The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

 behavior for certain values of the parameter

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

s. They are named after Mikhail Rabinovich

Mikhail Rabinovich

Mikhail Izrailevich Rabinovich is an influential physicist and neuroscientist working in the field of nonlinear dynamics and its applications...

 and Anatoly Fabrikant, who described them in 1979.

System description

The equations are:

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.

Equilibrium points

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

where

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). The correlation dimension

Correlation dimension

In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal 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

γ = 0.1

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.

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"