Heun function - Revision history

en>Rjwilmsi: /* References */Journal cites, Added 1 doi to a journal cite using AWB (10365)

2014-08-14T17:30:56Z

References: Journal cites, Added 1 doi to a journal cite using AWB (10365)

← Older revision		Revision as of 19:30, 14 August 2014
Line 1:		Line 1:
	~~[[File:RabFab800.png\|thumb\|upright]]~~		I'm Brook and was born on 25 September 1985. My hobbies are Art collecting and Stone collecting.<br><br>my webpage; [http://allakro.org/?document_srl=439315 Sexy Nebraska Women]
	~~[[File:Rabinovich Fabrikant 2314.png\|thumb\|upright]]~~
	~~[[File:Rabinovich Fabrikant 5212.png\|thumb\|upright]]~~
	~~The '''Rabinovich–Fabrikant equations'~~'~~' are a set of three coupled [[ordinary differential equations]] exhibiting [[Chaos theory\|chaotic]] behavior for certain values of the [[parameter]]s. They are named after [[Mikhail Rabinovich]]~~ and ~~[[Anatoly Fabrikant]], who described them in 1979~~.

	~~==System description==~~
	~~The equations~~ are~~:<ref name="Rabinovich"/>~~

	~~: <math>\dot{x} = y (z - 1 + x^2) + \gamma x \, </math>~~
	~~: <math>\dot{y} = x (3z + 1 - x^2) + \gamma y \, </math>~~
	~~: <math>\dot{z} = -2z (\alpha + xy), \, </math>~~

	~~where ''α'', ''γ'' are constants that control the evolution of the system. For some values of ''α''~~ and ~~''γ'', the system is chaotic, but for others it tends to a stable periodic orbit~~.

	~~Danca and Chen~~<~~ref name="DancaChen"/~~> note that the Rabinovich–Fabrikant system is difficult to analyse (due to the presence of quadratic and cubic terms) and that different attractors can be obtained for the same parameters by using different step sizes in the integration.

	~~===Equilibrium points===~~
	~~[[File:Rabinovich-Fabrikant equilibrium point existence regions.svg\|thumb\|Graph of the regions for which equilibrium points~~ <~~math~~>~~\tilde{\mathbf{x}}_{1,2,3,4}</math> exist.]]~~
	~~The Rabinovich–Fabricant system has five hyperbolic [[equilibrium points]], one at the origin and four dependent on the system parameters ''α'' and ''γ'':<ref name="DancaChen"/>~~

	~~: <math>\tilde{\mathbf{x}}_0 = (0,0,0)</math>~~
	~~: <math>\tilde{\mathbf{x}}_{1,2} = \left( \pm q_-, - \frac{\alpha}{q_-}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_-^2 \right)</math>~~
	~~: <math>\tilde{\mathbf{x}}_{3,4} = \left( \pm q_+, - \frac{\alpha}{q_+}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_+^2 \right)</math>~~

	~~where~~

	~~: <math>q_{\pm} = \sqrt{ \frac{ 1 \pm \sqrt{ 1- \gamma \alpha \left( 1- \frac{3 \gamma}{4\alpha} \right) } }{2 \left(1- \frac{3\gamma}{4\alpha}\right) }}</math>~~

	~~These equilibrium points only exist for certain values of ''α'' and ''γ'' > 0.~~

	~~===γ = 0.87, α = 1.1===~~
	~~An example of chaotic behavior is obtained for ''γ'' = 0.87 and ''&alpha~~;~~'' = 1.1 with initial conditions of (−1, 0, 0.5).<ref name="SprottJC"/> The~~ [[correlation dimension]] was found to be 2.19 ± 0.01.<ref name="Grassberger"/> The Lyapunov exponents, ''λ'' are approximately 0.1981, 0, −0.6581 and the [[Kaplan–Yorke dimension]], ''D''<sub>KY</sub> ≈ 2.3010<ref name="SprottJC"/>

	~~===γ = 0.1===~~
	~~Danca and Romera<ref name="DancaRomera"/> showed that for ''γ'' = 0.1, the system is chaotic for ''α'' = 0.98, but progresses on a stable [[limit cycle]] for ''α'' = 0.14.~~

	~~[[File:Rabinovich-Fabrikant LimitCicle.PNG\|thumb\|3D parametric plot of the solution of the Rabinovich-Fabrikant equations for ''α''=0.14 and ''γ''=0.1 (limit cycle is showed by the red curve)]]~~

	~~==See also==~~
	* [[List of chaotic maps]]

