\u4e09\u4e2a\u5f53\u65f6\u8d6b\u8d6b\u6709\u540d\u7684\u6570\u5b66\u5bb6\uff1a\u5fb7\u56fd\u7684\u7ef4\u5c14\u65af\u7279\u62c9\u65af(Karl Weierstrass)\uff0c\u6cd5\u56fd\u7684\u8d6b\u5bc6\u7279(Charles Hermite)\u5f53\u8bc4\u59d4\uff0c\u52a0\u4e0a\u7c73\u5854\u683c\u83b1\u592b\u52d2\u672c\u4eba\u7ec4\u6210\uff0c\u6b63\u597d\u7ec4\u6210\u4e09\u5de8\u5934\u3002<\/p>\n\n\n\n
\u8fd9\u6837\u7684\u6bd4\u8d5b\u660e\u663e\u5177\u6709\u201c\u7092\u4f5c\u201d\u548c\u7ed9\u65b0\u6742\u5fd7\u505a\u5e7f\u544a\u7684\u5acc\u7591\uff0c(\u4e8b\u5b9e\u4e0a\u5f53\u65f6\u5c31\u6709\u4e00\u4e9b\u6570\u5b66\u5bb6\u8fd9\u6837\u6279\u8bc4\u8fd9\u79cd\u6bd4\u8d5b\uff0c\u8c61\u5fb7\u56fd\u7684\u514b\u9686.\u5948\u514b(Leopold Kronecker))\u3002\u8fd9\u6b21\u6bd4\u8d5b\u5728\u5f53\u65f6\u8f70\u52a8\u4e00\u65f6\uff0c\u867d\u7136\u5956\u91d1\u4e0d\u9ad8\uff0c\u8fd9\u79cd\u5d07\u9ad8\u7684\u8363\u8a89\u662f\u5f53\u65f6\u7f55\u89c1\u7684\uff0c\u8981\u77e5\u9053\u745e\u5178\u66f4\u6709\u540d\u7684\u201c\u70b8\u836f\u5956\u201d\u8bfa\u8d1d\u5c14(Alfred Bernhard Nobel)\u5956\u662f\u5728\u51e0\u5e74\u540e\u76841896\u5e74\u624d\u5f00\u59cb\u8bc4\u9009\u7684\u3002<\/p>\n\n\n\n
\u4f46\u662f\u7531\u4e8e\u95ee\u9898\u7684\u56f0\u96be\u7a0b\u5ea6\uff0c\u5927\u591a\u6570\u4e00\u5f00\u59cb\u8dc3\u8dc3\u6b32\u8bd5\u7684\u6570\u5b66\u5bb6\u540e\u6765\u90fd\u77e5\u96be\u800c\u9000\uff0c\u6700\u540e\u53ea\u6709\u56db\u4e94\u4e2a\u6570\u5b66\u5bb6\u771f\u6b63\u4ea4\u4e86\u4ed6\u4eec\u7684\u7b54\u5377\u3002\u800c\u4f18\u80dc\u8005\u4e5f\u5e76\u4e0d\u96be\u9009\u51fa\uff0c\u867d\u7136\u8fd8\u662f\u6ca1\u6709\u4eba\u80fd\u5b8c\u6574\u5730\u89e3\u51b3\u4efb\u4f55\u4e00\u4e2a\u95ee\u9898\uff0c\u4f46\u662f\u6240\u6709\u8bc4\u59d4\u4e00\u81f4\u8ba4\u4e3a\u5176\u4e2d\u4e00\u4efd\u7b54\u5377\u5bf9\u4e8eN\u4f53\u95ee\u9898\u7684\u89e3\u51b3\u505a\u51fa\u4e86\u5173\u952e\u7684\u8d21\u732e\uff0c\u5e94\u8be5\u628a\u5956\u9881\u7ed9\u8fd9\u4f4d\u6570\u5b66\u5bb6,\u8fd9\u4f4d\u83b7\u80dc\u8005\u5c31\u662f\u6cd5\u56fd\u6570\u5b66\u5bb6\uff0c\u7269\u7406\u5b66\u5bb6\u5e9e\u52a0\u83b1(Jules Henri Poincare)\u3002<\/p>\n\n\n\n
\u5e9e\u52a0\u83b1\u5728\u73b0\u4ee3\u6570\u5b66\u5386\u53f2\u4e0a\u5360\u6709\u4e3e\u8db3\u8f7b\u91cd\u7684\u5730\u4f4d\uff0c\u4ed6\u66fe\u88ab\u79f0\u4e3a\u73b0\u4ee3\u6570\u5b66\u7684\u4e24\u4e2a\u5960\u57fa\u4eba\u4e4b\u4e00(\u53e6\u4e00\u4e2a\u662f\u9ece\u66fc(Bernhard Riemann))\uff0c\u4e5f\u6709\u4eba\u79f0\u4ed6\u4e3a\u5386\u53f2\u4e0a\u7cbe\u901a\u5f53\u65f6\u6240\u6709\u6570\u5b66\u7684\u6700\u540e\u4e24\u4e2a\u4eba\u4e4b\u4e00(\u53e6\u4e00\u4e2a\u5c31\u662f\u5e0c\u5c14\u4f2f\u7279))\u3002<\/p>\n\n\n\n
\u800c1885\u5e74\u7684\u5e9e\u52a0\u83b1\u53ea\u670931\u5c81\uff0c\u867d\u7136\u5df2\u521d\u9732\u950b\u8292\uff0c\u4f46\u8fd8\u662f\u4e00\u4f4d\u5e0c\u671b\u80fd\u591f\u4e00\u4e3e\u6210\u540d\u7684\u5e74\u8f7b\u6570\u5b66\u5bb6\uff0c\u6240\u4ee5\u8fd9\u6b21\u6bd4\u8d5b\u662f\u4e2a\u5927\u597d\u7684\u673a\u4f1a\uff0c\u8fd9\u4e5f\u8feb\u4f7f\u4ed6\u5148\u653e\u4e0b\u624b\u4e0a\u5176\u4ed6\u7684\u5de5\u4f5c\uff0c\u96c6\u4e2d\u7cbe\u529b\u6295\u5165\u5230\u5929\u4f53\u529b\u5b66\u548cN\u4f53\u95ee\u9898\u7684\u7814\u7a76\u4e2d\u3002<\/p>\n\n\n\n
\u5e9e\u52a0\u83b1\u83b7\u5956\u7684\u8bba\u6587\u201c\u5173\u4e8e\u4e09\u4f53\u95ee\u9898\u7684\u52a8\u6001\u65b9\u7a0b\u201d(Sur le probleme des trois corps et les equations de la dynamique) \u6700\u540e\u57281890\u5e74\u5728Acta Mathematica\u4e0a\u53d1\u8868\uff0c\u8bba\u6587\u957f\u8fbe270\u9875\uff0c\u5360\u4e86\u6574\u6574\u534a\u5377\u6742\u5fd7\u3002(\u5173\u4e8e\u8bba\u6587\u53d1\u8868\u7684\u4e00\u6bb5\u6545\u4e8b\u4e0b\u9762\u8fd8\u8981\u63d0\u5230)\u3002 \u8fd9\u7bc7\u91cd\u8981\u8bba\u6587\u4f7f\u539f\u6765\u5c31\u5df2\u6709\u4e0d\u5c0f\u540d\u6c14\u7684\u5e74\u8f7b\u5e9e\u52a0\u83b1\u66f4\u52a0\u8a89\u6ee1\u6574\u4e2a\u6b27\u6d32\u6570\u5b66\u754c\uff0c\u4e5f\u4f7f\u4ed6\u5f97\u5230\u4e86\u65b0\u7684\u70ed\u60c5\u548c\u52a8\u529b\u7ee7\u7eed\u8fdb\u884c\u4ed6\u5728\u8fd9\u7bc7\u8bba\u6587\u4e2d\u5f00\u59cb\u7684\u5de5\u4f5c\u3002<\/p>\n\n\n\n
