Critical phenomenon, crisis and transition to spatiotemporal chaos

1. Critical phenomenon, crisis and transition to spatiotemporal chaos Kaifen HE Inst. Low Ener. Nucl. Phys. Beijing Normal Univ., Beijing China

2. What is STC? • TC state: two orbits starting from adjacent points separate exponentially, Luaponov exponent. • Does STC refer to any chaotic state in space-time-dependent system? • In numerical simulations it has been found that a wave state can be temporally chaotic but spatially coherent. • With ‘STC’ we refer to those states where the spatial coherenceis destroyed. • To solve mechanism for the onset to STC, a key problem is: how the spatial coherence is destroyed. • How to judge an STC?

3. What causes the loss of spatial coherence ? • Landau’s picture ( ) and Ruelle-Takens picture ( ) of turbulence. Landau’s picture is not supported by experimental results, and Ruelle-Takens route can only explain chaotic motion in systems with few free dimensions. • Consequently, there must be another process which can destroy the spatial coherence----crisis? • There is evidence to show that there can be different routes to STC.

4. Model equation • We use the driven/damped nonlinear drift-wave equation as the model: • The unperturbed equation is non-integrable, with the driving and damping, it shows very rich phenomena (with pseudospectral method). • The unperturbed equation has solitary wave-like solution, a harmonic wave driving.

5. fully-developed turbulence?

7. Relation to nonlinear dynamics in time-dependent systems • Nonlinear dynamics of ST-system is closely related to that of the T-system. • A steady wave (SW) is a fixed point in the Fourier space in the moving frame following it. • One can make stability analysis for the fixed point under perturbation as we usually do in T-systems. • If studying response of an SW to a perturbation wave (PW)in the moving frame, the results agree qualitatively with that by solving the pde.

8. The equation in moving frame

9. Steady wave • The equation has SW solutions • With expansion • mode amplitudes and phases of an SW, can be solved in the reference frame following the SW from the SW equation

10. Mode equations of steady wave

11. Hystereses of steady ‘wave energy’ • Wave energy of SW forms groups of hystereses. • An SW state can be stable or unstable. • Numerical simulation shows that all the complicated behaviours of the system seem in connection with the groups of hystereses. • Does this fact suggests that the states at the negative tangency branchplays an important role in the complexity?

12. Bistability of wave energy

13. Hystereses of wave energy

14. Another two groups of hystereses in other ranges of

15. State diagram in parameter space, clearly They are associated with groups of hystereses respectively,and winding number bifurctions

16. An example of multi-hystereses here F is in the form of coherent structure of solitary wave Its SW shows multi-hystereses, which is associated with winding number bifurcation

17. 4th A driven/damped KdV equation shows multi-hystereses 3rd 2nd 1st

18. 1st winding number bifurcation along multi-hystereses 2nd 3rd

19. Perturbation wave • If an SW is perturbed by a PW, in the moving frame the PW motion is governed by (Linear dispersion) (Nonlinear dispersion) (Self-nonlinearity) s

20. Linear response of PW to SW

21. Fourier space of SSW and PW • The SW and PW can be expanded respectively in Fourier space: • In linear approximation of PW we assume

22. Eigen equation of PW • If the self-nonlinearity in the PW equation is neglected, one gets an eigen equation of PW: • From which the eigen motion of the PW modes can be solved. • In general PW eigenvalues are complex conjugated (the motion is allowed in both directions relative to SSW).

25. Schematic plot for the SSW and eigen motion of PW Due to reflection at the SW, the dispersion of PW is altered   

26. Hopf instability Resonance of two internal modes and Hopf instability

27. Resonance of one internal mode with applied frequency (Doppler shift) and Saddle instability In the middle branch of hysteresis the eigenfrequency zero----no characteristic frequency is allowed to pass through the system

28. variation of eigenfrequency along a hysteresis

29. Vanishing of mode eigenfrequency and saddle instability

30. Variation of eigenvalues along a hysteresis of the KdV model, one can also see vanishing of k=2 mode eigen Frequency.