	~~==References==~~
	~~<references>~~
	~~<ref name="Grassberger">{{cite journal~~
	~~\| first1 = P.~~
	~~\| last1 = Grassberger~~
	~~\| first2 = I.~~
	~~\| last2 = Procaccia~~
	~~\| title = Measuring the strangeness of strange attractors~~
	~~\| journal = Physica D~~
	~~\| year = 1983~~
	~~\| volume = 9~~
	~~\| pages = 189–208~~
	~~\| doi = 10.1016/0167-2789(83)90298-1~~
	~~\| bibcode=1983PhyD....9..189G~~
	~~}}</ref>~~
	~~<ref name="Rabinovich">{{cite journal~~
	~~\| first1 = Mikhail I.~~
	~~\| last1 = Rabinovich~~
	~~\| first2 = A. L.~~
	~~\| last2 = Fabrikant~~
	~~\| title = Stochastic Self-Modulation of Waves in Nonequilibrium Media~~
	~~\| journal = Sov. Phys. JETP~~
	~~\| year = 1979~~
	~~\| volume = 50~~
	~~\| pages = 311~~
	~~\|bibcode = 1979JETP...50..311R }}</ref>~~
	~~<ref name="SprottJC">{{cite book~~
	~~\| last = Sprott~~
	~~\| first = Julien C.~~
	~~\| title = Chaos and Time-series Analysis~~
	~~\| publisher = [[Oxford University Press]]~~
	~~\| year = 2003~~
	~~\| pages = 433~~
	~~\| isbn = 0-19-850840-9~~
	~~}}</ref>~~
	~~<ref name="DancaRomera">~~
	~~{{cite journal~~
	~~\| first1 = Marius-F.~~
	~~\| last1 = Danca~~
	~~\| first2 = Miguel~~
	~~\| last2 = Romera~~
	~~\| title = Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems~~
	~~\| journal = Dynamics of Continuous, Discrete and Impulsive Systems~~
	~~\| series = Series B~~: ~~Applications & Algorithms~~
	~~\| volume = 15~~
	~~\| year = 2008~~
	~~\| pages = 155–164~~
	~~\| publisher = Watam Press~~
	~~\| id = {{hdl\|10261~~/~~8868}}~~
	~~\| issn = 1492-8760~~
	~~}}</ref>~~
	~~<ref name="DancaChen">~~
	~~{{cite journal~~
	~~\| first1 = Marius-F.~~
	~~\| last1 = Danca~~
	~~\| first2 = Guanrong~~
	~~\| last2 = Chen~~
	~~\| title = Birfurcation and Chaos in a Complex Model of Dissipative Medium~~
	~~\| journal = International Journal of Bifurcation and Chaos~~
	~~\| volume = 14~~
	~~\| issue = 10~~
	~~\| year = 2004~~
	~~\| pages = 3409–3447~~
	~~\| publisher = World Scientiﬁc Publishing Company~~
	~~\| doi = 10.1142~~/~~S0218127404011430~~
	~~\|bibcode = 2004IJBC..~~.~~14.3409D }}</ref>~~
	</~~references>~~

	==~~External links==~~
	* Weisstein, Eric W. [http://mathworld.wolfram.com/Rabinovich-FabrikantEquation.html "Rabinovich–Fabrikant Equation."] From MathWorld—A Wolfram Web Resource.
	*Chaotics Models a more appropriate approach to the chaotic graph of the system [http://www.robert-doerner.de/Glossar/glossar.html#R "Rabinovich–Fabrikant Equation"]

	~~{{Chaos theory}}~~

	~~{{DEFAULTSORT:Rabinovich Fabrikant Equations}}~~
	~~[[Category:Chaotic maps]]~~
	~~[[Category:Equations]~~]

en>R.e.b.: /* See also */ lk

2011-08-29T17:38:50Z

See also: lk

New page

[[File:RabFab800.png|thumb|upright]]
[[File:Rabinovich Fabrikant 2314.png|thumb|upright]]
[[File:Rabinovich Fabrikant 5212.png|thumb|upright]]
The '''Rabinovich–Fabrikant equations''' are a set of three coupled [[ordinary differential equations]] exhibiting [[Chaos theory|chaotic]] behavior for certain values of the [[parameter]]s. They are named after [[Mikhail Rabinovich]] and [[Anatoly Fabrikant]], who described them in 1979.

