Oscillating localized structures in reaction-diffusion systems

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 14, Issue 12, 2004, Pages 4097-4104

  Descalzi, Orazio ^b,c   Hayase, Yumino ^d   Brand, Helmut R ^a  

^a UNIVERSITY OF BAYREUTH   (Germany)

^b UNIVERSIDAD DE LOS ANDES   (Chile)

^c Sistemas Complejos de Santiago   (Chile)

^d KYUSHU UNIVERSITY   (Japan)

Author keywords

Oscillating localized structures; Reaction diffusion models; Saddle node bifurcation

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; DIFFUSION; MATHEMATICAL MODELS; PROBLEM SOLVING; REYNOLDS NUMBER;

CHAOTIC BREATHING; OSCILLATING LOCALIZED STRUCTURES; REACTION-DIFFUSION MODELS; SADDLE-NODE BIFURCATION;

PATTERN RECOGNITION;

EID: 14844292164     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127404011806     Document Type: Article

References (27)

1. Brand, H. R. & Deissler, R. J. [1989] "Interaction of localized solutions for subcritical bifurcations," Phys. Rev. Lett. 63, 2801-2804.
2. Deissler, R. J. & Brand, H. R. [1990] "The effect of nonlinear gradient terms on localized states near a weakly inverted bifurcation," Phys. Lett. A146, 252-255.
3. Deissler, R. J. & Brand, H. R. [1991] "Interaction of two-dimensional localized solutions near a weakly inverted bifurcation," Phys. Rev. A44, R3411-R3414.
4. Deissler, R. J. & Brand, H. R. [1994] "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478-481.
5. Deissler, R. J. & Brand, H. R. [1995] "Two-dimensional localized solutions for subcritical bifurcations in systems with broken rotational symmetry," Phys. Rev. E51, R852-R855.
6. Deissler, R. J. & Brand, H. R. [1998] "Effect of nonlinear gradient terms on breathing localized solutions in the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 81, 3856-3859.
7. Descalzi, O., Argentina, M. & Tirapegui, E. [2002] "Stationary localized solutions in the subcritical complex Ginzburg-Landau equation," Int. J. Bifurcation and Chaos 12, 2459-2465.
8. Descalzi, O. [2003] "On the stability of localized structures in the complex Ginzburg-Landau equation," Physica A327, 23-28.
9. Descalzi, O., Argentina, M. & Tirapegui, E. [2003] "Saddle-node bifurcation: Appearance mechanism of pulses in the subcritical complex Ginzburg-Landau equation," Phys. Rev. E67, 015601-1(R)-015601-4(R).
10. Descalzi, O., Hayase, Y. & Brand, H. R. [2004] "Analytical approach to localized structures in a simple reaction-diffusion system," Phys. Rev. E69, 026121-1-026121-9.
11. Fauve, S. & Thual, O. [1990] "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284.
12. Hayase, Y. [1997] "Collision and self-replication of pulses in a reaction diffusion system," J. Phys. Soc. Jpn. 66, 2584-2587.
13. Hayase, Y. & Ohta, T. [2002] "Self-replication of a pulse in excitable reaction-diffusion systems," Phys. Rev. E66, 036218-1-036218-7.
14. Hayase, Y., Descalzi, O. & Brand, H. R. [2004] "Coexistence of stable particle and hole solutions for fixed parameters for a simple reaction-diffusion system," Phys. Rev. E69, 065201-1(R)-065201-4(R).
15. Koga, S. & Kuramoto, Y. [1980] "Localized patterns in reaction-diffusion systems," Prog. Theor. Phys. 63, 106-121.
16. Kosek, J. & Marek, M. [1995] "Collision-stable waves in excitable reaction-diffusion systems," Phys. Rev. Lett. 74, 2134-2137.
17. Lee, K. J. & Swinney, H. L. [1995] "Lamellar structures and self-replicating spots in a reaction-diffusion system," Phys. Rev. E51, 1899-1915.
18. Ohta, T., Mimura, M. & Kobayashi, R. [1989] "Higher-dimensional localized patterns in excitable media," Physica D34, 115-144.
19. Ohta, T., Hayase, Y. & Kobayashi, R. [1996] "Spontaneous formation of concentric wave in a two-component reaction-diffusion system," Phys. Rev. E54, 6074-6083.
20. Ohta, T., Kiyose, J. & Mimura, M. [1997] "Collision of propagating pulses in a reaction-diffusion system," J. Phys. Soc. Japan. 66, 1551-1558.
21. Petrov, V., Scott, S. K. & Showalter, K. [1994] "Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system," Phil. Trans. Roy. Soc. Lond. A347, 631-642.
22. Reynolds, W. N., Ponce-Dawson, S. & Pearson, J. E. [1997] "Self-replicating spots in reaction-diffusion systems," Phys. Rev. E56, 185-198.
23. Thual, O. & Fauve, S. [1988] "Localized structures generated by subcritical instabilities," J. Phys. France 49, 1829-1833.
24. Turing, A. M. [1952] "The chemical basis of morphogenesis," Phil Trans. R. Soc. London Ser. B327, 37-72.
25. van Saarloos, W. & Hohenberg, P. C. [1990] "Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation," Phys. Rev. Lett. 64, 749-752.
26. van Saarloos, W. & Hohenberg, P. C. [1992] "Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations," Physica D56, 303-367.
27. Winfree, A. T. [1980] The Geometry of Biological Time (Springer, NY).