×

On diffusion driven oscillations in coupled dynamical systems. (English) Zbl 0970.34029

Summary: The authors deal with the problem of destabilization of diffusively coupled identical systems. Following a question of S. Smale [J. Math. Biol. 3, 5-7 (1976; Zbl 0344.92009)], it is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillatory behavior. It is shown that if the diffusive medium consists of hyperbolically nonminimum phase systems and the diffusive factors exceed some threshold value, the origin of the overall system undergoes a Poincaré-Andronov-Hopf bifurcation resulting in oscillatory behavior.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
93C10 Nonlinear systems in control theory
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34H05 Control problems involving ordinary differential equations

Citations:

Zbl 0344.92009
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1109/9.90226 · Zbl 0758.93060 · doi:10.1109/9.90226
[2] DOI: 10.1109/9.100932 · Zbl 0758.93007 · doi:10.1109/9.100932
[3] Kocarev L. M., IEEE Trans. Circuits Syst. (1995)
[4] DOI: 10.2307/1969299 · Zbl 0061.18910 · doi:10.2307/1969299
[5] DOI: 10.1142/S0218127498000188 · Zbl 0938.93056 · doi:10.1142/S0218127498000188
[6] DOI: 10.1137/0109053 · Zbl 0108.01202 · doi:10.1137/0109053
[7] Tomberg E. A., Siberian Math. J. 30 (4) pp 180– (1989)
[8] DOI: 10.1098/rstb.1952.0012 · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
[9] We M., Notices of the Am. Math. Soc. 45 (1) pp 9– (1998)
[10] Yakubovich V. A., Siberian Math. J. 14 (5) pp 1100– (1973)
[11] DOI: 10.1007/BF00276172 · Zbl 0096.28902 · doi:10.1007/BF00276172
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.