#author("2022-08-14T22:51:35+09:00","default:Miyashita","Miyashita") #author("2024-02-27T11:57:59+09:00","default:Miyashita","Miyashita") * Van der Pol 振動子 メモ [#sf184cbc] **式 [#gc1d3c56] \[ \ {d^{2}x \over dt^{2}}-\varepsilon (1-x^{2}){dx \over dt}+x=0 \,\, (\varepsilon>0) \tag{1}\ \] 左辺第2項の速度に比例した負抵抗(\( -\varepsilon \))によって徐々に振動が成長する．~ **例1 [#o05a71d4] \(\varepsilon\) が小さいときの挙動．~ 左図は \(v-x\) 図で右は \(x-t\) 図．~ 徐々に振幅が大きくなっていくが，第２項の \( 1-x^2 \) の効果も強まって振幅はやがて落ち着く． #ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/vanderPol_eps1.gif,800x400) ~ ~ **例2 [#mf64dcdc] \(\varepsilon\) が大きいときの挙動．~ 開始すぐ定常サイクルにはまって以降同じ振動を繰り返す． #ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/vanderPol_eps2.gif,800x400) ~ ~ **参考文献 [#o02f18cb] #htmlinsert(amazon,transitional,"asins=4254136331") -[[波動と非線形問題30講 (物理学30講シリーズ 3)>https://www.amazon.co.jp/%25E6%25B3%25A2%25E5%258B%2595%25E3%2581%25A8%25E9%259D%259E%25E7%25B7%259A%25E5%25BD%25A2%25E5%2595%258F%25E9%25A1%258C30%25E8%25AC%259B-%25E7%2589%25A9%25E7%2590%2586%25E5%25AD%25A630%25E8%25AC%259B%25E3%2582%25B7%25E3%2583%25AA%25E3%2583%25BC%25E3%2582%25BA-%25E6%2588%25B8%25E7%2594%25B0-%25E7%259B%259B%25E5%2592%258C/dp/4254136331/ref=sr_1_1?__mk_ja_JP=%25E3%2582%25AB%25E3%2582%25BF%25E3%2582%25AB%25E3%2583%258A&crid=2T6VAOW40BJZL&dib=eyJ2IjoiMSJ9.W6bqcCqbbPpnXW1HTzr8VQ.aM9C6r5qIllY5oVG_V2cg0qg7ueEXvr9vImX6jk39GM&dib_tag=se&keywords=%25E9%259D%259E%25E7%25B7%259A%25E5%25BD%25A2%25E6%25B3%25A2%25E5%258B%2595+30&qid=1709002609&s=books&sprefix=%25E9%259D%259E%25E7%25B7%259A%25E5%25BD%25A2%25E6%25B3%25A2%25E5%258B%259530%252Cstripbooks%252C374&sr=1-1&_encoding=UTF8&tag=hydrocoast-22&linkCode=ur2&linkId=d239abafc8a296edd16a56c3a4063dc3&camp=247&creative=1211]]