Rabinovich–Fabrikant equations

The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behavior for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 1979.

System description

The equations are:^[1]

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

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

{\dot {z}}=-2z(\alpha +xy),\,

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^[2] 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

Graph of the regions for which equilibrium points

{\tilde {{\mathbf {x}}}}_{{1,2,3,4}}

exist.

The Rabinovich–Fabricant system has five hyperbolic equilibrium points, one at the origin and four dependent on the system parameters α and γ:^[2]

{\tilde {{\mathbf {x}}}}_{0}=(0,0,0)

{\tilde {{\mathbf {x}}}}_{{1,2}}=\left(\pm q_{-},-{\frac {\alpha }{q_{-}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{-}^{2}\right)

{\tilde {{\mathbf {x}}}}_{{3,4}}=\left(\pm q_{+},-{\frac {\alpha }{q_{+}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{+}^{2}\right)

where

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)}}}}

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).^[3] The correlation dimension was found to be 2.19 ± 0.01.^[4] The Lyapunov exponents, λ are approximately 0.1981, 0, −0.6581 and the Kaplan–Yorke dimension, D_KY ≈ 2.3010^[3]

γ = 0.1

Danca and Romera^[5] 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.

3D parametric plot of the solution of the Rabinovich-Fabrikant equations for α=0.14 and γ=0.1 (limit cycle is shown by the red curve)

References

↑ Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). "Stochastic Self-Modulation of Waves in Nonequilibrium Media". Sov. Phys. JETP. 50: 311. Bibcode:1979JETP...50..311R.
1 2 Danca, Marius-F.; Chen, Guanrong (2004). "Birfurcation and Chaos in a Complex Model of Dissipative Medium". International Journal of Bifurcation and Chaos. World Scientiﬁc Publishing Company. 14 (10): 3409–3447. Bibcode:2004IJBC...14.3409D. doi:10.1142/S0218127404011430.
1 2 Sprott, Julien C. (2003). Chaos and Time-series Analysis. Oxford University Press. p. 433. ISBN 0-19-850840-9.
↑ Grassberger, P.; Procaccia, I. (1983). "Measuring the strangeness of strange attractors". Physica D. 9: 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.
↑ Danca, Marius-F.; Romera, Miguel (2008). "Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems". Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications & Algorithms. Watam Press. 15: 155–164. ISSN 1492-8760. hdl:10261/8868.

External links

Weisstein, Eric W. "Rabinovich–Fabrikant Equation." From MathWorld—A Wolfram Web Resource.
Chaotics Models a more appropriate approach to the chaotic graph of the system "Rabinovich–Fabrikant Equation"

Chaos theory

Chaos theory	Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences

Chaotic maps (list)	Arnold tongue Arnold's cat map Baker's map Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Double scroll attractor Duffing equation Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Multiscroll attractor Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map

Chaos systems	Bouncing ball dynamics Chua's circuit Economic bubble Tilt-A-Whirl

Chaos theorists	Michael Berry Leon O. Chua Mitchell Feigenbaum Martin Gutzwiller Brosl Hasslacher Michel Hénon Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Henri Poincaré Otto Rössler David Ruelle Oleksandr Mykolaiovych Sharkovsky Floris Takens James A. Yorke

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