Character sum - Revision history

en>Trappist the monk: /* References */Fix CS1 deprecated coauthor parameter errors using AWB

2014-07-28T13:20:22Z

References: Fix CS1 deprecated coauthor parameter errors using AWB

← Older revision		Revision as of 15:20, 28 July 2014
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	~~{{no footnotes\|date=June 2013}}~~		Greetings! I am Marvella and I really feel comfortable when people use the full title. Playing baseball is the pastime he will by no means stop performing. California is where I've always been residing and I adore each working day living here. Bookkeeping is her working day job now.<br><br>Also visit my page ... [http://www.ninfeta.tv/blog/99493 http://www.ninfeta.tv/blog/99493]
	~~[[File:Forced Duffing equation Poincaré section.png\|thumb\|A [[Poincaré section]] of~~ the ~~forced Duffing equation suggesting chaotic behaviour]]~~
	~~The '''Duffing equation''', named after [[Georg Duffing]], is a [[non-linear]] second-order [[differential equation]] used to model certain [[damping\|damped]] and driven [[oscillator]]s~~. ~~The equation~~ is ~~given~~ by
	~~:<math>\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t)\,</math>~~

	~~where the (unknown) function ''x''=''x''(''t'') is the displacement at time ''t'', <math>\dot{x}</math> is the first [[derivative]] of ''x'' with respect to time, i.e~~. ~~[[velocity]], and <math>\ddot{x}</math>~~ is ~~the second time-derivative of ''x'~~'~~, i.e. [[acceleration]]. The numbers <math>\delta</math>, <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>~~ and ~~<math>\omega</math> are given constants.~~

	The equation describes the motion of a damped oscillator with a more complicated [[potential]] than in [[simple harmonic motion]] (which corresponds to the case β=δ=0); in physical terms, it models, for example, a [[spring pendulum]] whose spring's [[stiffness]] does not exactly obey [[Hooke's law]].

	~~The Duffing equation~~ is ~~an example of a dynamical system that exhibits [[chaos theory\|chaotic behavior]].~~
	~~Moreover the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour~~.

	~~==Parameters==~~
	* <~~math~~>~~\delta~~<~~/math~~> ~~controls the size of the [[damping]]~~.
	* <math>\alpha</math> controls the size of the [[stiffness]].
	* <math>\beta</math> controls the amount of non-linearity in the restoring force. ~~If <math>\beta=0</math>, the Duffing equation describes a damped and driven simple harmonic oscillator~~.
	* <math>\gamma</math> controls the [~~[amplitude]] of the periodic driving force. If <math>\gamma=0</math> we have a system without driving force.~~
	* <math>\omega</math> controls the [[frequency]] of the periodic driving force.

	~~==Methods of solution==~~
	~~[[File~~:~~Duffing oscillator limit cycle.gif\|thumb\|right\|300px\|Duffing oscillator limit cycle γ>0]]~~
	~~[[File:Duffing oscillator limit cycle phase animation.gif\|thumb\|right\|300px\|Duffing oscillator limit cycle phase animation γ>0]]~~


	~~[[File:Duffing oscillator chaos.gif\|thumb\|300px\|right\|Duffing oscillator chaos oscillation γ<0]]~~
	~~[[File:Duffing oscillator attractors animation.gif\|thumb\|right\|300px\|Duffing oscillator attractors animation γ<0]]~~

	~~In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:~~
	* Expansion in a [[Fourier series]] will provide an equation of motion to arbitrary precision.
	* The <math>x^3</~~math> term, also called the ''Duffing term'', can be approximated as small and the system treated as a [[perturbation theory\|perturbed]] simple harmonic oscillator.~~
	* The [[Frobenius method]] yields a complicated but workable solution.
	* Any of the various [[numerical analysis\|numeric methods]] such as [[Euler's method]] and [[Runge-Kutta]] can be used.

	~~In the special case of the undamped (<math>\delta = 0</math>) and undriven (<math>\gamma = 0<~~/~~math>) Duffing equation, an exact solution can be obtained using [[Jacobi's elliptic functions]]~~.

