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| The main goal on the current work is the proof of the existence of a geometrical Lorenz attractor in the Lorenz-Yudovich model:
$$
\begin{array}
\dot X = Y, \\
\dot Y = X - \lambda Y - X Z - X^3, \\
\dot Z = -\alpha Z + B X^2,
\end{array}
$$
where parameters $(\alpha, \lambda, B)$ can take arbitrary finite values.
The proof is based on the Shilnikov criterion [1]. According to it, in a system possessing two homoclinic orbits that tend to the same saddle equilibrium tangent to each other (a homoclinic butterfly-figure-eight) a Lorenz attractor is born if a saddle value $\sigma$ is zero and a separatrix value $A$ satisfies inequalities $0 < |A| < 2$ and $|A| \neq 1$.
For the Lorenz-Yudovich model the separatrix value was calculated and the existence of Lorenz attractor was established near the integrable case $(\alpha, \lambda, B) = (1, 0, 0)$.
[1] Shilnikov L.P. Theory of bifurcations and quasihyperbolic attractors. Rus. Math. Surv., 36:4, 1981. |
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