==System description==
The equations are:<ref name="Rabinovich"/>

: <math>\dot{x} = y (z - 1 + x^2) + \gamma x \, </math>
: <math>\dot{y} = x (3z + 1 - x^2) + \gamma y \, </math>
: <math>\dot{z} = -2z (\alpha + xy), \, </math>

where ''α'', ''γ'' are constants that control the evolution of the system. For some values of ''α'' and ''γ'', the system is chaotic, but for others it tends to a stable periodic orbit.

Danca and Chen<ref name="DancaChen"/> note that the Rabinovich–Fabrikant system is difficult to analyse (due to the presence of quadratic and cubic terms) and that different attractors can be obtained for the same parameters by using different step sizes in the integration.

===Equilibrium points===
[[File:Rabinovich-Fabrikant equilibrium point existence regions.svg|thumb|Graph of the regions for which equilibrium points <math>\tilde{\mathbf{x}}_{1,2,3,4}</math> exist.]]
The Rabinovich–Fabricant system has five hyperbolic [[equilibrium points]], one at the origin and four dependent on the system parameters ''α'' and ''γ'':<ref name="DancaChen"/>

: <math>\tilde{\mathbf{x}}_0 = (0,0,0)</math>
: <math>\tilde{\mathbf{x}}_{1,2} = \left( \pm q_-, - \frac{\alpha}{q_-}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_-^2 \right)</math>
: <math>\tilde{\mathbf{x}}_{3,4} = \left( \pm q_+, - \frac{\alpha}{q_+}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_+^2 \right)</math>

where

: <math>q_{\pm} = \sqrt{ \frac{ 1 \pm \sqrt{ 1- \gamma \alpha \left( 1- \frac{3 \gamma}{4\alpha} \right) } }{2 \left(1- \frac{3\gamma}{4\alpha}\right) }}</math>

These equilibrium points only exist for certain values of ''α'' and ''γ'' > 0.

===γ = 0.87, α = 1.1===
An example of chaotic behavior is obtained for ''γ'' = 0.87 and ''α'' = 1.1 with initial conditions of (−1, 0, 0.5).<ref name="SprottJC"/> The [[correlation dimension]] was found to be 2.19 ± 0.01.<ref name="Grassberger"/> The Lyapunov exponents, ''λ'' are approximately 0.1981, 0, −0.6581 and the [[Kaplan–Yorke dimension]], ''D''<sub>KY</sub> ≈ 2.3010<ref name="SprottJC"/>

===γ = 0.1===
Danca and Romera<ref name="DancaRomera"/> showed that for ''γ'' = 0.1, the system is chaotic for ''α'' = 0.98, but progresses on a stable [[limit cycle]] for ''α'' = 0.14.

[[File:Rabinovich-Fabrikant LimitCicle.PNG|thumb|3D parametric plot of the solution of the Rabinovich-Fabrikant equations for ''α''=0.14 and ''γ''=0.1 (limit cycle is showed by the red curve)]]

==See also==
* [[List of chaotic maps]]

==References==
<references>
<ref name="Grassberger">{{cite journal
| first1 = P.
| last1 = Grassberger
| first2 = I.
| last2 = Procaccia
| title = Measuring the strangeness of strange attractors
| journal = Physica D
| year = 1983
| volume = 9
| pages = 189–208
| doi = 10.1016/0167-2789(83)90298-1
| bibcode=1983PhyD....9..189G
}}</ref>
<ref name="Rabinovich">{{cite journal
| first1 = Mikhail I.
| last1 = Rabinovich
| first2 = A. L.
| last2 = Fabrikant
| title = Stochastic Self-Modulation of Waves in Nonequilibrium Media
| journal = Sov. Phys. JETP
| year = 1979
| volume = 50
| pages = 311
|bibcode = 1979JETP...50..311R }}</ref>
<ref name="SprottJC">{{cite book
| last = Sprott
| first = Julien C.
| title = Chaos and Time-series Analysis
| publisher = [[Oxford University Press]]
| year = 2003
| pages = 433
| isbn = 0-19-850840-9
}}</ref>
<ref name="DancaRomera">
{{cite journal
| first1 = Marius-F.
| last1 = Danca
| first2 = Miguel
| last2 = Romera
| title = Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems
| journal = Dynamics of Continuous, Discrete and Impulsive Systems
| series = Series B: Applications & Algorithms
| volume = 15
| year = 2008
| pages = 155–164
| publisher = Watam Press
| id = {{hdl|10261/8868}}
| issn = 1492-8760
}}</ref>
<ref name="DancaChen">
{{cite journal
| first1 = Marius-F.
| last1 = Danca
| first2 = Guanrong
| last2 = Chen
| title = Birfurcation and Chaos in a Complex Model of Dissipative Medium
| journal = International Journal of Bifurcation and Chaos
| volume = 14
| issue = 10
| year = 2004
| pages = 3409–3447
| publisher = World Scientiﬁc Publishing Company
| doi = 10.1142/S0218127404011430
|bibcode = 2004IJBC...14.3409D }}</ref>
</references>

