Global stability properties of a hyperbolic system arising in pattern formation

Nonlinear Analysis, Theory, Methods and Applications

Volumn 29, Issue 4, 1997, Pages 439-460

  Martel, Carlos ^a   Vega, José M ^a  

^a UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

Author keywords

Global stability; Hyperbolic system; Oscillatory instability; Pattern formation; Sub and super solutions

Indexed keywords

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); BOUNDARY CONDITIONS; DYNAMICS; NONLINEAR SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS;

HYPERBOLIC SYSTEMS; PATTERN FORMATION; QUASIPERIODIC SYSTEMS;

NONLINEAR EQUATIONS;

EID: 0031213051     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(96)00053-3     Document Type: Article

References (36)

1. CROSS M. C. & HOHENBERG P. C., Pattern formation outside of equilibrium, Rev. Modern Phys. 65, 851-1112 (1993).
2. STEINBERG V., FINEBERG J., MOSES E. & REHBERG I., Pattern selection and transition to turbulence in propagating waves, Physica D 37, 359-383 (1989).
3. BENSIMON D., KOLODNER P., SURKO C. M., WILLIAMS H. & CROQUETTE V., Competing and coexisting dynamical states of travelling-wave convection in an annulus, J. Fluid Mech. 217, 441-467 (1990).
4. KOLODNER P., Repeated transients of weakly nonlinear travelling-wave convection, Phys. Rev. E 47, 1038-1048 (1993).
5. KAPLAN E. & STEINBERG V., Measurement of reflection of travelling waves near the onset of binary-fluid convection, Phys. Rev. E 48, R661-R664 (1993).
6. PREDTECHENSKY A. A., McCORMICK W. D., SWIFT J. B., NOSZTICZIUS Z. & SWINNEY H. L., Onset of travelling waves in isothermal double diffusive convection, Phys. Rev. Lett. 72, 218-221 (1994).
7. 4He, J. Fluid Mech. 189, 263-286 (1988).
8. CROQUETTE V. & WILLIAMS H., Nonlinear competition between waves on convective rolls, Phys. Rev. A 39, 2765-2768 (1989).
9. CROQUETTE V. & WILLIAMS H., Nonlinear waves of the oscillatory instability on finite convective rolls, Physica D 37, 300-314 (1989).
10. ANDERECK C. D., LIU S. S. & SWINNEY H. L., Flow regimes in a circular Couette system with independently rotating cylinders, J. Fluid Mech. 164, 155-183 (1986).
11. LANGFORD W. F., TAGG R., KOSTELICH E. J., SWINNEY H. L. & GOLUBISTKY M., Primary instabilities and bicriticality in flow between counter-rotating cylinders, Physics Fluids 31, 776-785 (1988).
12. DUBOIS M., DAVIAUD F., RONSIN O. & BERGE P., Traveling waves in pure fluids heated along wires, Physica D 61, 140-146 (1992).
13. VINCE J. M. & DUBOIS M., Hot wire below the free surface of a liquid: structural and dynamical properties of a secondary instability, Europhys. Lett. 20, 505-510 (1992).
14. DAVIAUD F. & VINCE J. M., Traveling waves in a fluid layer subjected to a horizontal temperature gradient, Phys. Rev. E 48, 4432-4436 (1993).
15. JOETS A. & RIBOTTA R., Localized, time-dependent state in the convection of a nematic liquid crystal, Phys. Rev. Lett. 60, 2164-2167 (1988).
16. REHBERG I., RASENAT S. & STEINBERG V., Traveling waves and defect-initiated turbulence in electroconvecting nematics, Phys. Rev. Lett. 62, 756-759 (1989).
17. REHBERG I., WINKLER B. L., DE LA TORRE M., RASENAT S. & SCHÖPF W., Pattern formation in a liquid crystal, Advances in Solid State Physics 29 (Ed. V. ROESSLER), pp. 35-52. Vieweg, Braunschweig (1989).
18. DE LA TORRE M., Contribución al estudio de la formación de estructuras en fluidos anisótropes. Doctoral Thesis, Universidad Nacional de Educación a Distancia, Madrid (1990).
19. MARKSTEIN G. H., Nonsteady Flame Propagation. Plenum Press (1964).
20. FERGUSON C. R. & KECK J. C., Stand-off distances on a flat burner, Combustion and Flame 34, 85-98 (1979).
21. SABATHIER F., BOYER L. & CLAVIN P., Experimental study of weak turbulent premixed flames, Combustion and Flame 35, 139-153 (1979).
22. COULLET P., FAUVE S. & TIRAPEGUI E., Large scale instability of nonlinear standing waves, J. Physique Lett. 46, L787-L791 (1985).
23. CROSS M. C., Travelling and standing waves in binary-fluid convection in finite geometries, Phys. Rev. Lett. 57, 2935-2938 (1986).
24. MARTEL C., La inestabilidad oscilatoria y sus aplicaciones en Mecánica de Fluidos y Combustión. Doctoral Thesis, Universidad Politécnica de Madrid, Madrid (1995).
25. MARTEL C. & VEGA J. M., Finite size effects near the onset of the oscillatory instability, Nonlinearity 9, 1129-1171 (1996).
26. CROSS M. C. & KUO E. Y., One dimensional spatial structure near a Hopf bifurcation at finite wavenumber, Physica D 59, 90-120 (1992).
27. KNOBLOCH E. & DE LUCA J., Amplitude equations for traveling wave convection, Nonlinearity 3, 575-580 (1990).
28. ALVAREZ-PEREIRA C. & VEGA J. M., On the pulsating instability of two-dimensional flames, Eur. J. Appl. Math. 3, 55-73 (1992).
29. VEGA J. M., On the amplitude equations arising at the onset of the oscillatory instability in pattern formation, SIAM J. math. Analysis 24, 603-617 (1993).
30. MARTEL C. & VEGA J. M., Dynamics of a hyperbolic system that applies at the onset of the oscillatory instability, Nonlinearity (submitted).
31. COURANT R. & HILBERT D., Methods of Mathematical Physics, Vol. II. Interscience (1962).
32. HARDY G. H., LITTLEWOOD J. E. & POLYA G., Inequalities. Cambridge University Press (1967).
33. LADDE G. S., LAKSHMIKANTHAM V. & VATSALA A. S., Existence and asymptotic behavior of reaction-diffusion systems via coupled quasi-solutions, Nonlinear Analysis and Applications (Ed. V. LAKSHMIKAN-THAM). Academic Press (1982).
34. LEUNG A. & CLARK D., Bifurcations and large time behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. diff. Eqns 35, 113-127 (1980).
35. HERNÁNDEZ J., Some existence and stability results for solutions of reaction-diffusion systems with nonlinear boundary conditions, Nonlinear Differential Equations: Invariance, Stability and Bifurcation (Ed. P. DE MOTTONI). Academic Press (1981).
36. VEGA J. M., Invariant regions and global asymptotic stability in an isothermal catalyst, SIAM J. math. Analysis 19, 774-796 (1988).