Mathematical constants by continued fraction representation - Revision history

en>SuperNerd137: Organized numbers according to continued fractions.

2014-06-30T22:31:29Z

Organized numbers according to continued fractions.

← Older revision		Revision as of 00:31, 1 July 2014
Line 1:		Line 1:
	~~{{no footnotes\|date=March 2013}}~~		I'm Bernie and I live with my husband and our 2 children in Rasta, in the south part. My hobbies are Homebrewing, Art collecting and Coin collecting.<br><br>Also visit my homepage ... Fifa 15 Coin Generator; [http://www.jr-graph.fr/pixelpost/index.php?showimage=1 www.jr-graph.fr],
	~~In [[mathematics]], particularly in [[dynamical systems]], a~~ '''bifurcation diagram''' shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a [[Bifurcation theory\|bifurcation parameter]] in the system. It is usual to represent stable solutions with a solid line and ~~unstable solutions~~ with ~~a dotted line.~~

	~~==Bifurcations in 1D discrete dynamical systems==~~

	~~===Logistic map===~~
	~~{{See also\|Dynamical systems\|List of chaotic maps}}~~

	~~[[Image:LogisticMap BifurcationDiagram.png\|300px\|thumb\|right\|Bifurcation diagram of the [[logistic map]]]]~~
	~~[[Image:diagram bifurkacji anim small.gif\|300px\|thumb\|right\|Animation showing the formation of bifurcation diagram]]~~
	~~[[Image:Circle map bifurcation.jpeg\|thumb\|right\|Bifurcation diagram of the [[circle map]]. Black regions correspond to [[Arnold tongues]].]]~~

	~~An example is the bifurcation diagram of the [[logistic map]]:~~

	~~:<math> x_{n+1}=rx_n(1-x_n). \,</math>~~

	~~The bifurcation parameter ''r'' is shown on the horizontal axis of the plot~~ and ~~the vertical axis shows the possible long-term population values of the logistic function.~~

	~~The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a [[period-doubling bifurcation]].~~
	~~The ratio of the lengths of successive intervals between values of ''r'' for which bifurcation occurs [[convergent series\|converges]] to the [[first Feigenbaum constant]].~~

	~~===Real quadratic map===~~
	~~{{See also\|Complex quadratic polynomial}}~~
	~~The map is <math>x_{n+1}=x_n^~~2~~-c</math>.~~

	~~==Symmetry breaking~~ in ~~bifurcation sets==~~

	~~[[Image:Asymbif.gif\|300px\|thumb\|right\|Symmetry breaking~~ in ~~[[pitchfork bifurcation]] as~~ the ~~parameter epsilon is varied~~. ~~epsilon = 0 is the case of symmetric pitchfork bifurcation.]]~~

	~~In a dynamical system such as~~

	~~:<math> \ddot {x} + f(x;\mu) + \epsilon g(x) = 0</math>~~,

	which is [[structurally stable]] when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \epsilon </math>, the case <math> \epsilon = 0</math> is the symmetric pitchfork bifurcation. ~~When~~ <~~math~~> ~~\epsilon \neq 0~~ <~~/math~~>~~, we say we have a pitchfork with ''broken symmetry~~.~~'' This is illustrated in the animation on the right~~.

	~~== See also ==~~

	* [[Bifurcation theory]]
	* [[Feigenbaum constants]]
	* [[Phase portrait]]
	* [[Complex_quadratic_polynomial#Critical_curves\|Skeleton of bifurcation diagram]]

	~~==References==~~
	*Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
	*Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.

	~~==External links==~~
	* [http://~~yuval~~.~~bar~~-or.~~org~~/index.php?~~item~~=~~4 Logistic Map Simulation]. A Java applet simulating the Logistic Map by Yuval Baror.~~
	* [http://www.~~egwald.com/nonlineardynamics/logisticsmapchaos~~.~~php The Logistic Map and Chaos]~~