\u4ece1892\u5e74\u52301899\u5e74\uff0c\u5e9e\u52a0\u83b1\u9646\u7eed\u51fa\u7248\u4e86\u4ed6\u7684\u4e09\u5927\u5377\u5b8f\u4f1f\u5de8\u8457\u201c\u5929\u4f53\u529b\u5b66\u7684\u65b0\u65b9\u6cd5\u201d(Les MethodsNouvelles de la Mecanique Celeste)\u3002\u4ed6\u7684\u83b7\u5956\u8bba\u6587\u548c\u8fd9\u4e09\u5377\u4e66\u53ef\u4ee5\u8bf4\u5960\u5b9a\u4e86\u73b0\u4ee3\u5929\u4f53\u529b\u5b66\uff0c\u52a8\u529b\u7cfb\u7edf\uff0c\u5fae\u5206\u65b9\u7a0b\u5b9a\u6027\u7406\u8bba\uff0c\u751a\u81f3\u6df7\u6c8c\u7406\u8bba\u7684\u57fa\u7840\uff0c\u5c3d\u7ba1\u5927\u591a\u6570\u4ed6\u7684\u601d\u60f3\u76f4\u5230\u51e0\u5341\u5e74\u540e\u624d\u88ab\u5e7f\u5927\u7684\u6570\u5b66\u5de5\u4f5c\u8005\u6240\u9886\u609f\u8fdb\u800c\u53d1\u5c55\u6210\u73b0\u4ee3\u7684\u6570\u5b66\u7406\u8bba\u3002<\/p>\n\n\n\n
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\u5341\u4e5d\u4e16\u7eaa\u672b\u671f\u77e5\u8bc6\u754c\u5bf9\u4e8e\u79d1\u5b66\u6280\u672f\u7684\u8fdb\u5c55\u662f\u975e\u5e38\u4e50\u89c2\u7684\uff0c\u4eba\u4eec\u5bf9\u4e8e\u7269\u7406\u4e16\u754c\u7684\u7406\u89e3\u662f\u7ed9\u5b9a\u73b0\u5b9e\u7684\u72b6\u6001\uff0c\u4eba\u7c7b\u662f\u6709\u80fd\u529b\u9884\u6d4b\u672a\u6765\u7684\u3002<\/p>\n\n\n\n
\u800c\u8fd9\u79cd\u6df7\u6c8c\u7406\u8bba\u7684\u4e0d\u786e\u5b9a\u6027\u6070\u6070\u548c\u5f53\u65f6\u8fd9\u79cd\u601d\u60f3\u4e0d\u76f8\u543b\uff0c<\/p>\n\n\n\n
\u6240\u4ee5\u5373\u4f7f\u662f\u5e9e\u52a0\u83b1\u8fd9\u6837\u4f1f\u5927\u7684\u79d1\u5b66\u5bb6\u4e5f\u5bf9\u4e8e\u63d0\u51fa\u8fd9\u79cd\u5927\u80c6\u7684\u601d\u60f3\u6301\u4fdd\u7559\u6001\u5ea6\u3002\u4f46\u6211\u4eec\u4e5f\u4e0d\u80fd\u8d23\u5907\u5e9e\u52a0\u83b1\u7684\u5c40\u9650\u6027\uff0c\u4e8b\u5b9e\u4e0a\u5bf9\u4e8e\u793e\u4f1a\u4e0a\u7684\u4e00\u822c\u516c\u4f17\uff0c\u6df7\u6c8c\u7406\u8bba\u662f\u5230\u4e8c\u5341\u4e16\u7eaa\u516b\u5341\u5e74\u4ee3\u540e\uff0c\u8ba1\u7b97\u673a\u88ab\u666e\u904d\u4f7f\u7528\u540e\u624d\u88ab\u771f\u6b63\u63a5\u53d7\u7684\u3002\u4eca\u5929\u4eba\u4eec\u8fd8\u662f\u627f\u8ba4\u5e9e\u52a0\u83b1\u662f\u7b2c\u4e00\u4e2a\u6709\u6df7\u6c8c\u7406\u8bba\u57fa\u672c\u60f3\u6cd5\u7684\u4eba\u3002\u800c\u5728\u79d1\u5b66\u53f2\u4e0a\uff0c\u5e9e\u52a0\u83b1\u4e5f\u88ab\u8ba4\u4e3a\u6700\u6709\u80fd\u529b\u548c\u673a\u4f1a\u521b\u9020\u72ed\u4e49\u76f8\u5bf9\u8bba\u7684\u7406\u8bba\uff0c\u4e5f\u8bb8\u8fd8\u662f\u4ed6\u6027\u683c\u4e0a\u548c\u54f2\u5b66\u601d\u60f3\u4e0a\u7684\u5f31\u70b9\uff0c\u6700\u540e\u771f\u6b63\u5927\u80c6\u5730\u505a\u5230\u8fd9\u4e00\u70b9\u7684\u662f\u4eca\u5929\u5bb6\u55bb\u6237\u6653\u7684\u7231\u56e0\u65af\u5766(Albert Einstein)\u3002<\/p>\n\n\n\n
\u4e0b\u9762\u6211\u4eec\u5c31\u6765\u7b80\u5355\u770b\u4e00\u770b\u5e9e\u52a0\u83b1\u5728\u8fd9\u4e00\u65f6\u671f\u7684\u5de5\u4f5c\u7a76\u7adf\u7ed9N\u4f53\u95ee\u9898\u7684\u89e3\u51b3\u5e26\u6765\u4e86\u4ec0\u4e48\u8fdb\u5c55\u3002<\/p>\n\n\n\n
\u7b2c\u4e00\uff0c\u5e9e\u52a0\u83b1\u8bc1\u660e\u4e86\u5bf9\u4e8eN\u4f53\u95ee\u9898\u5728N\u5927\u4e8e\u4e8c\u65f6\uff0c\u4e0d\u5b58\u5728\u7edf\u4e00\u7684\u7b2c\u4e00\u79ef\u5206(uniform first integral)\u3002\u4e5f\u5c31\u662f\u8bf4\u5373\u4f7f\u662f\u4e00\u822c\u7684\u4e09\u4f53\u95ee\u9898\uff0c\u4e5f\u4e0d\u53ef\u80fd\u901a\u8fc7\u53d1\u73b0\u5404\u79cd\u4e0d\u53d8\u91cf\u6700\u7ec8\u964d\u4f4e\u95ee\u9898\u7684\u81ea\u7531\u5ea6\uff0c\u628a\u95ee\u9898\u5316\u7b80\u6210\u66f4\u7b80\u5355\u53ef\u4ee5\u89e3\u51fa\u6765\u7684\u95ee\u9898\uff0c\u8fd9\u6253\u7834\u4e86\u5f53\u65f6\u5f88\u591a\u4eba\u5e0c\u671b\u627e\u5230\u4e09\u4f53\u95ee\u9898\u4e00\u822c\u7684\u663e\u5f0f\u89e3\u7684\u5e7b\u60f3\u3002<\/p>\n\n\n\n
\u5728\u4e00\u767e\u5e74\u540e\u5b66\u4e60\u5fae\u5206\u65b9\u7a0b\u8bfe\u7684\u4eba\u5927\u591a\u5728\u7b2c\u4e8c\u4e2a\u661f\u671f\u5c31\u4ece\u8001\u5e08\u90a3\u91cc\u77e5\u9053\u7edd\u5927\u591a\u6570\u5fae\u5206\u65b9\u7a0b\u662f\u6ca1\u6cd5\u627e\u5230\u5b9a\u91cf\u7684\u89e3\u7684\uff0c\u4f46\u4e00\u822c\u90fd\u80fd\u4ece\u5b9a\u6027\u7406\u8bba\u4e2d\u4e86\u89e3\u66f4\u591a\u89e3\u7684\u6027\u8d28\uff0c\u751a\u81f3\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u673a\u201c\u770b\u5230\u201d\u89e3\u7684\u5f62\u72b6\u884c\u4e3a\u3002\u800c\u5728\u5e9e\u52a0\u83b1\u7684\u5e74\u4ee3\uff0c\u5927\u591a\u6570\u6570\u5b66\u5bb6\u66f4\u70ed\u8877\u4e8e\u7528\u4ee3\u6570\u6216\u5e42\u51fd\u6570\u65b9\u6cd5\u627e\u5230\u89e3\uff0c\u4f7f\u7528\u5b9a\u6027\u65b9\u6cd5\u548c\u51e0\u4f55\u65b9\u6cd5\u6765\u8ba8\u8bba\u5fae\u5206\u65b9\u7a0b\u5c31\u662f\u8d77\u6e90\u4e8e\u5e9e\u52a0\u83b1\u5bf9\u4e8eN\u4f53\u95ee\u9898\u7684\u7814\u7a76\uff0c\u8fd9\u5f7b\u5e95\u6539\u53d8\u4eba\u4eec\u7814\u7a76\u5fae\u5206\u65b9\u7a0b\u7684\u57fa\u672c\u60f3\u6cd5\u3002<\/p>\n\n\n\n
\u7b2c\u4e8c\uff0c\u4e3a\u4e86\u7814\u7a76N\u4f53\u95ee\u9898\uff0c\u5e9e\u52a0\u83b1\u53d1\u660e\u4e86\u8bb8\u591a\u5168\u65b0\u7684\u6570\u5b66\u5de5\u5177\u3002\u4f8b\u5982\u4ed6\u5b8c\u6574\u5730\u63d0\u51fa\u4e86\u4e0d\u53d8\u79ef\u5206(invariant integrals) \u7684\u6982\u5ff5\uff0c\u5e76\u4e14\u4f7f\u7528\u5b83\u8bc1\u660e\u4e86\u8457\u540d\u7684\u56de\u5f52\u5b9a\u7406(recurrence theorem)\u3002\u53e6\u4e00\u4e2a\u4f8b\u5b50\u662f\u4ed6\u4e3a\u4e86\u7814\u7a76\u5468\u671f\u89e3\u7684\u884c\u4e3a\uff0c\u5f15\u8fdb\u4e86\u7b2c\u4e00\u56de\u5f52\u6620\u8c61(first return map)\u7684\u6982\u5ff5\uff0c\u5728\u540e\u6765\u7684\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\u4e2d\u88ab\u79f0\u4e3a\u5e9e\u52a0\u83b1\u6620\u8c61\u3002\u8fd8\u6709\u8c61\u7279\u5f81\u6307\u6570(characteristic expontents)\uff0c\u89e3\u5bf9\u53c2\u6570\u7684\u8fde\u7eed\u4f9d\u8d56\u6027(continuous dependence of solutions with respect to parameters)\u7b49\u7b49\u3002<\/p>\n\n\n\n
\u6240\u6709\u8fd9\u4e9b\u90fd\u6210\u4e3a\u4e86\u73b0\u4ee3\u5fae\u5206\u65b9\u7a0b\u548c\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\u4e2d\u7684\u57fa\u672c\u6982\u5ff5\u3002<\/p>\n\n\n\n
\u7b2c\u4e09\u70b9\uff0c\u4e5f\u8bb8\u662f\u6700\u91cd\u8981\u7684\u4e00\u70b9\uff0c\u662f\u5e9e\u52a0\u83b1\u901a\u8fc7\u7814\u7a76\u6240\u8c13\u7684\u6e10\u8fdb\u89e3(asymptotic solutions)\uff0c\u540c\u5bbf\u8f68\u9053(homoclinic orbits) \u548c\u5f02\u5bbf\u8f68\u9053(hetroclinic orbits)\uff0c\u53d1\u73b0\u5373\u4f7f\u5728\u7b80\u5355\u7684\u4e09\u4f53\u95ee\u9898\u4e2d\uff0c\u5728\u8fd9\u6837\u7684\u540c\u5bbf\u8f68\u9053\u6216\u8005\u5f02\u5bbf\u8f68\u9053\u9644\u8fd1\uff0c\u65b9\u7a0b\u7684\u89e3\u7684\u72b6\u51b5\u4f1a\u975e\u5e38\u590d\u6742\uff0c\u4ee5\u81f3\u4e8e\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u521d\u59cb\u6761\u4ef6\uff0c\u51e0\u4e4e\u662f\u6ca1\u6709\u529e\u6cd5\u9884\u6d4b\u5f53\u65f6\u95f4\u8d8b\u4e8e\u65e0\u7a77\u65f6\uff0c\u8fd9\u4e2a\u8f68\u9053\u7684\u6700\u7ec8\u547d\u8fd0\u3002<\/p>\n\n\n\n
\u4e8b\u5b9e\u4e0a\u534a\u4e2a\u4e16\u7eaa\u540e\uff0c\u540e\u6765\u7684\u6570\u5b66\u5bb6\u4eec\u53d1\u73b0\u8fd9\u79cd\u73b0\u8c61\u5728\u4e00\u822c\u52a8\u529b\u7cfb\u7edf\u4e2d\u662f\u5e38\u89c1\u7684\uff0c\u4ed6\u4eec\u628a\u5b83\u53eb\u505a\u7a33\u5b9a\u6d41\u5f62(stable manifold)\u548c\u4e0d\u7a33\u5b9a\u6d41\u5f62(unstable manifold)\u6b63\u6001\u76f8\u4ea4(intersects transversally)\u6240\u5f15\u8d77\u7684\u540c\u5bbf\u4ea4\u9519\u7f51(homoclinic tangle)\uff0c\u800c\u8fd9\u79cd\u5bf9\u4e8e\u8f68\u9053\u7684\u957f\u65f6\u95f4\u884c\u4e3a\u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u6570\u5b66\u5bb6\u548c\u7269\u7406\u5b66\u5bb6\u79f0\u4e4b\u4e3a\u6df7\u6c8c(chaos)\u3002<\/p>\n\n\n\n
\u5e9e\u52a0\u83b1\u7684\u53d1\u73b0\u53ef\u4ee5\u8bf4\u662f\u6df7\u6c8c\u7406\u8bba\u7684\u6700\u65e9\u8d77\u6e90\u4e86\u3002<\/p>\n\n\n\n
(\u5e9e\u52d2\u7ef4—–\u51af\u6cfd\u57f9\u5c14—–\u8428\u745e—–\u590f\u5fd7\u5b8f—–\u54e5\u7ef4\u5c14)<\/p>\n\n\n\n
