1. Driven autoresonant three-oscillator interactions Oded Yaakobi 1,2 Lazar Friedland 2 Zohar Henis 1 1 Soreq Research Center, Yavne, Israel. 2 The Hebrew University of Jerusalem, Jerusalem, Israel. Email: yaakobio@yahoo.com O. Yaakobi, L. Friedland and Z. Henis, Phys. Rev. E (accepted).

2. Three waves interactions Plasma physics –Laser plasma interactions: »Stimulated Brillouin Scattering (SBS) »Stimulated Raman Scattering (SRS) Nonlinear optics –Optical Parametric Amplifier/Generator (OPA/OPG) –Brillouin scattering, Raman scattering Hydrodynamics Acoustic waves Frequencies matching (energy): Wave vectors matching (momentum): Controlling three waves interactions is an important goal of both basic and applied physics research.

3. Three oscillators interactions

4. Research goal: Study a control scheme of three oscillators interactions using an external drive. Definitions:

5. Threshold phenomena

6. Adiabatic approximation Definitions:Approximated equations neglecting : Nonlinear frequency shift

7. Small nonlinear frequency shift Assumption: Approximated equations: Definitions: Small nonlinear frequency shift Range of validity

8. Autoresonant quasi steady state Assumptions: Quasi steady state: Small nonlinear frequency shift & quasi steady state

9. Threshold analysis Quasi-steady-state asymptotic result: Constraint: Asymptotic phase mismatches:

10. Threshold analysis Dimensional equations: Necessary condition:

11. Threshold analysis Necessary condition for autoresonant quasi steady state: Computed threshold (numerical)

12. Linear frequencies mismatch

13. Dissipation Small nonlinear frequency shift: Necessary condition: Exponential decay:

14. Phase mismatches deviations

15. Linearization Quasi-steady-state: Expansion: Assumptions: Exact equations:

16. Linearization Linearized equations: Differentiating with respect to :

17. Numerical results comparison

18. WKB approximation

19. First order terms satisfy:

20. Singular value decomposition

21. First order terms satisfy: Multiplying with :

22. Asymptotic form of matrices

23. Quasi steady state stability Small deviations from the quasi-steady-state do not increase with time.

24. Quasi steady state stability Small deviations from the quasi steady state do not increase with time.

25. Small nonlinear frequency shift & quasi steady state Large nonlinear frequency shift Assumptions:

26. Conclusions Controlling three oscillators interactions using autoresonance is demonstrated. Analytic expressions for autoresonant time dependent amplitudes are obtained. Conditions for autoresonant trapping are analyzed in terms of coupling parameter, driving parameter, dissipation and linear frequencies mismatch. The autoresonant quasi-steady state is linearly stable.

27. Outlook Generalization of the theory to driven three-wave interactions is of interest.