==External links==
* Weisstein, Eric W. [http://mathworld.wolfram.com/Rabinovich-FabrikantEquation.html "Rabinovich–Fabrikant Equation."] From MathWorld—A Wolfram Web Resource.
*Chaotics Models a more appropriate approach to the chaotic graph of the system [http://www.robert-doerner.de/Glossar/glossar.html#R "Rabinovich–Fabrikant Equation"]

{{Chaos theory}}

{{DEFAULTSORT:Rabinovich Fabrikant Equations}}
[[Category:Chaotic maps]]
[[Category:Equations]]

← Older revision		Revision as of 19:30, 14 August 2014
Line 1:		Line 1:
	~~[[File:RabFab800.png\|thumb\|upright]]~~		I'm Brook and was born on 25 September 1985. My hobbies are Art collecting and Stone collecting.<br><br>my webpage; [http://allakro.org/?document_srl=439315 Sexy Nebraska Women]
	~~[[File:Rabinovich Fabrikant 2314.png\|thumb\|upright]]~~
	~~[[File:Rabinovich Fabrikant 5212.png\|thumb\|upright]]~~
	~~The '''Rabinovich–Fabrikant equations'~~'~~' are a set of three coupled [[ordinary differential equations]] exhibiting [[Chaos theory\|chaotic]] behavior for certain values of the [[parameter]]s. They are named after [[Mikhail Rabinovich]]~~ and ~~[[Anatoly Fabrikant]], who described them in 1979~~.

	~~==System description==~~
	~~The equations~~ are~~:<ref name="Rabinovich"/>~~

	~~: <math>\dot{x} = y (z - 1 + x^2) + \gamma x \, </math>~~
	~~: <math>\dot{y} = x (3z + 1 - x^2) + \gamma y \, </math>~~
	~~: <math>\dot{z} = -2z (\alpha + xy), \, </math>~~

	~~where ''α'', ''γ'' are constants that control the evolution of the system. For some values of ''α''~~ and ~~''γ'', the system is chaotic, but for others it tends to a stable periodic orbit~~.

	~~Danca and Chen~~<~~ref name="DancaChen"/~~> note that the Rabinovich–Fabrikant system is difficult to analyse (due to the presence of quadratic and cubic terms) and that different attractors can be obtained for the same parameters by using different step sizes in the integration.

	~~===Equilibrium points===~~
	~~[[File:Rabinovich-Fabrikant equilibrium point existence regions.svg\|thumb\|Graph of the regions for which equilibrium points~~ <~~math~~>~~\tilde{\mathbf{x}}_{1,2,3,4}</math> exist.]]~~
	~~The Rabinovich–Fabricant system has five hyperbolic [[equilibrium points]], one at the origin and four dependent on the system parameters ''α'' and ''γ'':<ref name="DancaChen"/>~~

	~~: <math>\tilde{\mathbf{x}}_0 = (0,0,0)</math>~~
	~~: <math>\tilde{\mathbf{x}}_{1,2} = \left( \pm q_-, - \frac{\alpha}{q_-}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_-^2 \right)</math>~~
	~~: <math>\tilde{\mathbf{x}}_{3,4} = \left( \pm q_+, - \frac{\alpha}{q_+}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_+^2 \right)</math>~~

	~~where~~

	~~: <math>q_{\pm} = \sqrt{ \frac{ 1 \pm \sqrt{ 1- \gamma \alpha \left( 1- \frac{3 \gamma}{4\alpha} \right) } }{2 \left(1- \frac{3\gamma}{4\alpha}\right) }}</math>~~

	~~These equilibrium points only exist for certain values of ''α'' and ''γ'' > 0.~~

	~~===γ = 0.87, α = 1.1===~~
	~~An example of chaotic behavior is obtained for ''γ'' = 0.87 and ''&alpha~~;~~'' = 1.1 with initial conditions of (−1, 0, 0.5).<ref name="SprottJC"/> The~~ [[correlation dimension]] was found to be 2.19 ± 0.01.<ref name="Grassberger"/> The Lyapunov exponents, ''λ'' are approximately 0.1981, 0, −0.6581 and the [[Kaplan–Yorke dimension]], ''D''<sub>KY</sub> ≈ 2.3010<ref name="SprottJC"/>