	~~[[Category:Chaos theory]]~~
	~~[[Category:Bifurcation theory]]~~

	~~[[de:Bifurkationsdiagramm]~~]

en>Gandalf61: rv - R \ A means real and not algebraic so was correct before

2014-01-26T21:33:34Z

rv - R \ A means real and *not* algebraic so was correct before

← Older revision		Revision as of 23:33, 26 January 2014
Line 1:		Line 1:
	~~Historically folks~~, ~~mostly women have had~~ a ~~tendency to believe which~~ the ~~2nd they begin to lift weights,~~ or ~~make progress~~ in the ~~weights they are lifting that they may become muscle-bound plus bulky~~. ~~This couldn't be farther within the truth~~. ~~In order to place on fat, whether it be muscle or fat, you have to take~~ in ~~more calories than what you're burning~~.~~<br><br>Additionally to~~ the ~~ones reported, you need to equally find specialist advice since this will assist you program balanced diets inside more detail in addition to helping you get inspired~~. ~~These professionals know how to prepare~~ [~~http~~:~~//safedietsthatwork~~.~~blogspot~~.~~com diet plans~~] ~~depending on what kind~~ of ~~activities you do and what kind of food you like.~~<br><br>~~For~~ the ~~men, plus females, which need to change their 12 pack into a 6 pack, Tom Venuto's Burn~~ the ~~Fat program comes inside handy. Tom shares twenty years~~ of ~~experience in his diet plan, plus muscle booster~~. He shows ~~how to turn several fat into muscle, plus shed~~ the ~~remainder~~ of ~~harmful fat for superior~~. ~~This really~~ is ~~ideal for those whom would very not take diet pills, or supplements, plus keeping~~ a ~~metabolism at bay~~. ~~Tom is a specialist, plus knowledged inside "physical" science when it comes~~ to the ~~human body~~. ~~Burn the Fat, Feed the Muscle 337 page book by Tom Ventuno has been rated excellent on those whom have experienced Tom's weight loss plan plus muscle builder initial hand~~. ~~For providing a life-time diet which does thus much~~ in ~~just a couple of days a week results~~ in the ~~rightful place on~~ the ~~list~~ of ~~the best ten diet plans~~.<br><br>~~I very recommend eating vegetarian diet for fat loss. This really~~ is ~~by far the many all-natural method for healthy weight reduction considering you're not ingesting anything that could harm~~ the ~~health~~, but ~~yet you're getting all imperative nutrients the body needs~~ for ~~optimal mobile growth plus maximum fat reduction. It's~~ the ~~number one way to lose weight, indeed.~~<br><br>~~To be fairly honest with we~~ is ~~that~~ the ~~diets which are promoted are all trash~~. ~~Yes. Garbage. None of those fad diet do not work. I guess it could start creating sense to you why~~ we ~~couldn~~'~~t lose fat~~. ~~Also we should learn which all those diets are~~ the ~~cause~~ of ~~we Yo-Yo Dieting~~, ~~losing fat~~ and ~~gaining more inside couple of weeks~~, ~~being cranky plus stressful etc~~.~~<br><br>It has been medically proven that there are cardiovascular benefits to a glass of wine~~, ~~a beer or perhaps a cocktail by raising the superior cholesterol levels plus thinning the blood. The health~~ and ~~diet issues are raised by drinking alcohol~~ to ~~excess.<br><br>One that has serious restrictions on what we can eat~~, ~~or 1 which knows which you're human~~ and ~~reside in the real world~~, ~~plus therefore~~, ~~enables we to cheat a little~~?		{{no footnotes\|date=March 2013}}
			In [[mathematics]], particularly in [[dynamical systems]], a '''bifurcation diagram''' shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a [[Bifurcation theory\|bifurcation parameter]] in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.

			==Bifurcations in 1D discrete dynamical systems==

			===Logistic map===
			{{See also\|Dynamical systems\|List of chaotic maps}}

			[[Image:LogisticMap BifurcationDiagram.png\|300px\|thumb\|right\|Bifurcation diagram of the [[logistic map]]]]
			[[Image:diagram bifurkacji anim small.gif\|300px\|thumb\|right\|Animation showing the formation of bifurcation diagram]]
			[[Image:Circle map bifurcation.jpeg\|thumb\|right\|Bifurcation diagram of the [[circle map]]. Black regions correspond to [[Arnold tongues]].]]

			An example is the bifurcation diagram of the [[logistic map]]:

			:<math> x_{n+1}=rx_n(1-x_n). \,</math>

			The bifurcation parameter ''r'' is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function.

