Global analytic integrability of the Rabinovich system

Abstract

a b s t r a c tThe Rabinovich system can be written asẋ = hy − v 1 x + yz,ẏ = hx − v 2 y − xz anḋ z = −v 3 z + xy with h, v 1 , v 2 and v 3 being real parameters. In this paper we characterize its global analytic first integrals.

Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

“…In this section, we also recall some geometrical properties of the system (1) [12]. The global analytic first integrals of the Rabinovich system are obtained in [40].…”

“…Therefore, by applying the same procedure it obtains that the expression N (u 0 (t)) is a combination of the elementary functions cos(ω 0 t), sin(ω 0 t), cos(3ω 0 t), sin(3ω 0 t) in the both cases. So, the first approximation u 1 (t) has the form by Equation ( 39) and the first-order approximate analytic solution ū(t) has the form by Equation (40).…”

Unraveling the Chaos-Land and Its Organization in the Rabinovich System

“…Quantities h, ν 1 and ν 2 are the parameters of the system: the value of h is proportional to the pump field, whereas ν 1 and ν 2 are the normalized damping decrements in the parametrically excited waves k and κ, respectively. After the original investigation in [8], the studies of this system have further been continued in [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], see also the contribution by S. Kuznetsov in this volume [25].…”

“…Quantities h, ν 1 and ν 2 are the parameters of the system: the value of h is proportional to the pump field, whereas ν 1 and ν 2 are the normalized damping decrements in the parametrically excited waves k and κ, respectively. After the original investigation in [8], the studies of this system have further been continued in [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], see also the contribution by S. Kuznetsov in this volume [25]. Although initial numerical simulations have revealed the presence of a Lorenz-like chaotic behavior in the Rabinovich system, the exact boundaries of static, periodic and chaotic dynamics in the parametric space have not been identified.…”

On Asymptotically Stabilizing the Rabinovich Dynamical System

“…Among the studied topics related to the Rabinovich system, we recall a few of them together with a partial list of references, namely: integrals and invariant manifolds [25,26], analytic integrability [16], time delay [8], dynamical description on the Poincaré ball [15], topological structure [7], Hamiltonian dynamics [4,9,12,23], and many others.…”