Trajectory of a solution with parameter values and and initial conditions , , and , using the default ODE solver in MATLAB. Colors vary from blue to yellow with time.Trajectory of a solution with parameter values and and initial conditions , , and , using the default ODE solver in Mathematica. Colors vary from orange-red to magenta-red with time. Notice the drastic change in the solutions with respect to the solution obtained with MATLAB.A chaotic attractor found with parameter values and and initial conditions , , and , using the default ODE solver in Mathematica. Colors vary from orange-red to magenta-red with time. Notice that colors do not follow any order, reflecting the chaotic dynamics of the solution.
The Rabinovich–Fabrikant equations are a set of three coupled
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
hidden attractor was discovered in the Rabinovich–Fabrikant system.[3]
Equilibrium points
Graph of the regions for which equilibrium points exist.
The Rabinovich–Fabrikant system has five hyperbolic
equilibrium points, one at the origin and four dependent on the system parameters α and γ:[2]
where
These equilibrium points only exist for certain values of α and γ > 0.
γ = 0.87, α = 1.1
An example of chaotic behaviour is obtained for γ = 0.87 and α = 1.1 with initial conditions of (−1, 0, 0.5),[4] see trajectory on the right. The correlation dimension was found to be 2.19 ± 0.01.[5] The Lyapunov exponents, λ are approximately 0.1981, 0, −0.6581 and the Kaplan–Yorke dimension, DKY ≈ 2.3010[4]
γ = 0.1
Danca and Romera[6] 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)
^ ab
Danca, Marius-F.; Chen, Guanrong (2004). "Birfurcation and Chaos in a Complex Model of Dissipative Medium". International Journal of Bifurcation and Chaos. 14 (10). World Scientific Publishing Company: 3409–3447.
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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. 15. Watam Press: 155–164.
