NTUNL 板


LINE

[2.7] How are maps related to flows (differential equations)? Every differential equation gives rise to a map, the time one map, defined b y advancing the flow one unit of time. This map may or may not be useful. If the differential equation contains a term or terms periodic in time, then t he time T map (where T is the period) is very useful--it is an example of a Poincar□ section. The time T map in a system with periodic terms is also ca lled a stroboscopic map, since we are effectively looking at the location in phase space with a stroboscope tuned to the period T. This map is useful be cause it permits us to dispense with time as a phase space coordinate: the r emaining coordinates describe the state completely so long as we agree to co nsider the same instant within every period. In autonomous systems (no time-dependent terms in the equations), it may als o be possible to define a Poincar□ section and again reduce the phase space dimension by one. Here the Poincar□ section is defined not by a fixed time interval, but by successive times when an orbit crosses a fixed surface in phase space. (Surface here means a manifold of dimension one less than the p hase space dimension). However, not every flow has a global Poincar□ section (e.g. any flow with a n equilibrium point), which would need to be transverse to every possible or bit. Maps arising from stroboscopic sampling or Poincar□ section of a flow are n ecessarily invertible, because the flow has a unique solution through any po int in phase space--the solution is unique both forward and backward in time . However, noninvertible maps can be relevant to differential equations: Poi ncar□ maps are sometimes very well approximated by noninvertible maps. For example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the logistic map, x -> -a+x^2. It is often (though not always) possible to go backwards, from an invertible map to a differential equation having the map as its Poincar□ map. This is called a suspension of the map. One can also do this procedure approximatel y for maps that are close to the identity, giving a flow that approximates t he map to some order. This is extremely useful in bifurcation theory. Note that any numerical solution procedure for a differential initial value problem which uses discrete time steps in the approximation is effectively a map. This is not a trivial observation; it helps explain for example why a continuous-time system which should not exhibit chaos may have numerical sol utions which do--see [2.15]. -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --



※ 發信站: 批踢踢實業坊(ptt.csie.ntu.edu.tw)
◆ From: 140.112.102.146







like.gif 您可能會有興趣的文章
icon.png[問題/行為] 貓晚上進房間會不會有憋尿問題
icon.pngRe: [閒聊] 選了錯誤的女孩成為魔法少女 XDDDDDDDDDD
icon.png[正妹] 瑞典 一張
icon.png[心得] EMS高領長版毛衣.墨小樓MC1002
icon.png[分享] 丹龍隔熱紙GE55+33+22
icon.png[問題] 清洗洗衣機
icon.png[尋物] 窗台下的空間
icon.png[閒聊] 双極の女神1 木魔爵
icon.png[售車] 新竹 1997 march 1297cc 白色 四門
icon.png[討論] 能從照片感受到攝影者心情嗎
icon.png[狂賀] 賀賀賀賀 賀!島村卯月!總選舉NO.1
icon.png[難過] 羨慕白皮膚的女生
icon.png閱讀文章
icon.png[黑特]
icon.png[問題] SBK S1安裝於安全帽位置
icon.png[分享] 舊woo100絕版開箱!!
icon.pngRe: [無言] 關於小包衛生紙
icon.png[開箱] E5-2683V3 RX480Strix 快睿C1 簡單測試
icon.png[心得] 蒼の海賊龍 地獄 執行者16PT
icon.png[售車] 1999年Virage iO 1.8EXi
icon.png[心得] 挑戰33 LV10 獅子座pt solo
icon.png[閒聊] 手把手教你不被桶之新手主購教學
icon.png[分享] Civic Type R 量產版官方照無預警流出
icon.png[售車] Golf 4 2.0 銀色 自排
icon.png[出售] Graco提籃汽座(有底座)2000元誠可議
icon.png[問題] 請問補牙材質掉了還能再補嗎?(台中半年內
icon.png[問題] 44th 單曲 生寫竟然都給重複的啊啊!
icon.png[心得] 華南紅卡/icash 核卡
icon.png[問題] 拔牙矯正這樣正常嗎
icon.png[贈送] 老莫高業 初業 102年版
icon.png[情報] 三大行動支付 本季掀戰火
icon.png[寶寶] 博客來Amos水蠟筆5/1特價五折
icon.pngRe: [心得] 新鮮人一些面試分享
icon.png[心得] 蒼の海賊龍 地獄 麒麟25PT
icon.pngRe: [閒聊] (君の名は。雷慎入) 君名二創漫畫翻譯
icon.pngRe: [閒聊] OGN中場影片:失蹤人口局 (英文字幕)
icon.png[問題] 台灣大哥大4G訊號差
icon.png[出售] [全國]全新千尋侘草LED燈, 水草

請輸入看板名稱,例如:BabyMother站內搜尋

TOP