On Deterministic Chaos: Disclosing the Missing Mathematics from the Lorenz-Haken Equations
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On Deterministic Chaos: Disclosing the Missing Mathematics from the Lorenz-Haken Equations

Authors: Belkacem Meziane

Abstract:

The original 3D Lorenz-Haken equations -which describe laser dynamics- are converted into 2-second-order differential equations out of which the so far missing mathematics is extracted. Leaning on high-order trigonometry, important outcomes are pulled out: A fundamental result attributes chaos to forbidden periodic solutions, inside some precisely delimited region of the control parameter space that governs self-pulsing.

Keywords: chaos, Lorenz-Haken equations, laser dynamics, nonlinearities

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References:


[1] E. N. Lorenz, “Deterministic nonperiodic flow”, J. Atmos. Sci. 20 1963, 130.
[2] H. Haken, “Analogy between higher instabilities in fluids and lasers”, Phys. Lett. A 53, 1975, 77.
[3] S. Smale “Mathematical problems for the next century” Math. Intell. 20, 1998 7–15.
[4] B. Meziane, “Isomorphic transformation of the Lorenz equations into a single-control-parameter structure”, Int. J. Eng. Res. Sci. 2, 2016, 70-8.
[5] B. Meziane, “Lorenz-Haken dynamics-analytical framework: from symmetric to asymmetric trajectories”, Phys. Scr. 94, 2019, 125217.