31. The physics of saddle instability • In the moving frame, saddle instability occurs when a mode eigen frequency is 0. • That is, in the lab frame the mode eigen frequency is . • It indicates that the mode eigenfrequency is resonant with the applied frequency . • However, due to scattering at the SSW, the PW mode eigenfrequency has been changed nonlinearly.

32. Saddle steady wave • When an SW is unstable due to saddle instability, we call it as saddle steady wave (SSW). • SSW locate at the negative tangency branch of the hystereses. • The internal modes one-by-one become resonant with the applied frequency, which causes groups of hystereses and superstructures in the state diagram.

33. Reconnection of eigenvalues and Hopf bifurcation ------real parts

34. Reconnection of eigenvalues in Hopf birfurcation ------- imaginary parts

35. The new frequency of the Hopf bifurcation appearing in the energy and wave patterns

36. Nonlinear N-mode and P-mode • Linear N-mode and P-mode: in a stream of velocity v, due to Doppler shift, • N-mode: • There is evidence to show that: • Saddle-bifurcation corresponds to transition of nonlinear N-mode to P-mode; • Hopf-bifurcation corresponds to exchange of energy types of a pair of N-mode and P-mode

37. Linear dispersion in the moving frame set We have When varying from 0-1, we come across the critical point at which the eigenfrequency of mode k changes sign for our parameter, the positions for k=1,2,3,4 …are 0.78,0.465,0.279,0.179… corresponding to the superstructures What happens when nonlinear dispersion is included?

38. Nonlinear dispersion • In linear case ( ), mode eigenfrequency is only allowed to transport in one direction. • When nonlinear dispersion is considered, the eigenfrequency is allowed to transport in both directions relative to the SW. • One may use eigenvector to define a N-mode and P-mode

39. If define mode eigenvector as Then N-mode and P-mode can be defined according to or According to this definition, the middle branch just corresponds to the situation when the two eigenvectors of the resonance mode k=2 are equal. In the lower branch: N, upper branch: P

40. Eigenvectors of k=2 mode along the 1st hysteresis for the KdV model

41. Stable and unstable orbits of SSW • Set • by neglecting the self-nonlinear term, one can find two stable and two unstable orbits of an SSW from the PW equation. • An important feature of the SO/UO orbits are: they correspond to constant mode phases, respectively, and the difference between two stable/unstable orbits is .

42. Stable orbit Unstable orbit Saddle point SO and UO of a saddle point

43. Stable orbits and unstable orbits of SSW

44. Effect of relative phase difference between PW and SSW • This result indicates that there are two (group of ) relative phase differences between the PW modes and SSW, at which the PW mode amplitudes are excited by the SSW, there are another two groups …… • This knowledge is very important for us to understand the turbulent behaviors.

45. Self-nonlinearity of PW is included

46. Perturbation wave including self-nonlinear term

47. Mode equations of PW

48. Gap soliton in periodic potential • In a system with periodic potential, forbidden gap appears in the linear spectrum. Linear wave with the frequency in the gap is not allowed to pass through the system. • However, if the dielectric constant is assumed to depend on the local field intensity (nonlinear), the radiation can be transmitted through the system with an envelop of a soliton shape.

49. Forbidden gap in the linear spectrum in a system with periodic potential. When nonlinearity is considered, the radiation is allowed to pass through, it forms a gap soliton.

50. Gap solitary wave and coexisting steady waves • In our case we have a periodic potential in the system, there is a forbidden gap in the linear spectrum (for saddle instability, no any characteristic frequency is allowed to transmit into the system), what happens when self-nonlinearity of PW is included? • Similar to gap soliton, we find that may build up a coherent structure on periodic potential of SSW; • The structure added on the SSW to form a new SW coexisting with the old SSW; • The new SW is a gap solitary wave in the sense that its energy locates about in the gap of the hysteresis.