Multistability in networks of weakly coupled bistable units. (English) Zbl 0891.34060
Summary: We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a ‘propagation failure’ phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.
MSC:
34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |
94C05 | Analytic circuit theory |
34D20 | Stability of solutions to ordinary differential equations |
34C99 | Qualitative theory for ordinary differential equations |
93D99 | Stability of control systems |
34D10 | Perturbations of ordinary differential equations |
70K20 | Stability for nonlinear problems in mechanics |
Keywords:
weakly coupled bistable units; high multiplicity of stable steady states; diffusive networksReferences:
[1] | Purwins, H. G.; Klempt, G.; Berkemeier, J.: Temporal and spatial structures of nonlinear dynamical systems. Festkörperproblem (Advances in solid state physics) 27, 27-61 (1987) |
[2] | Radehaus, C.; Dohmen, R.; Willebrand, H.; Niedemostheide, F. J.: Model for current patterns in physical systems with two charge-carriers. Phys. rev. A 42, 7426-7446 (1990) |
[3] | Domb, C.; Lebowitz, J. L.: Phase transitions and critical phenomena. (1983) · Zbl 1063.82002 |
[4] | Bowden, C. M.; Ciftan, M.; Robl, J. R.: Optical bistability. (1981) |
[5] | De Kepper, P.; Boissonade, J.; Epstein, I. R.: Chloride iodide reaction – a versatile system for the study of nonlinear dynamic behaviour. J. phys. Chem. 94, 6525-6536 (1990) |
[6] | Murray, J. D.: Mathematical biology. Biomathematics 19 (1989) · Zbl 0682.92001 |
[7] | Aubry, S.; Abramovici, G.: Chaotic trajectories in the standard map: the concept of anti-integrability. Physica D 43, 199-219 (1990) · Zbl 0713.58014 |
[8] | Keener, J. P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. math. 47, 556-572 (1987) · Zbl 0649.34019 |
[9] | Firth, W. J.: Optical memory and spatial chaos. Phys. rev. Lett. 61, 329-332 (1988) |
[10] | Keener, J. P.: The effects of discrete gap junction coupling on propagation in myocardium. J. theor. Biol. 148, 49-82 (1991) |
[11] | Laplante, J. P.; Emeux, T.: Propagation failure in arrays of coupled bistable chemical reactors. J. phys. Chem. 96, 4931-4934 (1992) |
[12] | Emeux, T.; Nicolis, G.: Propagating waves in discrete bistable reaction-diffusion systems. Physica D 67, 237-244 (1993) · Zbl 0787.92010 |
[13] | R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, in press. · Zbl 0811.70017 |
[14] | Erneux, T.; Mandel, P.: Temporal aspects of absorptive optical bistability. Phys. rev. A 28, 896-909 (1983) |
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