\u592a\u9633\u7cfb\u6240\u6709\u884c\u661f\u90fd\u4ee5\u592a\u9633\u4e3a\u53c2\u7167\u7269\u505a\u7740\u5468\u671f\u8fd0\u52a8\uff0c\u7136\u800c\u5728\u5b87\u5b99\u4e2d\u5e76\u975e\u6240\u6709\u661f\u7403\u90fd\u80fd\u4fdd\u6301\u8fd9\u79cd\u5468\u671f\u8fd0\u52a8\uff0cN\u4f53\u95ee\u9898\u672c\u6765\u5c31\u662f\u88ab\u7528\u6765\u4f5c\u4e3a\u661f\u7403\u8fd0\u52a8\u7684\u6a21\u578b\uff0c\u6211\u4eec\u53ef\u4ee5\u731c\u60f3\u5728N\u4f53\u95ee\u9898\u67d0\u4e9b\u89e3\u91cc\u4f1a\u6709\u78b0\u649e\u53d1\u751f\u3002<\/p>\n\n\n\n
\u4e8b\u5b9e\u4e0a\u5373\u4f7f\u5728\u4e8c\u4f53\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e24\u4e2a\u8d28\u70b9\u7684\u76f8\u5bf9\u4f4d\u7f6e\u603b\u5728\u4e00\u6761\u76f4\u7ebf\u4e0a\u7684\u8bdd\uff0c\u5b83\u4eec\u662f\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5c31\u78b0\u649e\u5728\u4e00\u8d77\u7684\uff0c\u8fd9\u6837\u8fd9\u4e2a\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u5728\u8fd9\u4e00\u65f6\u523b\u5c31\u5931\u53bb\u610f\u4e49\u4e86\u56e0\u4e3a\u65b9\u7a0b\u53f3\u9762\u67d0\u4e9b\u9879\u7684\u5206\u6bcd\u6210\u4e86\u96f6\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u6211\u4eec\u79f0\u65b9\u7a0b\u6709\u4e00\u4e2a\u5947\u70b9(singularity)\uff0c\u800c\u8fd9\u4e2a\u5947\u70b9\u5c31\u662f\u4e00\u4e2a\u78b0\u649e(collision)\u3002\u4f46\u662f\u5728\u5fae\u5206\u65b9\u7a0b\u7684\u7406\u8bba\u4e2d\uff0c\u5947\u70b9\u5e76\u4e0d\u90fd\u662f\u8fd9\u6837\u7684\u78b0\u649e\u7c7b\u578b\u7684\u3002<\/p>\n\n\n\n
\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u662f\u65b9\u7a0b $x’=x^2$ \u7684\u975e\u96f6\u89e3\u603b\u662f\u5728\u6709\u9650\u65f6\u95f4\u91cc\u5c31\u8dd1\u5230\u65e0\u7a77\u8fdc\u53bb\u4e86\uff0c\u8fd9\u79cd\u73b0\u8c61\u6211\u4eec\u79f0\u4e4b\u4e3a\u7206\u7834(blowup)\u3002<\/p>\n\n\n\n
\u5bf9\u4e8eN\u4f53\u95ee\u9898\u6211\u4eec\u4e00\u822c\u66f4\u5173\u5fc3\u8d28\u70b9\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u6240\u4ee5\u5982\u679c\u81f3\u5c11\u5176\u4e2d\u4e24\u4e2a\u8d28\u70b9\u4e4b\u95f4\u7684\u76f8\u5bf9\u8ddd\u79bb\u5728\u6709\u9650\u65f6\u95f4\u91cc\u5c31\u8dd1\u5230\u65e0\u7a77\u8fdc\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8bf4\u7206\u7834\u5728N\u4f53\u95ee\u9898\u4e2d\u51fa\u73b0\u3002<\/p>\n\n\n\n
\u5982\u679c\u8fd9\u662f\u771f\u7684\u5b87\u5b99\u7684\u8bdd\uff0c\u90a3\u5c31\u610f\u5473\u7740\u8fd9\u4e2a\u5b87\u5b99\u5728\u4e00\u6bb5\u65f6\u95f4\u540e\u5728\u6ca1\u6709\u78b0\u649e\u7684\u60c5\u51b5\u5c31\u6d88\u5931\u5230\u65e0\u7a77\u8fdc\u7684\u5c3d\u5934\u53bb\u4e86\u3002<\/p>\n\n\n\n
\u8fd9\u4f3c\u4e4e\u5927\u5927\u6709\u6096\u4e00\u822c\u4eba\u5bf9\u4e8e\u5b87\u5b99\u7684\u8ba4\u8bc6\uff0c\u4f46\u662f\u4eceN\u4f53\u95ee\u9898\u7684\u65b9\u7a0b\u6765\u770b\uff0c\u8fd9\u4f3c\u4e4e\u5e76\u4e0d\u592a\u53ef\u80fd\u53d1\u751f\uff0c\u56e0\u4e3a\u5f53\u65b9\u7a0b\u4e2d\u7684\u8ddd\u79bb\u9879\u53d8\u5927\u540e\uff0c\u8ddd\u79bb\u53d8\u5316\u7684\u901f\u5ea6\u5c31\u5c0f\u4e86\uff0c\u7206\u7834\u4f3c\u4e4e\u5c31\u4e0d\u4f1a\u53d1\u751f\u4e86\u3002\u5bf9\u4e8e\u4e8c\u4f53\u95ee\u9898\uff0c\u5927\u5bb6\u5f88\u5bb9\u6613\u8bc1\u660e\u7206\u7834\u4e0d\u4f1a\u53d1\u751f\u3002<\/p>\n\n\n\n
\u5f53N\u5927\u4e8e\u4e8c\u5462\uff1f\u5927\u5bb6\u53ef\u4ee5\u4ece\u5e9e\u52a0\u83b1\u7684\u5de5\u4f5c\u4e2d\u770b\u51fa\uff0c\u6211\u4eec\u7684\u56de\u7b54\u5e94\u8be5\u8c28\u614e\u4e00\u4e9b……<\/p>\n\n\n\n
\u5386\u53f2\u4e0a\u5173\u4e8eN\u4f53\u95ee\u9898\u4e2d\u5947\u70b9\u7684\u7814\u7a76\uff0c\u662f\u7531\u548c\u5e9e\u52a0\u83b1\u540c\u65f6\u4ee3\u7684\u53e6\u4e00\u4f4d\u6cd5\u56fd\u6570\u5b66\u5bb6\u5e9e\u52d2\u7ef4(Paul Painleve)\u5f00\u59cb\u7684\u3002<\/p>\n\n\n\n