			The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a [[period-doubling bifurcation]].
			The ratio of the lengths of successive intervals between values of ''r'' for which bifurcation occurs [[convergent series\|converges]] to the [[first Feigenbaum constant]].

			===Real quadratic map===
			{{See also\|Complex quadratic polynomial}}
			The map is <math>x_{n+1}=x_n^2-c</math>.

			==Symmetry breaking in bifurcation sets==

			[[Image:Asymbif.gif\|300px\|thumb\|right\|Symmetry breaking in [[pitchfork bifurcation]] as the parameter epsilon is varied. epsilon = 0 is the case of symmetric pitchfork bifurcation.]]

			In a dynamical system such as

			:<math> \ddot {x} + f(x;\mu) + \epsilon g(x) = 0</math>,

			which is [[structurally stable]] when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \epsilon </math>, the case <math> \epsilon = 0</math> is the symmetric pitchfork bifurcation. When <math> \epsilon \neq 0 </math>, we say we have a pitchfork with ''broken symmetry.'' This is illustrated in the animation on the right.

			== See also ==

			* [[Bifurcation theory]]
			* [[Feigenbaum constants]]
			* [[Phase portrait]]
			* [[Complex_quadratic_polynomial#Critical_curves\|Skeleton of bifurcation diagram]]

			==References==
			*Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
			*Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.

			==External links==
			* [http://yuval.bar-or.org/index.php?item=4 Logistic Map Simulation]. A Java applet simulating the Logistic Map by Yuval Baror.
			* [http://www.egwald.com/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos]

			[[Category:Chaos theory]]
			[[Category:Bifurcation theory]]

			[[de:Bifurkationsdiagramm]]

en>Markhurd: Add || to fill table; highlight the math markup issues

2011-11-25T05:51:21Z

Add || to fill table; highlight the math markup issues

New page

Historically folks, mostly women have had a tendency to believe which the 2nd they begin to lift weights, or make progress in the weights they are lifting that they may become muscle-bound plus bulky. This couldn't be farther within the truth. In order to place on fat, whether it be muscle or fat, you have to take in more calories than what you're burning. Additionally to the ones reported, you need to equally find specialist advice since this will assist you program balanced diets inside more detail in addition to helping you get inspired. These professionals know how to prepare [http://safedietsthatwork.blogspot.com diet plans] depending on what kind of activities you do and what kind of food you like. For the men, plus females, which need to change their 12 pack into a 6 pack, Tom Venuto's Burn the Fat program comes inside handy. Tom shares twenty years of experience in his diet plan, plus muscle booster. He shows how to turn several fat into muscle, plus shed the remainder of harmful fat for superior. This really is ideal for those whom would very not take diet pills, or supplements, plus keeping a metabolism at bay. Tom is a specialist, plus knowledged inside "physical" science when it comes to the human body. Burn the Fat, Feed the Muscle 337 page book by Tom Ventuno has been rated excellent on those whom have experienced Tom's weight loss plan plus muscle builder initial hand. For providing a life-time diet which does thus much in just a couple of days a week results in the rightful place on the list of the best ten diet plans. I very recommend eating vegetarian diet for fat loss. This really is by far the many all-natural method for healthy weight reduction considering you're not ingesting anything that could harm the health, but yet you're getting all imperative nutrients the body needs for optimal mobile growth plus maximum fat reduction. It's the number one way to lose weight, indeed. To be fairly honest with we is that the diets which are promoted are all trash. Yes. Garbage. None of those fad diet do not work. I guess it could start creating sense to you why we couldn't lose fat. Also we should learn which all those diets are the cause of we Yo-Yo Dieting, losing fat and gaining more inside couple of weeks, being cranky plus stressful etc. It has been medically proven that there are cardiovascular benefits to a glass of wine, a beer or perhaps a cocktail by raising the superior cholesterol levels plus thinning the blood. The health and diet issues are raised by drinking alcohol to excess. One that has serious restrictions on what we can eat, or 1 which knows which you're human and reside in the real world, plus therefore, enables we to cheat a little?

Mathematical constants by continued fraction representation - Revision history

en>SuperNerd137: Organized numbers according to continued fractions.

en>Gandalf61: rv - R \ A means real and *not* algebraic so was correct before

en>Markhurd: Add || to fill table; highlight the math markup issues

en>Gandalf61: rv - R \ A means real and not algebraic so was correct before