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Coupling leads to chaos. (English) Zbl 1133.34027
From the abstract: The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension \(4\). On the other hand, we will prove that in any generic unfolding \(X_\mu\) of an \(n\)-dimensional nilpotent singularity of codimension \(n\) there are bifurcation curves of \((n-1)\)-dimensional nilpotent singularities of codimension \(n-1\) which are in turn generically unfolded by \(X_\mu\). Arguments conclude recalling that any generic unfolding of the \(3\)-dimensional nilpotent singularity of codimension \(3\) exhibits strange attractors.

MSC:
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Software:
Matlab; Mathematica
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[1] Alexander, J.C., Spontaneous oscillations in two 2-component cells coupled by diffusion, J. math. biol., 23, 205-219, (1986) · Zbl 0588.92002
[2] Alexander, J.C., A period-doubling bubble in the dynamics of two coupled oscillators, (), 208-220
[3] Aronson, D.G.; Doedel, E.J.; Othmer, H.G., An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators, Phys. D, 25, 20-104, (1987) · Zbl 0624.34029
[4] Dolnik, M.; Zhabotinsky, A.M.; Epstein, I.R., Resonant suppression of Turing patterns by periodic illumination, Phys. rev. E, 63, 026101, (2001), (10 pages)
[5] Dumortier, F.; Ibáñez, S., Nilpotent singularities in generic 4-parameter families of 3-dimensional vector fields, J. differential equations, 127, 590-647, (1996) · Zbl 0865.58032
[6] Dumortier, F.; Ibáñez, S.; Kokubu, H., New aspects in the unfolding of the nilpotent singularity of codimension three, Dyn. syst., 16, 63-95, (2001) · Zbl 0993.37026
[7] Dumortier, F.; Ibáñez, S.; Kokubu, H., COCOON bifurcation in three-dimensional reversible vector fields, Nonlinearity, 19, 305-328, (2006) · Zbl 1107.34039
[8] Golubitsky, M.; Stewart, I., The symmetry perspective, Progr. math., vol. 200, (2002), Birkhäuser Basel
[9] Homburg, A.J., Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15, 1029-1050, (2002) · Zbl 1017.37012
[10] Hoppenstead, F.C.; Izhikevich, E.M., Weakly connected neural networks, Appl. math. sci., vol. 126, (1998), Springer New York
[11] Howard, L.N., Nonlinear oscillations, (), 1-67
[12] Ibáñez, S.; Rodríguez, J.A., Sil’nikov bifurcations in generic 4-unfoldings of a codimension-4 singularity, J. differential equations, 120, 411-428, (1995) · Zbl 0839.34050
[13] Ibáñez, S.; Rodríguez, J.A., Shil’nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on \(\mathbb{R}^3\), J. differential equations, 208, 147-175, (2005) · Zbl 1080.34029
[14] Wolfram Research Inc., {\scmathematica} 5
[15] MathWorks Inc., {\scmatlab} 5.3
[16] Pumariño, A.; Rodríguez, J.A., Coexistence and persistence of strange attractors, Lecture notes in math., vol. 1658, (1997), Springer Berlin · Zbl 0877.58041
[17] Pumariño, A.; Rodríguez, J.A., Coexistence and persistence of infinitely many strange attractors, Ergodic theory dynam. systems, 21, 1511-1523, (2001) · Zbl 1073.37514
[18] Smale, S., A mathematical model of two cells via Turing’s equation, (), 15-26 · Zbl 0333.92002
[19] Schreiber, I.; Marek, M., Strange attractors in coupled reaction – diffusion cells, Phys. D, 5, 258-272, (1982)
[20] Turing, A., The chemical basis of morphogenesis, Philos. trans. roy. soc. B, 237, 37-72, (1952) · Zbl 1403.92034
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