For faster navigation, this Iframe is preloading the Wikiwand page for 杜芬振子.

Our magic isn't perfect

You can help our automatic cover photo selection by reporting an unsuitable photo.

The cover is visually disturbing

The cover is not a good choice

Thank you for helping!

Your input will affect cover photo selection, along with input from other users.

< back
- Rich
- Minimal
Serif
Sans

Justify Text
Get Wikiwand
Note: preferences and languages are saved separately in https mode
{{::langAbbreviation}}
- Read On Wikipedia
- Edit
- History
- Talk Page
- Print
- Download PDF

{{::$root.activation.text}} {{::$root.activation.toolbarText}}

杜芬振子

受驱的杜芬方程的庞加莱截面表明混沌行为。

杜芬振子（英語：Duffing oscillator）是一个描写受驱振动的振动子，由非线性微分方程表示^[1]

杜芬方程列式如下：

${\frac {d^{2}x(t)}{dt^{2))}+2\gamma {\frac {dx(t)}{dt))+\alpha *x(t)+\beta *x(t)^{3}=\delta *cos(\omega *t)$

其中

γ控制阻尼度
α控制韧度
β控制动力的非线性度
δ驱动力的振幅
ω驱动力的圆频率

数值解

杜芬方程没有解析解，但可用龙格－库塔法求得数值解。

当γ>0，杜芬振子呈现极限环振动；

Duffing oscillator limit cycle

Duffing oscillator limit cycle phase animation

当γ<0，系统进入混沌态，相图呈吸引子形态。

Duffing oscillator chaos

Duffing oscillator attractors animation

参考文献

^ Richard H. Enns George C. McCGuire, Nonlinear Physics, p209-217, Birkhauser,1997

外部链接

（英文）杜芬振子（页面存档备份，存于互联网档案馆）
（英文）杜芬方程（页面存档备份，存于互联网档案馆）

查论编吸引子

洛伦茨吸引子若斯叻吸引子杜芬振子范德波尔振荡器陈氏吸引子吕氏吸引子蔡氏电路多卷波混沌吸引子拉比诺维奇-法布里康特方程

查论编极限环

俄勒冈振子方程布鲁塞尔振子杜芬振子范德波尔振荡器洛特卡-沃爾泰拉方程糖酵解振子造父变星型振子

查论编混沌理论

混沌理论	Anosov diffeomorphism（英语：Anosov diffeomorphism）分岔理論蝴蝶效应 Chaos theory in organizational development（英语：Chaos theory in organizational development）复杂混沌控制（英语：Control of chaos）動態系统混沌的边缘分形可预测性（英语：Predictability）量子混沌（英语：Quantum chaos）聖菲研究所混沌同步化（英语：Synchronization of chaos）意外后果（英语：Unintended consequences）

混沌映射(列表（英语：List of chaotic maps）)	Arnold tongue（英语：Arnold tongue）猫映射（英语：Arnold's cat map） Baker映射（英语：Baker's map） Complex quadratic map（英语：Complex quadratic polynomial） Complex squaring map（英语：Complex squaring map） Coupled map lattice（英语：Coupled map lattice）双摆三体问题多卷波混沌吸引子杜芬振子 Duffing map（英语：Duffing map） Dyadic transformation（英语：Dyadic transformation） Dynamical billiards（英语：Dynamical billiards） outer（英语：Outer billiard） Exponential map（英语：Exponential map (discrete dynamical systems)）高斯映射 Gingerbreadman map（英语：Gingerbreadman map）厄农映射马蹄映射 Ikeda映射（英语：Ikeda map） Interval exchange map（英语：Interval exchange transformation） Kaplan–Yorke映射（英语：Kaplan–Yorke map）逻辑斯谛映射洛伦茨吸引子拉比诺维奇-法布里康特方程若斯叻吸引子标准映射（英语：Standard map） Swinging Atwood's machine（英语：Swinging Atwood's machine）帐篷映射 Tinkerbell映射（英语：Tinkerbell map）范德波尔振荡器 Zaslavskii映射（英语：Zaslavskii map）

