In this paper we study the bifurcation of the homogeneous fixed point of a lattice of n diffusively coupled logistic maps. An analytical computation of the reduced map on the centermanifold is performed by taking into account the symmetries of the system. If n is even, a subcritical flip bifurcation causes a symmetry breaking of the homogeneous pattern whichproduces a traveling (rotating) wave with velocity 1 and time period 2. For odd n, since the bifurcation has a two dimensional normal form, we limit ourselves to consider only the simplest case (n = 3). In this case, a supercritical flip bifurcation is observed; three less symmetric periodic orbits of time period 2 are generated by the breaking of the homogeneous orbit. However, the bifurcation is rather degenerate and we have numerical hints that a second family of asymmetric periodic points is generated. Some details, pertaining to the dynamicsof the truncated map on the two dimensional center manifold for n = 3, are also presented.

Bifurcation of Homogeneous Solutions in a Chain of Logistic Maps / Giberti, Claudio; Vernia, Cecilia. - In: ATTI DEL SEMINARIO MATEMATICO E FISICO DEL'UNIVERSITÀ DI MODENA E REGGIO EMILIA. - ISSN 1825-1269. - STAMPA. - LIII:(2005), pp. 173-206.

Bifurcation of Homogeneous Solutions in a Chain of Logistic Maps

GIBERTI, Claudio;VERNIA, Cecilia
2005

Abstract

In this paper we study the bifurcation of the homogeneous fixed point of a lattice of n diffusively coupled logistic maps. An analytical computation of the reduced map on the centermanifold is performed by taking into account the symmetries of the system. If n is even, a subcritical flip bifurcation causes a symmetry breaking of the homogeneous pattern whichproduces a traveling (rotating) wave with velocity 1 and time period 2. For odd n, since the bifurcation has a two dimensional normal form, we limit ourselves to consider only the simplest case (n = 3). In this case, a supercritical flip bifurcation is observed; three less symmetric periodic orbits of time period 2 are generated by the breaking of the homogeneous orbit. However, the bifurcation is rather degenerate and we have numerical hints that a second family of asymmetric periodic points is generated. Some details, pertaining to the dynamicsof the truncated map on the two dimensional center manifold for n = 3, are also presented.
LIII
173
206
Bifurcation of Homogeneous Solutions in a Chain of Logistic Maps / Giberti, Claudio; Vernia, