Two-phase resonant patterns in forced oscillatory systems: boundaries, mechanisms and forms. (English) Zbl 1062.37050
Summary: We use the forced complex Ginzburg-Landau (CGL) equation to study resonance in oscillatory systems periodically forced at approximately twice the natural oscillation frequency. The CGL equation has both resonant spatially uniform solutions and resonant two-phase standing-wave pattern solutions such as stripes or labyrinths. The spatially uniform solutions form a tongue-shaped region in the parameter plane of the forcing amplitude and frequency. But the parameter range of resonant standing-wave patterns does not coincide with the tongue of spatially uniform oscillations.
On one side of the tongue the boundary of resonant patterns is inside the tongue and is formed by the nonequilibrium Ising Bloch bifurcation and the instability to traveling waves. On the other side of the tongue the resonant patterns extend outside the tongue forming a parameter region in which standing-wave patterns are resonant but uniform oscillations are not. The standing-wave patterns in that region appear similar to those inside the tongue but the mechanism of their formation is different. The formation mechanism is studied using a weakly nonlinear analysis near a Hopf-Turing bifurcation.
The analysis also gives the existence and stability regions of the standing-wave patterns outside the resonant tongue. The analysis is supported by numerical solutions of the forced complex Ginzburg-Landau equation.
On one side of the tongue the boundary of resonant patterns is inside the tongue and is formed by the nonequilibrium Ising Bloch bifurcation and the instability to traveling waves. On the other side of the tongue the resonant patterns extend outside the tongue forming a parameter region in which standing-wave patterns are resonant but uniform oscillations are not. The standing-wave patterns in that region appear similar to those inside the tongue but the mechanism of their formation is different. The formation mechanism is studied using a weakly nonlinear analysis near a Hopf-Turing bifurcation.
The analysis also gives the existence and stability regions of the standing-wave patterns outside the resonant tongue. The analysis is supported by numerical solutions of the forced complex Ginzburg-Landau equation.
MSC:
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
| 92B05 | General biology and biomathematics |
Keywords:
resonant pattern; forced oscillatory system; Ginzburg-Landau equation; Ising Bloch bifurcation; standing-wave patterns; Hopf-Turing bifurcationReferences:
| [1] | Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0507.34003 |
| [2] | Jensen, M. H.; Bak, P.; Bohr, T., Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps, Phys. Rev. A, 1960-1969 (1984) |
| [3] | Glass, L.; Sun, J., Periodic forcing of a limit-cycle oscillator : Fixed points, Arnold tongues, and the global organization of bifurcations, Phys. Rev. E, 50, 6, 5077-5084 (1994) |
| [4] | Hilborn, R. C., Chaos and Nonlinear Dynamics (1994), Oxford University Press: Oxford University Press New York · Zbl 0804.58002 |
| [5] | Glazier, J. A.; Libchaber, A., Quasi-periodicity and dynamical systems: an experimentalist’s view, IEEE Trans. Circuits Syst., 790-809 (1988) |
| [6] | Eiswirth, M.; Ertl, G., Forced oscillations of a self-oscillating surface reaction, Phys. Rev. Lett., 60, 15, 1526-1529 (1988) |
| [7] | Coullet, P.; Lega, J.; Houchmanzadeh, B.; Lajzerowicz, J., Breaking chirality in nonequilibrium systems, Phys. Rev. Lett., 65, 1352-1355 (1990) |
| [8] | Coullet, P.; Emilsson, K., Strong resonances of spatially distrubuted oscillatiors: a laboratory to study patterns and defects, Physica D, 61, 119-131 (1992) · Zbl 0787.70016 |
| [9] | Petrov, V.; Ouyang, Q.; Swinney, H. L., Resonant pattern formation in a chemical system, Nature, 388, 655-657 (1997) |
| [10] | Lin, A. L.; Hagberg, A.; Ardelea, A.; Bertram, M.; Swinney, H. L.; Meron, E., Four-phase patterns in forced oscillatory systems, Phys. Rev. E, 62, 3790-3798 (2000) |
| [11] | Chate, H.; Pikovsky, A.; Rudzick, O., Forcing oscillatory media: phase kinks vs. synchronization, Physica D, 131, 17-30 (1999) · Zbl 1076.70514 |
| [12] | Lin, A. L.; Bertram, M.; Martinez, K.; Swinney, H. L.; Ardelea, A.; Carey, G. F., Resonant phase patterns in a reaction-diffusion system, Phys. Rev. Lett., 84, 4240-4243 (2000) |
| [13] | Vanag, V. K.; Yang, L. F.; Dolnik, M.; Zhabotinsky, A. M.; Epstein, I. R., Oscillatory cluster patterns in a homogeneous chemical system with global feedback, Nature, 406, 6794, 389-391 (2000) |
| [14] | Vanag, V. K.; Zhabotinsky, A. M.; Epstein, I. R., Pattern formation in the Belousov-Zhabotinsky reaction with photochemical global feedback, J. Phys. Chem. A, 104, 49, 11566-11577 (2000) |
| [15] | Vanag, V. K.; Zhabotinsky, A. M.; Epstein, I. R., Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction, Phys. Rev. Lett., 86, 3, 552-555 (2001) |
| [16] | Park, H.-K., Frequency locking in spatially extended systems, Phys. Rev. Lett., 86, 1130-1133 (2001) |
| [17] | Gomila, D.; Colet, P.; Oppo, G. L.; Miguel, M. S., Stable droplets and growth laws close to the modulational instability of a domain wall, Phys. Rev. Lett., 87, 19, 194101 (2001) |
| [18] | Gallego, R.; Walgraef, D.; Miguel, M. San.; Toral, R., Transition from oscillatory to excitable regime in a system forced at three times its natural frequency, Phys. Rev. E, 64, 5, 056218 (2001) |
| [19] | Kim, J.; Lee, J.; Kahng, B., Harmonic forcing of an extended oscillatory system: Homogeneous and periodic solutions, Phys. Rev. E, 65, 4, 046208 (2002) |
| [20] | Yochelis, A.; Hagberg, A.; Meron, E.; Lin, A. L.; Swinney, H. L., Development of standing-wave labyrinthine patterns, SIADS, 1, 2, 236-247 (2002) · Zbl 1004.35064 |
| [21] | Martinez, K.; Lin, A. L.; Kharrazian, R.; Sailer, X.; Swinney, H. L., Resonance in periodically inhibited reaction-diffusion systems, Physica D, 2, 168-169 (2002) · Zbl 1004.35015 |
| [22] | Meron, E., Phase fronts and synchronization patterns in forced oscillatory systems, Discrete Dyn. Nat. Soc., 4, 3, 217-230 (2000) · Zbl 0973.35025 |
| [23] | Gambaudo, J. M., Perturbation of a Hopf bifurcation by an external time-periodic forcing, J. Diff. Eq., 57, 172-199 (1985) · Zbl 0516.34042 |
| [24] | Elphick, C.; Iooss, G.; Tirapegui, E., Normal form reduction for time-periodically driven differential equations, Phys. Lett. A, 120, 459-463 (1987) |
| [25] | Elphick, C.; Hagberg, A.; Meron, E., A phase front instability in periodically forced oscillatory systems, Phys. Rev. Lett., 80, 22, 5007-5010 (1998) |
| [26] | Elphick, C.; Hagberg, A.; Meron, E., Multiphase patterns in periodically forced oscillatory systems, Phys. Rev. E, 59, 5, 5285-5291 (1999) |
| [27] | A. Yochelis, Pattern formation in periodically forced oscillatory systems, Ph.D. Thesis, Ben-Gurion University, 2003.; A. Yochelis, Pattern formation in periodically forced oscillatory systems, Ph.D. Thesis, Ben-Gurion University, 2003. |
| [28] | Hagberg, A.; Meron, E., Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations, Nonlinearity, 7, 805-835 (1994) · Zbl 0804.35058 |
| [29] | Hagberg, A.; Meron, E., From labyrinthine patterns to spiral turbulence, Phys. Rev. Lett., 72, 15, 2494-2497 (1994) |
| [30] | Sarker, S.; Trullinger, S. E.; Bishop, A. R., Solitary-wave solution for a complex one-dimensional field, Phys. Lett. A, 59, 4, 255-258 (1976) |
| [31] | Elphick, C.; Hagberg, A.; Meron, E.; Malomed, B., On the origin of traveling pulses in bistable systems, Phys. Lett. A, 230, 33-37 (1997) |
| [32] | Skryabin, D. V.; Yulin, A.; Michaelis, D.; Firth, W. J.; Oppo, G. L.; Peschel, U.; Lederer, F., Perturbation theory for domain walls in the parametric Ginzburg-Landau equation, Phys. Rev. E, 64, 5, 056618 (2001) |
| [33] | C. Elphick, Dynamical approach to the Ising-Bloch transition in 3 + 1 dimensional time periodically driven dynamical systems, Physica D, submitted for publication.; C. Elphick, Dynamical approach to the Ising-Bloch transition in 3 + 1 dimensional time periodically driven dynamical systems, Physica D, submitted for publication. |
| [34] | Newell, A. C.; Whitehead, J. A., Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38, 279-303 (1969) · Zbl 0187.25102 |
| [35] | Segel, L. A., Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38, 203-224 (1969) · Zbl 0179.57501 |
| [36] | Cross, M. C.; Hohenberg, P. C., Pattern formation outside of equilibrium, Rev. Mod. Phys., 65, 3, 851-1112 (1993) · Zbl 1371.37001 |
| [37] | Keener, J. P., Secondary bifurcation in nonlinear diffusion-reaction equations, Stud. Appl. Math., 55, 3, 187-211 (1976) · Zbl 0336.35057 |
| [38] | Kidachi, H., On mode interactions in reaction diffusion equation with nearly degenerate bifurcations, Prog. Theor. Phys., 63, 4, 1152-1169 (1980) · Zbl 1059.35513 |
| [39] | Crawford, J. D., Introduction to bifurcation theory, Rev. Mod. Phys., 63, 4, 991-1037 (1991) |
| [40] | A. Yochelis, C. Elphick, A. Hagberg, E. Meron, Frequency locking in extended systems: the impact of a Turing mode, in press.; A. Yochelis, C. Elphick, A. Hagberg, E. Meron, Frequency locking in extended systems: the impact of a Turing mode, in press. · Zbl 1062.37050 |
| [41] | Or-Guil, M.; Bode, M., Propagation of Turing-Hopf fronts, Physica A, 249, 174-178 (1998) |
| [42] | Park, H.-K.; Br, M., Spiral destabilization by resonant forcing, Europhys. Lett., 65, 873-879 (2004) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
