1. Delayed feedback control of chaos: bifurcation analysis N. Janson (Loughborough) Collaborators: A. Balanov and E. Schöll (TUB) Technische Universität Berlin Loughborough University

2. Methods for control of deterministic chaos a continuous external perturbation R. Lima et. al., Phys. Rev. A 41 (1990) 726. a time-discrete conditioned perturbation (OGY) E. Ott et al., Phys. Rev. Lett. 64 (1990) 1196. delayed feedback loop K. Pyragas, Phys. Lett. A 170 (1992) 421. a continuous external perturbation R. Lima et. al., Phys. Rev. A 41 (1990) 726. a time-discrete conditioned perturbation (OGY) E. Ott et al., Phys. Rev. Lett. 64 (1990) 1196. delayed feedback loop K. Pyragas, Phys. Lett. A 170 (1992) 421. Three main methods :

3. Delayed feedback control of chaos Delayed feedback control of chaos = =T =T How does it work?

4. Chaos control: Example I Chaos control: Example I S. Bielawski et al, Phys. Rev. E 49 (1994) R971 CO 2 laser without control with control EXPERIMENT

5. Electrochemical oscillators P. Paramanade et. al., Phys. Rev. E 59 (1999) 5266 control no control Chaos control: Example II Chaos control: Example II EXPERIMENT

6. Chaos control: Example III Chaos control: Example III NUMERICAL RESULTS G. Franceschini et al, Phys. Rev. E 60 (1999) 5426 Semiconductor structure no control control

7. Chaos control: Example IV Chaos control: Example IV EXPERIMENT C. Beta et. al., Phys. Rev. E 67 (2003) 046224 Catalytic CO oxidation on platinum control no control

8. Delayed feedback control of chaos: main known facts. Delayed feedback control of chaos: main known facts. (more than 700 citations) K. Pyragas, Phys. Lett. A 170, 421 (1992) 1) At fixed = T, there is a range of K for which the orbit is stabilized. 2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized. 3) Feedback can induce multistability and death of oscillations M.E. Bleich et al, Phys. Lett. A 210, 87 (1996) W. Just et. al., Phys. Rev. Lett. 203, 78 (1997) H. Nakajima, Phys. Lett. A 232, 207 (1997) W. Just et al, Phys. Lett A 254, 158 (1999) limitations of delayed feedback W. Just et. al., Phys. Rev. Lett. 8, 562 (1998) What if is not exactly equal to T, but close to it? 1)The orbit changes its shape and period. 2)A method to estimate the orbit period from the given two values of.

9. General bifurcation picture in the plane of parameters of delayed feedback K - K -

10. System for the study System for the study Rössler system a=0.2, b=0.2, m=6.5 period-one limit cycle period-two limit cycle Unstable orbits:

11. Bifurcation diagram (main result) Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

12. Multistability and hysteretic transitions Multistability and hysteretic transitions 1D bifurcation diagrams

13. Multistability and hysteretic transitions Multistability and hysteretic transitions Control force increases increases decreases decreases

14. General structure of bifurcation diagram General structure of bifurcation diagram Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

15. Hopf bifurcation stable fixed point Multi-leaf structure Multi-leaf structure Size of period-1 orbit Orbit is stable Orbit is unstable

16. Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos Bifurcation diagram: Leaf #0 Bifurcation diagram: Leaf #0

17. Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos Bifurcation diagram: Leaf #1 Bifurcation diagram: Leaf #1

18. Bifurcation diagram: Leaf #2 Bifurcation diagram: Leaf #2 Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

19. Leaf #2 : transition to chaos =13.5 =13.5 Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

20. Bifurcation diagram: Leaf #3 Bifurcation diagram: Leaf #3 Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

21. Bifurcation diagram: Leaf #4 Bifurcation diagram: Leaf #4 Hopf bifurcation period-doubling bifurcation crisis of attractors subcritical Neimark-Sacker bifurcation supercritical Neimark-Sacker bifurcation stable fixed pointchaos

22. Period of period-1 periodic orbit for any ( I) Period of period-1 periodic orbit for any ( I) In W. Just et. al., Phys. Rev. Lett. 8, 562 (1998) it is shown: (K, ) – period of the resulting cycle in the system under control (K, ) – period of the resulting cycle in the system under control T – period of the unperturbed orbit T – period of the unperturbed orbit – a parameter characterizing the effect of the control force on the system – a parameter characterizing the effect of the control force on the system dynamics for the K or dynamics for the particular orbit. Does not depend on K or - derivative of (K, ) over - derivative of (K, ) over Facts: 1. (K,T)=T 2.Control force F(t)=x(t- )-x(t)=0 not only for =T, but also for any =nT, n =1,2,… 3.Thus, (K,nT)=T

23. Period of period-1 periodic orbit for any ( II) =nT Substitute =nT into: Take into account the facts from the previous slide: And obtain an expression for the derivative of with respect to :

24. Expand into a Taylor series around =nT : Period of period-1 periodic orbit for any ( III) Substitute this into (*): (*) and obtain an approximate expression for the period-1 orbit of the system under control for the case of close to nT : (**)

25. Period of period-m periodic orbit for any Period of period-m periodic orbit for any Thus, for the cycle of the period mT, m=1,2,… Facts: 1.It is not likely that a period-m orbit has a period exactly mT. 2.However, it is likely that the period of period-m orbit is close to mT. (K,mT)=T (K,mT)=T 4.For a different orbit, will be different. 5.Observation: for a period- m orbit, m becomes m. 6.Eq. (**) should hold for period-m orbits approximately.

26. K=0.13 K=0.8 K=2.0 by formula numerically Period of period-1 orbits: numerics vs analytics Period of period-1 orbits: numerics vs analytics n - the number of the leaf of the bifurcation diagram. Numerical fit for is 0.35.

27. K=0.8 by formula numerically Period of period-2 and period-4 orbits: Period of period-2 and period-4 orbits: numerics vs analytics m=2; n=0,1,2 period-2 orbit m=4; n=0,1 period-4 orbit n - the number of the leaf of the bifurcation diagram m – the integer period of the periodic orbit n=0 n=1n=2 n=0 n=1 K=2.0 Numerical fit for is 0.35.

28. Limit cycles classification Limit cycles classification n

29. Main conclusions: Main conclusions: The bifurcation diagram in (,K) parameter space has essentially The bifurcation diagram in (,K) parameter space has essentially multi-leaf structure The lager the period of the orbit, the smaller the domain of its The lager the period of the orbit, the smaller the domain of its stabilization is. The same limit cycle can have several domains of stability The same limit cycle can have several domains of stability Increase of both and K leads to severe multistability Increase of both and K leads to severe multistability A.G. Balanov, N.B. Janson, E. Scholl Delayed feedback control of chaos: Bifurcation analysis, Phys. Rev. E 71, 016222 (2005).

30. Delayed feedback control of chaos: bifurcation analysis N. Janson (Loughborough) Collaborators: A. Balanov and E. Schöll (TUB) Technische Universität Berlin Loughborough University