1. e-mail: belkacem.meziane@univ-artois.fr
UNITE DE CATALYSE ET DE CHIMIE DU SOLIDE *- UMR CNRS 8181
Predicting the unpredictable with an isomorphic single-control parameter structure of the Laser-Lorenz equations
Belkacem Meziane
Université d’Artois, Faculté des Sciences Jean Perrin, Rue Jean Souvraz,
SP18, 62307, Lens Cedex, France

3. The Lorenz-Haken equations
Lamb self-consistent analysis
P(t) (source) Maxwell’s equations Field E(t)
Schrödinger equation
Excitation parameter 2C quantifies the pumping mechanism with respect to its level at lasing threshold, and
Normalized quantities

4. = 2C > Back to Lorenz notations and main characteristics (2a) (2b)
Where
= 2C
Instability threshold Bad-cavity condition
>
→ Unstable solutions

5. Three control parameters ( , , )
govern the dynamics of Eqs (2)
Ex = 10 , = 8/3, = Lorenz attractor
= 3, = 0, = 10
Period 1 limit cycle
=3 , = 0,15, = 11
Period-doubling

6. Self-organized solution-recurrence
Period 1
Period 2

7. Chaotic attractor
defines an isomorphic class of solutions !
Including the particular set = , =1

8. ( , , ) ( , )
Furthermore, simulations reveal :
( , )
( , )

9. Simulations far above the instability threshold
Simulations at the instability threshold, obtained with other values

10. Predicting the unpredictible
Dynamic chart 1

11. Dynamic chart 2

12. Dynamic chart 3

13. Conclusion
Lorenz Equations
ISOMORPHISM
3 control parameters single control parameter
, ,

14. Solutionnnnnn