	~~==References==~~
	~~{{reflist}}~~

	~~== External links ==~~
	* [http://scholarpedia.~~org~~/~~article~~/~~Duffing_oscillator Duffing oscillator on Scholarpedia]~~
	* [http://~~mathworld~~.~~wolfram~~.~~com~~/~~DuffingDifferentialEquation.html MathWorld page]~~

	~~{{Chaos theory}}~~

	~~[[Category:Ordinary differential equations]~~]

	~~{{mathapplied-stub}}~~

en>Yobot: /* Sums over ranges */WP:CHECKWIKI error fixes (22) using AWB (9842)

2014-01-08T11:57:59Z

Sums over ranges: WP:CHECKWIKI error fixes (22) using AWB (9842)

← Older revision		Revision as of 13:57, 8 January 2014
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	~~Nice~~ to ~~meet you~~, ~~my title~~ is ~~Refugia~~. ~~Years~~ in the ~~past~~ we ~~moved to North Dakota~~. ~~One~~ of the ~~issues she enjoys most is~~ to ~~do aerobics~~ and ~~now she is trying to earn money with it~~. ~~Hiring has been my occupation for some time~~ but I'~~ve currently utilized for another one~~.<br><br>~~my blog:~~ [http://~~youulike~~.~~com~~/~~blogs~~/~~114396~~/~~390615~~/~~items-to-know-when-confronted-wi youulike~~.com]		{{no footnotes\|date=June 2013}}
			[[File:Forced Duffing equation Poincaré section.png\|thumb\|A [[Poincaré section]] of the forced Duffing equation suggesting chaotic behaviour]]
			The '''Duffing equation''', named after [[Georg Duffing]], is a [[non-linear]] second-order [[differential equation]] used to model certain [[damping\|damped]] and driven [[oscillator]]s. The equation is given by
			:<math>\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t)\,</math>

			where the (unknown) function ''x''=''x''(''t'') is the displacement at time ''t'', <math>\dot{x}</math> is the first [[derivative]] of ''x'' with respect to time, i.e. [[velocity]], and <math>\ddot{x}</math> is the second time-derivative of ''x'', i.e. [[acceleration]]. The numbers <math>\delta</math>, <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> and <math>\omega</math> are given constants.

			The equation describes the motion of a damped oscillator with a more complicated [[potential]] than in [[simple harmonic motion]] (which corresponds to the case β=δ=0); in physical terms, it models, for example, a [[spring pendulum]] whose spring's [[stiffness]] does not exactly obey [[Hooke's law]].

			The Duffing equation is an example of a dynamical system that exhibits [[chaos theory\|chaotic behavior]].
			Moreover the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.

			==Parameters==
			* <math>\delta</math> controls the size of the [[damping]].
			* <math>\alpha</math> controls the size of the [[stiffness]].
			* <math>\beta</math> controls the amount of non-linearity in the restoring force. If <math>\beta=0</math>, the Duffing equation describes a damped and driven simple harmonic oscillator.
			* <math>\gamma</math> controls the [[amplitude]] of the periodic driving force. If <math>\gamma=0</math> we have a system without driving force.
			* <math>\omega</math> controls the [[frequency]] of the periodic driving force.

			==Methods of solution==
			[[File:Duffing oscillator limit cycle.gif\|thumb\|right\|300px\|Duffing oscillator limit cycle γ>0]]
			[[File:Duffing oscillator limit cycle phase animation.gif\|thumb\|right\|300px\|Duffing oscillator limit cycle phase animation γ>0]]


			[[File:Duffing oscillator chaos.gif\|thumb\|300px\|right\|Duffing oscillator chaos oscillation γ<0]]
			[[File:Duffing oscillator attractors animation.gif\|thumb\|right\|300px\|Duffing oscillator attractors animation γ<0]]

			In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:
			* Expansion in a [[Fourier series]] will provide an equation of motion to arbitrary precision.
			* The <math>x^3</math> term, also called the ''Duffing term'', can be approximated as small and the system treated as a [[perturbation theory\|perturbed]] simple harmonic oscillator.
			* The [[Frobenius method]] yields a complicated but workable solution.
			* Any of the various [[numerical analysis\|numeric methods]] such as [[Euler's method]] and [[Runge-Kutta]] can be used.

			In the special case of the undamped (<math>\delta = 0</math>) and undriven (<math>\gamma = 0</math>) Duffing equation, an exact solution can be obtained using [[Jacobi's elliptic functions]].

			==References==
			{{reflist}}

			== External links ==
			* [http://scholarpedia.org/article/Duffing_oscillator Duffing oscillator on Scholarpedia]
			* [http://mathworld.wolfram.com/DuffingDifferentialEquation.html MathWorld page]

			{{Chaos theory}}

			[[Category:Ordinary differential equations]]

			{{mathapplied-stub}}

en>Jsondow: Corrected punctuation

2012-03-26T11:43:52Z

Corrected punctuation

New page

Nice to meet you, my title is Refugia. Years in the past we moved to North Dakota. One of the issues she enjoys most is to do aerobics and now she is trying to earn money with it. Hiring has been my occupation for some time but I've currently utilized for another one.<br><br>my blog: [http://youulike.com/blogs/114396/390615/items-to-know-when-confronted-wi youulike.com]