混沌系统	Bouncing ball dynamics（英语：Bouncing ball dynamics）蔡氏電路多卷波混沌吸引子泡沫经济 Tilt-A-Whirl（英语：Tilt-A-Whirl）

混沌理论学家	迈克尔·贝里蔡少棠陈关荣米切爾·費根鮑姆古茨威勒 Brosl Hasslacher（英语：Brosl Hasslacher） Michel Hénon（英语：Michel Hénon）愛德華·諾頓·勞侖次亞歷山大·李亞普諾夫本華·曼德博儒勒·昂利·庞加莱 Otto Rössler（英语：Otto Rössler）达维德·吕埃勒 Oleksandr Mykolaiovych Sharkovsky（英语：Oleksandr Mykolaiovych Sharkovsky） Floris Takens（英语：Floris Takens）詹姆士·约克古茨威勒

分类

{{bottomLinkPreText}} {{bottomLinkText}}

This page is based on a Wikipedia article written by contributors (read/edit).
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

Cover photo is available under {{::mainImage.info.license.name || 'Unknown'}} license. Cover photo is available under {{::mainImage.info.license.name || 'Unknown'}} license. Credit: (see original file).

杜芬振子

{{current.index+1}} of {{items.length}}

Date: {{current.info.dateOriginal || 'Unknown'}}
Date: {{(current.info.date | date:'mediumDate') || 'Unknown'}}
Credit:
- Uploaded by: {{current.info.uploadUser}} on {{current.info.uploadDate | date:'mediumDate'}}
License: {{current.info.license.usageTerms || current.info.license.name || current.info.license.detected || 'Unknown'}}
License: {{current.info.license.usageTerms || current.info.license.name || current.info.license.detected || 'Unknown'}}
View file on Wikipedia

Suggest as cover photo

Would you like to suggest this photo as the cover photo for this article?

Yes, this would make a good choice No, never mind

Thank you for helping!

Your input will affect cover photo selection, along with input from other users.

Thanks for reporting this video!

{{result.lang}} {{result.T}}

No matching articles found

Search for articles containing: {{search.query}}

This browser is not supported by Wikiwand :(
Wikiwand requires a browser with modern capabilities in order to provide you with the best reading experience.
Please download and use one of the following browsers:

An extension you use may be preventing Wikiwand articles from loading properly.

If you're using HTTPS Everywhere or you're unable to access any article on Wikiwand, please consider switching to HTTPS (https://www.wikiwand.com).

Switch Wikiwand to HTTPS

An extension you use may be preventing Wikiwand articles from loading properly.

If you are using an Ad-Blocker, it might have mistakenly blocked our content. You will need to temporarily disable your Ad-blocker to view this page.

✕

This article was just edited, click to reload

This article has been deleted on Wikipedia (Why?)

Back to homepage

Please click Add in the dialog above

Please click Allow in the top-left corner,
then click Install Now in the dialog

Please click Open in the download dialog,
then click Install

Please click the "Downloads" icon in the Safari toolbar, open the first download in the list,
then click Install

{{::$root.activation.text}}

Install Wikiwand

Install on Chrome Install on Firefox

Don't forget to rate us

Tell your friends about Wikiwand!

Gmail Facebook Twitter Link

Enjoying Wikiwand?

Tell your friends and spread the love:

Share on Gmail Share on Facebook Share on Twitter Share on Buffer

All languages

{{::lang.NameEnglish}} - {{::lang.NameNative}}

Follow Us

Don't forget to rate us

Our magic isn't perfect

You can help our automatic cover photo selection by reporting an unsuitable photo.

This photo is visually disturbing This photo is not a good choice

Thank you for helping!

Your input will affect cover photo selection, along with input from other users.

X

Get ready for Wikiwand 2.0 🎉! the new version arrives on September 1st! Don't want to wait?