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\u5e9e\u52d2\u7ef4\u5bf9\u4e8eN\u4f53\u95ee\u9898\u4e2d\u5947\u70b9\u7684\u7814\u7a76\uff0c\u4e5f\u548c\u524d\u9762\u63d0\u5230\u7684\u745e\u5178\u548c\u632a\u5a01\u56fd\u738b\u5965\u65af\u5361\u4e8c\u4e16\u7684\u540d\u5b57\u8054\u7cfb\u5728\u4e00\u8d77\u30021895\u5e74\u5965\u65af\u5361\u4e8c\u4e16\u9080\u8bf7\u5e9e\u52d2\u7ef4\u5230\u65af\u7279\u54e5\u5c14\u6469\u5927\u5b66(University of Stockholm)\u8bb2\u5b66\uff0c\u5e76\u4eb2\u81ea\u5230\u8bb2\u6f14\u5385\u548c\u6559\u6388\u5b66\u751f\u4eec\u4e00\u8d77\u8046\u542c\u5e9e\u52d2\u7ef4\u7684\u7cbe\u5f69\u6f14\u8bb2\u3002\u8fd9\u4f4d\u7231\u597d\u6570\u5b66\u7684\u653f\u6cbb\u5bb6\u548c\u53e6\u4e00\u4f4d\u7231\u597d\u653f\u6cbb\u7684\u6570\u5b66\u5bb6\u7684\u76f8\u9022\u786e\u662f\u6570\u5b66\u53f2\u4e0a\u4e00\u6bb5\u4f73\u8bdd\u3002\u5e9e\u52d2\u7ef4\u5728\u65af\u7279\u54e5\u5c14\u6469\u505a\u4e8623\u6b21\u7684\u7cfb\u5217\u6f14\u8bb2\uff0c\u7ed9\u540e\u4eba\u7559\u4e0b\u4e86\u957f\u8fbe\u4e94\u516d\u767e\u9875\u7684\u6f14\u8bb2\u7b14\u8bb0\uff0c\u800c\u4ed6\u7684\u4e3b\u9898\u5c31\u662f\u5fae\u5206\u65b9\u7a0b\u4e2d\u8d85\u8d8a\u51fd\u6570\u53ca\u5176\u5bf9N\u4f53\u95ee\u9898\u5947\u70b9\u7684\u5e94\u7528\u3002\u5e9e\u52d2\u7ef4\u8bc1\u660e\u4e86\u5728\u4e09\u4f53\u95ee\u9898\uff0c\u5947\u70b9\u5fc5\u987b\u662f\u78b0\u649e\u89e3\uff0c\u4e5f\u5c31\u662f\u8bf4\u7206\u7834\u662f\u4e0d\u53ef\u80fd\u5728\u4e09\u4f53\u95ee\u9898\u4e2d\u51fa\u73b0\u7684\u3002\u4f46\u662f\u5728\u4ed6\u7b14\u8bb0\u7684\u7b2c588\u9875\uff0c\u4ed6\u731c\u6d4b\u5f53N\u5927\u4e8e\u4e09\u65f6\uff0cN\u4f53\u95ee\u9898\u5b58\u5728\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\u3002\u8fd9\u5c31\u662f\u540e\u4eba\u79f0\u4e3a\u5e9e\u52d2\u7ef4\u731c\u60f3\u7684\u8457\u540d\u95ee\u9898\u3002\u5e9e\u52d2\u7ef4\u60f3\u8c61\u4e86\u8fd9\u6837\u4e00\u79cd\u53ef\u80fd\u6027\uff1a<\/p>\n\n\n\n
\u4e00\u4e2a\u8d28\u70b9\u5728\u5176\u4ed6\u8d28\u70b9\u4e4b\u95f4\u5f98\u5f8a\uff0c\u5f53\u5b83\u51e0\u4e4e\u548c\u67d0\u4e2a\u8d28\u70b9\u649e\u4e0a\u65f6\uff0c\u53c8\u6070\u597d\u8eb2\u5f00\uff0c\u8fc7\u4e00\u4f1a\u513f\u53c8\u548c\u53e6\u4e00\u4e2a\u8d28\u70b9\u51e0\u4e4e\u649e\u4e0a\uff0c\u8fd9\u6837\u5468\u800c\u590d\u59cb\u4f46\u6240\u6709\u8fd9\u4e00\u5207\u90fd\u53d1\u751f\u5728\u6709\u9650\u65f6\u95f4\u5185\u3002\u7136\u800c\u8fd9\u79cd\u5947\u7279\u7684\u632f\u8361\u5f62\u7206\u7834\u662f\u5426\u53ef\u80fd\u53d1\u751f\u5462\uff1f\u8fd9\u6b63\u662f\u5e9e\u52d2\u7ef4\u7684\u731c\u60f3\u3002<\/p>\n\n\n\n
\u5e9e\u52d2\u7ef4\u62c9\u5f00\u4e86\u957f\u8fbe\u8fd1\u767e\u5e74\u7684\u53d1\u73b0N\u4f53\u95ee\u9898\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\u7684\u5e0f\u5e55\u3002\u505a\u51fa\u7b2c\u4e00\u4e2a\u91cd\u8981\u8d21\u732e\u7684\u662f\u745e\u5178\u5929\u6587\u5b66\u5bb6\u51af\u6cfd\u57f9\u5c14(Hugo von Zeipel)\uff0c\u4ed6\u6b63\u662f1895\u5e74\u5e9e\u52d2\u7ef4\u6f14\u8bb2\u65f6\u5728\u5ea7\u7684\u4e00\u4e2a\u5e74\u4ec522\u5c81\u7684\u542c\u4f17\u3002\u5728\u4ed61908\u5e74\u4e00\u7bc7\u4ec5\u56db\u9875\u7684\u8bba\u6587\u4e2d\uff0c\u4ed6\u8bc1\u660e\u4e86N\u4f53\u95ee\u9898\u5982\u679c\u6709\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\uff0c\u90a3\u4e48\u8d28\u70b9\u95f4\u76f8\u4e92\u8ddd\u79bb\u4e00\u5b9a\u8981\u5728\u6709\u9650\u65f6\u95f4\u5185\u53d8\u6210\u65e0\u754c\uff0c\u4e5f\u5c31\u662f\u8bf4\u7206\u7834\u4e00\u5b9a\u8981\u53d1\u751f\uff0c\u5c3d\u7ba1\u8fd9\u79cd\u7206\u7834\u53ef\u80fd\u662f\u5e9e\u52d2\u7ef4\u6240\u63cf\u8ff0\u7684\u5947\u7279\u7684\u632f\u8361\u5f62\u7206\u7834\u3002\u51af\u6cfd\u57f9\u5c14\u7684\u7ed3\u679c\u81f3\u5c11\u8bf4\u660e\u4e0d\u4f1a\u6709\u5176\u4ed6\u66f4\u5947\u602a\u7684\u5947\u70b9\u5728N\u4f53\u95ee\u9898\u4e2d\u51fa\u73b0\uff0c\u8fd9\u4e00\u7ed3\u679c\u57281971\u5e74\u88ab\u7f8e\u56fd\u897f\u5317\u5927\u5b66\u6559\u6388\u8428\u745e(Donald Saari)\u63a8\u5e7f\u5230\uff1a\u5982\u679c\u6709\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\uff0c\u8d28\u70b9\u5fc5\u987b\u6709\u5f88\u5f3a\u70c8\u7684\u632f\u8361\uff0c\u8fd9\u6837\u4eba\u4eec\u5c31\u66f4\u5173\u5fc3\u632f\u8361\u5f62\u7206\u7834\u5230\u5e95\u4f1a\u4e0d\u4f1a\u53d1\u751f\u7684\u95ee\u9898\u4e86\u3002<\/p>\n\n\n\n
