1. Networks of Passive Oscillators
Vishwesh V. Kulkarni, Marc Riedel, and Guy-Bart Stan

2. Outline
Passive oscillators using passive systems and static nonlinearities
Modified MIMO Lure’ system
Supercritical Hopf/Pitchfork bifurcation
Networks of such passive oscillators
Identical or non-identical
How to choose the interconnection ?
We establish a class of suitable - - -

3. Passive Oscillator - Stable, LTI Static Stiffening Nonlinearity
If the system possesses a globally stable limit cycle that attracts all solutions except those in the origin’s stable manifold.

4. Main Problem: Oscillator Network- -
Given: ODE model specifying and
Qn: How to choose the coupling so that is oscillatory?

5. Passivity and Dissipativity

6. Dissipativity w.r.t. a Special w(u,y)
supply
local activation
global dissipation
For example:

7. Loop-Shift Transformed System
where +
Find a positivity preserving multiplier
& finite normed N  stability. … Zames-Falb (1968)

8. Zames-Falb MultipliersIm
s-plane Re

9. Passive Oscillator -
If the system possesses a stable global limit cycle that attracts
all solutions except those in the stable origin’s manifold.

10. Bifurcation for Oscillations: Hopf
Global oscillations through Hopf bifurcation for if there exists a ZF multiplier such that
is strongly passive for ; and
has two eigenvalues on the axis at
s-plane X
stable
unstable X
(Stan-Sepulchre, 2007)

11. Bifurcation for Oscillations: Pitchfork
Global oscillations through pitchfork bifurcation for
if there exists a ZF multiplier such that
is strongly passive for ; and
has an eigenvalue on the axis at
Fitzhuh-Nagumo Oscillator:
… slow adaption added to
enforce the relaxation phase
(Stan-Sepulchre, 2007)

12. Networks of Passive Oscillators-
Forcing
Input
Oscillatory
Output
Network of oscillators
i-th Oscillator
If the system possesses a stable global limit cycle that attracts all solutions except those in the stable origin’s manifold.

13. Feedback System Representation- -
Given: ODE model specifying and
Qn: How to choose the coupling so that is oscillatory?

14. Redrawn Oscillator Network- -
repeated static monotone
L2-stability of this system using relevant multipliers
Substitution of those conditions in the Stan-Sepulchre results

15. KS Multipliers is a positive operator (Kulkarni-Safonov, 2002)
Impulse response of a KS multiplier is given by
is a positive operator (Kulkarni-Safonov, 2002)
Hence, is L2-stable if a KS multiplier s.t.

16. Networked Oscillators: Bifurcation

17. Networked Oscillations

18. Summary
Passive oscillators using passive systems and static nonlinearities
Modified MIMO Lure’ system
Supercritical Hopf/Pitchfork bifurcation
Networks of such passive oscillators
How to choose the interconnection ?
We establish a class of suitable
Direct extension of Stan-Sepulchre
Asymmetry & non-identical oscillators considered
We hope to reduce the global results to local results - -

19. Questions?

20. Thank You!
Research Supported by NSF CAREER Award