	~~===γ = 0.1===~~
	~~Danca and Romera<ref name="DancaRomera"/> showed that for ''γ'' = 0.1, the system is chaotic for ''α'' = 0.98, but progresses on a stable [[limit cycle]] for ''α'' = 0.14.~~

	~~[[File:Rabinovich-Fabrikant LimitCicle.PNG\|thumb\|3D parametric plot of the solution of the Rabinovich-Fabrikant equations for ''α''=0.14 and ''γ''=0.1 (limit cycle is showed by the red curve)]]~~

	~~==See also==~~
	* [[List of chaotic maps]]

	~~==References==~~
	~~<references>~~
	~~<ref name="Grassberger">{{cite journal~~
	~~\| first1 = P.~~
	~~\| last1 = Grassberger~~
	~~\| first2 = I.~~
	~~\| last2 = Procaccia~~
	~~\| title = Measuring the strangeness of strange attractors~~
	~~\| journal = Physica D~~
	~~\| year = 1983~~
	~~\| volume = 9~~
	~~\| pages = 189–208~~
	~~\| doi = 10.1016/0167-2789(83)90298-1~~
	~~\| bibcode=1983PhyD....9..189G~~
	~~}}</ref>~~
	~~<ref name="Rabinovich">{{cite journal~~
	~~\| first1 = Mikhail I.~~
	~~\| last1 = Rabinovich~~
	~~\| first2 = A. L.~~
	~~\| last2 = Fabrikant~~
	~~\| title = Stochastic Self-Modulation of Waves in Nonequilibrium Media~~
	~~\| journal = Sov. Phys. JETP~~
	~~\| year = 1979~~
	~~\| volume = 50~~
	~~\| pages = 311~~
	~~\|bibcode = 1979JETP...50..311R }}</ref>~~
	~~<ref name="SprottJC">{{cite book~~
	~~\| last = Sprott~~
	~~\| first = Julien C.~~
	~~\| title = Chaos and Time-series Analysis~~
	~~\| publisher = [[Oxford University Press]]~~
	~~\| year = 2003~~
	~~\| pages = 433~~
	~~\| isbn = 0-19-850840-9~~
	~~}}</ref>~~
	~~<ref name="DancaRomera">~~
	~~{{cite journal~~
	~~\| first1 = Marius-F.~~
	~~\| last1 = Danca~~
	~~\| first2 = Miguel~~
	~~\| last2 = Romera~~
	~~\| title = Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems~~
	~~\| journal = Dynamics of Continuous, Discrete and Impulsive Systems~~
	~~\| series = Series B~~: ~~Applications & Algorithms~~
	~~\| volume = 15~~
	~~\| year = 2008~~
	~~\| pages = 155–164~~
	~~\| publisher = Watam Press~~
	~~\| id = {{hdl\|10261~~/~~8868}}~~
	~~\| issn = 1492-8760~~
	~~}}</ref>~~
	~~<ref name="DancaChen">~~
	~~{{cite journal~~
	~~\| first1 = Marius-F.~~
	~~\| last1 = Danca~~
	~~\| first2 = Guanrong~~
	~~\| last2 = Chen~~
	~~\| title = Birfurcation and Chaos in a Complex Model of Dissipative Medium~~
	~~\| journal = International Journal of Bifurcation and Chaos~~
	~~\| volume = 14~~
	~~\| issue = 10~~
	~~\| year = 2004~~
	~~\| pages = 3409–3447~~
	~~\| publisher = World Scientiﬁc Publishing Company~~
	~~\| doi = 10.1142~~/~~S0218127404011430~~
	~~\|bibcode = 2004IJBC..~~.~~14.3409D }}</ref>~~
	</~~references>~~

	==~~External links==~~
	* Weisstein, Eric W. [http://mathworld.wolfram.com/Rabinovich-FabrikantEquation.html "Rabinovich–Fabrikant Equation."] From MathWorld—A Wolfram Web Resource.
	*Chaotics Models a more appropriate approach to the chaotic graph of the system [http://www.robert-doerner.de/Glossar/glossar.html#R "Rabinovich–Fabrikant Equation"]

	~~{{Chaos theory}}~~

	~~{{DEFAULTSORT:Rabinovich Fabrikant Equations}}~~
	~~[[Category:Chaotic maps]]~~
	~~[[Category:Equations]~~]