\u5728\u516d\u5341\u5e74\u4ee3\u5230\u516b\u5341\u5e74\u4ee3\u7684\u65f6\u95f4\u91cc\uff0c\u8bb8\u591a\u6570\u5b66\u5bb6\u6784\u9020\u51faN\u4f53\u95ee\u9898\u7684\u4e00\u4e9b\u7279\u6b8a\u89e3\uff0c\u5176\u4e2d\u6709\u4e9b\u89e3\u5177\u6709\u5f3a\u70c8\u7684\u632f\u8361\u6027\uff0c\u4f46\u662f\u5b83\u4eec\u4ecd\u7136\u4e0d\u662f\u771f\u6b63\u7684\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\u3002<\/p>\n\n\n\n
\u4f8b\u5982\u4fc4\u56fd\u6570\u5b66\u5bb6\u65af\u7279\u5c3c\u79d1\u592b(K. Sitnikov)\u57281960\u5e74\u6784\u9020\u4e86\u8fd9\u6837\u4e00\u4e2a\u4e09\u4f53\u95ee\u9898\u7684\u89e3\uff1a\u4e24\u4e2a\u7b49\u8d28\u91cf\u7684\u8d28\u70b9A\u548cB\u5728\u4e00\u4e2a\u5e73\u9762\u4e0a\u4e92\u76f8\u73af\u7ed5\u7740\u5728\u692d\u5706\u8f68\u9053\u4e0a\u8fd0\u52a8\uff0c\u7b2c\u4e09\u4e2a\u8d28\u91cf\u5f88\u5c0f\u7684\u8d28\u70b9C\u5728\u548c\u5e73\u9762\u5782\u76f4\u7684\u4e00\u6761\u76f4\u7ebf\u4e0a\u6765\u56de\u632f\u52a8\uff0c\u53ea\u8981\u9009\u62e9\u5408\u9002\u7684\u521d\u59cb\u503c\uff0cC\u5c31\u4f1a\u65e0\u7a77\u6b21\u6765\u56de\u7a7f\u8d8aAB\u6240\u5728\u5e73\u9762\uff0c\u800c\u4e14\u632f\u52a8\u7684\u5e45\u5ea6\u8d8a\u6765\u8d8a\u5927\uff0c\u6700\u540e\u5728\u65e0\u7a77\u7684\u65f6\u95f4\u5185\u632f\u5e45\u4f1a\u8d8b\u4e8e\u65e0\u7a77\u5927\u3002\u56e0\u4e3a\u8fd9\u91cc\u9700\u8981\u7684\u65f6\u95f4\u662f\u65e0\u7a77\uff0c\u6240\u4ee5\u8fd9\u4e2a\u89e3\u5e76\u4e0d\u662f\u4e00\u4e2a\u5947\u70b9\u89e3\u3002<\/p>\n\n\n\n
\u8fd9\u6837\u7684\u89e3\u4f3c\u4e4e\u6709\u6096\u5e38\u7406\uff0c\u4f46\u662f\u6570\u5b66\u5bb6\u4eec\u7528\u4ed6\u4eec\u5de7\u5999\u7684\u6784\u601d\u8bc1\u660e\u4e86\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u8fd9\u786e\u5b9e\u4f1a\u53d1\u751f\u3002<\/p>\n\n\n\n
1975\u5e74\u4e24\u4f4d\u7f8e\u56fd\u6570\u5b66\u5bb6\u9a6c\u745f\u5c14(John Mather)\u548c\u9ea6\u5409\u5c14(Richard McGehee)\u6784\u9020\u4e86\u4e00\u4e2a\u5171\u7ebf\u7684\u56db\u4f53\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u4e2a\u89e3\u786e\u5b9e\u4f1a\u5728\u6709\u9650\u7684\u65f6\u95f4\u5185\u4f7f\u67d0\u4e24\u4e2a\u8d28\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u8fbe\u5230\u65e0\u7a77\u3002\u4f46\u662f\u8fd9\u4ecd\u7136\u4e0d\u662f\u5e9e\u52d2\u7ef4\u6240\u5e0c\u671b\u7684\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\uff0c\u56e0\u4e3a\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u5728\u8fbe\u5230\u6700\u7ec8\u8fd9\u4e2a\u975e\u78b0\u649e\u7684\u5947\u70b9\u4e4b\u524d\uff0c\u56db\u4e2a\u5171\u7ebf\u7684\u8d28\u70b9\u4e92\u76f8\u4e4b\u95f4\u5fc5\u987b\u5148\u6709\u65e0\u7a77\u591a\u6b21\u7684\u78b0\u649e\uff0c\u6240\u4ee5\u975e\u78b0\u649e\u7684\u5947\u70b9\u5e76\u4e0d\u662f\u8fd9\u4e2a\u8f68\u9053\u4e2d\u7b2c\u4e00\u4e2a\u5947\u70b9\u3002\u5728\u8fd9\u4e4b\u524d\uff0c\u4eba\u4eec\u5df2\u7ecf\u53d1\u73b0\u5982\u679c\u4ec5\u4ec5\u662f\u4e24\u4e2a\u8d28\u70b9\u78b0\u649e\u7684\u8bdd\uff0c\u90a3\u4e48\u8fd9\u4e2a\u89e3\u53ef\u4ee5\u7528\u5f39\u6027\u53cd\u5f39\u5728\u78b0\u649e\u540e\u7ee7\u7eed\u4e0b\u53bb(\u518d\u591a\u8d28\u70b9\u540c\u65f6\u78b0\u649e\u5c31\u4e0d\u884c\u4e86)\u3002\u6240\u4ee5\u9a6c\u745f\u5c14\u548c\u9ea6\u5409\u5c14\u5229\u7528\u8fd9\u4e2a\u6027\u8d28\u8bbe\u8ba1\u4e86\u8fd9\u6837\u4e00\u4e2a\u5947\u7279\u7684\u4f8b\u5b50\u3002\u8fd9\u4ee5\u540e\u8bb8\u591a\u4eba\u5bf9\u4e8e\u6700\u540e\u53d1\u73b0\u771f\u6b63\u7684\u975e\u78b0\u649e\u7684\u5947\u70b9\u89e3\u66f4\u6709\u4fe1\u5fc3\u4e86\uff0c\u7136\u800c\u5728\u8fd9\u4e4b\u540e\u7684\u5341\u51e0\u5e74\u91cc\uff0c\u8fdb\u5c55\u662f\u5fae\u5f31\u7684\uff0c\u6709\u4eba\u8bc1\u660e\u4e86\u5982\u679c\u8fd9\u6837\u7684\u4f8b\u5b50\u786e\u5b9e\u5b58\u5728\u4e8e\u56db\u4f53\u95ee\u9898\u4e2d\uff0c\u90a3\u4e48\u8fd9\u56db\u4e2a\u8d28\u70b9\u5e94\u8be5\u51e0\u4e4e\u662f\u5728\u4e00\u6761\u76f4\u7ebf\u4e0a\u7684\u3002\u4e5f\u5c31\u662f\u8bf4\u771f\u6b63\u7684\u975e\u78b0\u649e\u7684\u5947\u70b9\u4e5f\u8bb8\u5c31\u8ddf\u9a6c\u745f\u5c14\u548c\u9ea6\u5409\u5c14\u7684\u4f8b\u5b50\u5f88\u63a5\u8fd1\uff0c\u7136\u800c\u65f6\u81f3\u4eca\u65e5\uff0c\u8fd9\u6837\u7684\u56db\u4f53\u95ee\u9898\u7684\u4f8b\u5b50\u8fd8\u662f\u6ca1\u6709\u4eba\u80fd\u627e\u51fa\u6765\u3002<\/p>\n\n\n\n
