Rabinovich–Fabrikant equations

The equations are:

: $\backslash dot\{x\}\; =\; y\; (z\; -\; 1\; +\; x^2)\; +\; \backslash gamma\; x\; \backslash ,$

: $\backslash dot\{y\}\; =\; x\; (3z\; +\; 1\; -\; x^2)\; +\; \backslash gamma\; y\; \backslash ,$

: $\backslash dot\{z\}\; =\; -2z\; (\backslash alpha\; +\; xy),\; \backslash ,$

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 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, see on the right an example of a solution obtained by two different solvers for the same parameter values and initial conditions. Also, recently, a hidden attractor was discovered in the Rabinovich–Fabrikant system.{{cite journal

|author1=Danca M.-F. |author2=Kuznetsov N. |author3=Chen G. |

year = 2017 |

title = Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system |

journal = Nonlinear Dynamics |

volume = 88 |

issue = 1 |

pages = 791–805 |

doi = 10.1007/s11071-016-3276-1|arxiv=1511.07765 |s2cid=119303488 }}

File:Rabinovich-Fabrikant equilibrium point existence regions.svg

The Rabinovich–Fabrikant system has five hyperbolic equilibrium points, one at the origin and four dependent on the system parameters α and γ:

: $\backslash tilde\{\backslash mathbf\{x\}\}\_0\; =\; (0,0,0)$

: $\backslash tilde\{\backslash mathbf\{x\}\}\_\{1,2\}\; =\; \backslash left(\; \backslash pm\; q\_-,\; \backslash mp\; \backslash frac\{\backslash alpha\}\{q\_-\},\; 1-\; \backslash left(1-\backslash frac\{\backslash gamma\}\{\backslash alpha\}\backslash right)q\_-^2\; \backslash right)$

: $\backslash tilde\{\backslash mathbf\{x\}\}\_\{3,4\}\; =\; \backslash left(\; \backslash pm\; q\_+,\; \backslash mp\; \backslash frac\{\backslash alpha\}\{q\_+\},\; 1-\; \backslash left(1-\backslash frac\{\backslash gamma\}\{\backslash alpha\}\backslash right)q\_+^2\; \backslash right)$

where

: $q\_\{\backslash pm\}\; =\; \backslash sqrt\{\; \backslash frac\{\; 1\; \backslash pm\; \backslash sqrt\{\; 1-\; \backslash gamma\; \backslash alpha\; \backslash left(\; 1-\; \backslash frac\{3\; \backslash gamma\}\{4\backslash alpha\}\; \backslash right)\; \}\; \}\{2\; \backslash left(1-\; \backslash frac\{3\backslash gamma\}\{4\backslash alpha\}\backslash right)\; \}\}$

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

An example of chaotic behaviour is obtained for γ = 0.87 and α = 1.1 with initial conditions of (−1, 0, 0.5), see trajectory on the right. The correlation dimension was found to be 2.19 ± 0.01. The Lyapunov exponents, λ are approximately 0.1981, 0, −0.6581 and the Kaplan–Yorke dimension, D_KY ≈ 2.3010

Danca and Romera 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

• List of chaotic maps

{{cite journal

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| journal = Physica D

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| pages = 189–208

| doi = 10.1016/0167-2789(83)90298-1

| bibcode=1983PhyD....9..189G

}}

{{cite journal

| first1 = Mikhail I.

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| 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 }}

{{cite book

| last = Sprott

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| title = Chaos and Time-series Analysis

| publisher = Oxford University Press

| year = 2003

| pages = 433

| isbn = 0-19-850840-9

}}

{{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

| hdl = 10261/8868

| issn = 1492-8760

}}

{{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 }}

• 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