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In this work, we study the integrability, as well as the dynamics of the Lorenz System. This include a very useful identity:\[ β z^2(σ t)+y^2(βσ t)=ρ x^2(β t)+ν e-2βσ t, \]where ν is a constant. And we will see some applications of this identity.
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Yiting Yao (2024) studied this question.
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