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\u4e8b\u5b9e\u4e0a\u590f\u5fd7\u5b8f\u8bc1\u660e\u7684\u521d\u7a3f\u4e2d\u786e\u5b9e\u5b58\u5728\u8868\u8ff0\u4e0a\u7684\u7f3a\u9677\uff0c\u67d0\u4e9b\u5173\u952e\u7684\u8bc1\u660e\u4e5f\u6709\u503c\u5f97\u63a8\u6572\u4e4b\u5904\u3002\u5728\u8fd9\u7bc7\u51e0\u4e4e\u957f\u8fbe\u767e\u9875\u7684\u6587\u7ae0\u88ab\u6295\u7a3f\u5230\u6700\u8457\u540d\u7684\u6570\u5b66\u6742\u5fd7\u201c\u6570\u5b66\u5e74\u520a\u201d(Annals of Mathematics)\u4e24\u5e74\u540e\uff0c\u590f\u5fd7\u5b8f\u5f97\u5230\u4e86\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u590d\uff0c\u5ba1\u7a3f\u7684\u4eba\u65e0\u6cd5\u5224\u65ad\u4ed6\u7684\u8bc1\u660e\u662f\u5426\u6b63\u786e\uff0c\u4f46\u786e\u5b9e\u6307\u51fa\u4e86\u5176\u4e2d\u7684\u4e00\u4e9b\u95ee\u9898\u3002\u590f\u5fd7\u5b8f\u5e76\u4e0d\u6c14\u9981\uff0c\u4ed6\u7ee7\u7eed\u6539\u8fdb\u8865\u5145\u4ed6\u7684\u8bc1\u660e\uff0c\u53c8\u628a\u4fee\u6539\u7a3f\u6295\u4e86\u4e0a\u53bb\u3002<\/p>\n\n\n\n
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\u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0c\u5728\u590f\u5fd7\u5b8f\u6b63\u5728\u4e3a\u4ed6\u7684\u8bba\u6587\u505a\u540e\u671f\u4fee\u6539\u7684\u65f6\u5019\uff0c\u53e6\u4e00\u4f4d\u7f8e\u56fd\u6570\u5b66\u5bb6\u54e5\u7ef4\u5c14(Joseph Gerver)\u5728\u542c\u5230\u590f\u5fd7\u5b8f\u7684\u7ed3\u679c\u540e\uff0c\u6df1\u53d7\u9707\u52a8\u3002<\/p>\n\n\n\n
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\u5c3d\u7ba1N\u4f53\u95ee\u9898\u4e2d\u8457\u540d\u7684\u5e9e\u52d2\u7ef4\u731c\u60f3\u5df2\u7ecf\u5728\u4e0a\u4e2a\u4e16\u7eaa\u7ed3\u675f\u524d\u88ab\u6210\u529f\u5730\u89e3\u51b3\u4e86\uff0cN\u4f53\u95ee\u9898\u672c\u8eab\u8fd8\u662f\u6709\u592a\u591a\u7684\u795e\u79d8\u9886\u57df\u503c\u5f97\u65b0\u4e16\u7eaa\u7684\u5e74\u8f7b\u6570\u5b66\u5bb6\u4eec\u63a2\u7d22\u3002\u4e5f\u8bb8\u5728\u4eca\u540e\u51e0\u4e2a\u4e16\u7eaa\u91cc\uff0cN\u4f53\u95ee\u9898\u5c06\u4ecd\u7136\u662f\u65b0\u7684\u6570\u5b66\u548c\u65b0\u7684\u601d\u60f3\u7684\u6e90\u6cc9\uff0c\u5c31\u8c61\u8fc7\u53bb\u7684\u4e09\u767e\u5e74\u4e00\u6837\u3002\u4f8b\u5982\u8fd1\u5e74\u6765\u5728\u4e09\u4f53\u95ee\u9898\u5468\u671f\u89e3\u65b9\u9762\u53c8\u6709\u6700\u65b0\u8fdb\u5c55\uff0c\u4e00\u79cd\u4e09\u4e2a\u8d28\u70b9\u5728\u4e00\u4e2a\u5e73\u97628\u5b57\u5f62\u8f68\u9053\u4e0a\u5468\u671f\u8fd0\u52a8\u7684\u89e3\u88ab\u6cd5\u56fd\u6570\u5b66\u5bb6\u9648\u601d\u7eb3(Alain Chenciner)\u548c\u7f8e\u56fd\u6570\u5b66\u5bb6\u8499\u54e5\u9a6c\u5229(Richard Montgomery)\u53d1\u73b0\uff0c\u800c\u4e14\u8fdb\u4e00\u6b65\u7684\u8ba1\u7b97\u673a\u6570\u503c\u6a21\u62df\u8fd8\u53d1\u73b0\u4e86\u66f4\u591a\u7684\u5177\u6709\u5404\u79cd\u5947\u7279\u8f68\u9053\u7684\u5468\u671f\u89e3\u3002<\/p>\n\n\n\n
\u5bf9\u4e8eN\u4f53\u95ee\u9898\u7684\u5386\u53f2\uff0c\u5386\u53f2\u8fdb\u5c55\uff0c\u548c\u76f8\u5e94\u6570\u5b66\u53d1\u5c55\u7684\u8fc7\u7a0b\uff0c\u4e00\u672c\u66f4\u8be6\u7ec6\u7684\u901a\u4fd7\u8bfb\u7269\u662f\u7f8e\u56fd\u666e\u6797\u65af\u987f\u5927\u5b66\u51fa\u7248\u793e1996\u5e74\u51fa\u7248\u7684\u201c\u5929\u4f53\u7684\u9082\u9005\uff0c\u7a33\u5b9a\u6027\u548c\u6df7\u6c8c\u7684\u8d77\u6e90\u201d(Celestial Encounters, the origins of chaos and stability)\u4e00\u4e66\uff0c\u4f5c\u8005\u662fFlorin Diacu \u548c Philip Holmes\u3002\u5bf9\u4e8e\u672c\u6587\u6240\u6d89\u53ca\u7684\u5185\u5bb9\uff0c\u6b64\u4e66\u6709\u975e\u5e38\u8be6\u7ec6\u800c\u53c8\u901a\u4fd7\u6613\u61c2\u7684\u4ecb\u7ecd\uff0c\u540c\u65f6\u4e66\u4e2d\u8fd8\u63d0\u53ca\u4e86\u6211\u4eec\u8fd9\u7bc7\u6587\u7ae0\u6ca1\u6709\u8c08\u5230\u7684\u7b26\u53f7\u52a8\u529b\u7cfb\u7edf\u4e0e\u6df7\u6c8c(symbolic