1) The document discusses a nonlinear microwave oscillator system that can exhibit chaotic behavior based on a time-delayed feedback loop architecture operating in the 2-4 GHz frequency band. 2) It presents the experimental setup of the system using a voltage controlled oscillator (VCO), splitter, mixer, and delay line. Mathematical models are developed to simulate the system dynamics. 3) Experiments show the system can transition from periodic to chaotic behavior as a nonlinearity parameter is increased. Synchronization is also achieved by bidirectionally coupling two similar microwave oscillator systems.

1. Nonlinear Microwave Oscillators: Dynamics and Synchronization Hien Dao (Chemical Physics Program) John C. Rodgers (IREAP) Thomas E. Murphy (ECE & IREAP)

5. VCO splitter  d mixer   Bias at operating point H ( s ) low pass filter gain delay Experimental setup

6. Voltage Controlled Oscillator (VCO) is a device that converts an input analog voltage into a signal whose frequency is linearly proportional to the magnitude of voltage VCO Tuning signal v (t) RF signal and with  is named tuning sensitivity (VCO gain) and  0 is bias frequency. Slowly varying phase  0 =2.65 GHz  =175 MHz/v

7. v VCO  d Mixer Splitter delay Nonlinear function is created using delay-line frequency discriminator output

8. VCO Splitter  d Mixer bidirectional coupler combiner

9. VCO splitter  d mixer   Bias at operating point H ( s ) low pass filter gain delay Experimental setup

11. Mathematical model for tuning signal system equation  H (s)  nonlinearity gain delay low pass filter v(t) f cutoff =3 MHz  s  1.2  s  Varying from 0.5-9.5  5 ns   175 MHz/V  Value Parameters

12.  =1.6  =2.2  =6.5 Experiment Simulation

13. Phase portrait of system plotting phase of envelope signal versus its derivative can tell us about dynamics of system  =1.6  =2.2  =6.5

14. Simulation Experiment Bifurcation diagram of system Vtune [V] Vtune [V] -2 -1 2 1 0 -2 -1 2 1 0 1 2 3 4 6 5 7 

15. Vtune [V] Vtune [V] increasing  decreasing  Historesis effect 1 2 3 4 6 5 7  -2 -1 2 1 0 -2 -1 2 1 0

16. Vtune [V] Maximum Lyapunov exponent -2 -1 2 1 0 1 2 3 4 6 5 7  -0.2 0.4 0.2 -0.1 0.1 0 0.3 0.5

17. Synchronization of coupled microwave oscillators

18. Chaotic synchronization had been achieved by coupling two optoelectronic systems. How to couple two microwave systems and what kind of synchronization we should observe? x 1 (t): x 1 (t) – x 2 (t): x 2 (t):

19. Two systems are coupled bi-directionally in microwave band, VCO Splitter  d Mixer   Bias VCO Splitter  d Mixer   Bias H(s) H(s)  is coupling strength v 1 (t) v 2 (t)  Behavior depends on whether the VCO difference frequency exceeds the filter bandwidth

21. RF signal collected from scope Analytic signal Hilbert transform Analytic signal Where is Hilbert transform of Using Hilbert transform to estimate phase RF signal Constant amplitude Phase varies around a bias value  0

22. VCO Tuning signal RF signal  =1.2 and  =0.1

23. t (  s) 0 100 50 200 250 150 0 100 200

24. VCO Tuning signal RF signal  =2.1 and  =0.1

25. 0 3000 6000 0 100 50 200 250 150 t (  s)