dynamics and chaos), \u5929\u4f53\u8fd0\u52a8\u7684\u7a33\u5b9a\u6027\u95ee\u9898\u548c\u8457\u540d\u7684KAM\u7406\u8bba\u3002\u5927\u5b66\u7406\u5de5\u79d1\u5b66\u751f\u5b8c\u5168\u6709\u80fd\u529b\u8bfb\u61c2\u8fd9\u672c\u5f15\u4eba\u5165\u80dc\u7684\u5c0f\u4e66\uff0c\u8fdb\u800c\u901a\u8fc7\u9605\u8bfb\u8fd9\u672c\u4e66\u4e86\u89e3\u5230\u8bb8\u591a\u4e8c\u5341\u4e16\u7eaa\u6570\u5b66\u548c\u5929\u4f53\u529b\u5b66\u6700\u91cd\u8981\u7684\u4e00\u4e9b\u8fdb\u5c55\uff0c\u4e5f\u80fd\u5bf9N\u4f53\u95ee\u9898\u8fd9\u4e00\u6700\u57fa\u672c\u7684\u7269\u7406\u95ee\u9898\u6709\u521d\u6b65\u7684\u8ba4\u8bc6\u3002<\/p>\n\n\n\n
\u5173\u4e8e\u5e9e\u52a0\u83b1\u5728\u4e09\u4f53\u95ee\u9898\u4e0a\u7684\u5de5\u4f5c\uff0c\u53ef\u53c2\u770b\u7f8e\u56fd\u6570\u5b66\u4f1a\u548c\u4f26\u6566\u6570\u5b66\u4f1a1997\u5e74\u8054\u5408\u51fa\u7248\u7684\u201c\u5e9e\u52a0\u83b1\u548c\u4e09\u4f53\u95ee\u9898\u201d(Poincare and the three body problem)[1]\u4e00\u4e66\u3002\u4f5c\u8005\u662fJune Barrow-Green\uff0c\u4e00\u4f4d\u4e13\u95e8\u4ece\u4e8b\u6570\u5b66\u53f2\u7814\u7a76\u7684\u5b66\u8005\u3002\u4ece\u4e2d\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u8bb8\u591a\u5f53\u65f6\u7684\u539f\u59cb\u5386\u53f2\u8d44\u6599\u548c\u66f4\u591a\u7684\u5386\u53f2\u9038\u4e8b\u3002\u5173\u4e8e\u5e9e\u52d2\u7ef4\u731c\u60f3\u53ca\u590f\u5fd7\u5b8f\u7684\u89e3\u51b3\u65b9\u6cd5\uff0c\u8bfb\u8005\u4e5f\u53ef\u4ee5\u53c2\u770b\u4e24\u7bc7\u82f1\u6587\u7684\u77ed\u6587[2]\u548c[6]\u3002<\/p>\n\n\n\n
\u672c\u6587\u7684\u5927\u591a\u6570\u5185\u5bb9\u4e5f\u53d6\u6750\u4e8e\u4e0a\u8ff0\u53c2\u8003\u6587\u732e\u3002<\/p>\n\n\n\n
\u53c2\u8003\u6587\u732e\uff1a<\/p>\n\n\n\n
[1] Barrow-Green, June, Poincar\/'{e} and the three body problem. History of Mathematics, 11.
American Mathematical Society & London Mathematical Society, 1997.<\/p>\n\n\n\n
[2] Diacu, Florin N., Painlev\/'{e}’s conjecture. Mathematical Intelligencer, 15 (1993), no. 2, 6–12.<\/p>\n\n\n\n
[3] Diacu, Florin; Holmes, Philip, Celestial encounters. The origins of chaos and stability.
Princeton University Press, Princeton, NJ, 1996.<\/p>\n\n\n\n
[4] Hilbert, David, Mathematical Problems. Bulletin of American Mathematical Society 8 (1902), 437–479.
Reprinted in Bulletin of American Mathematical Society 37 (2000), no. 4, 407–436.<\/p>\n\n\n\n
[5] Montgomery, Richard, A new solution to the three-body problem.
Notices of American Mathematical Society 48 (2001), no. 5, 471–481.<\/p>\n\n\n\n
[6] Saari, Donald G.; Xia, Zhihong, Off to infinity in finite time.
Notices of American Mathematical Society 42 (1995), no. 5, 538–546.<\/p>\n","protected":false},"excerpt":{"rendered":"
N\u4f53\u95ee\u9898\u7684\u7b2c\u4e00\u4e2a\u5b8c\u6574\u7684\u6570\u5b66\u63cf\u8ff0(\u8c61\u4f7f\u7528\u4e0a\u9762\u8fd9\u6837\u7684\u5fae\u5206\u65b9\u7a0b)\u662f\u51fa\u73b0\u5728\u725b\u987f\u7684\u201c\u81ea\u7136\u54f2\u5b66\u7684\u6570\u5b66\u539f\u7406\u201d(Philosophiae Naturalis Prinicipia Mathematica\uff0c1687\u5e74\u51fa\u7248)\u4e00\u4e66\u4e2d\u3002<\/p>\n","protected":false},"author":1,"featured_media":10007,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[64],"tags":[],"class_list":["post-8427","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fun"],"_links":{"self":[{"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/posts\/8427","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8427"}],"version-history":[{"count":39,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/posts\/8427\/revisions"}],"predecessor-version":[{"id":11209,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/posts\/8427\/revisions\/11209"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=\/wp\/v2\/media\/10007"}],"wp:attachment":[{"href":"https:\/\/www.7mgh.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.7mgh.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}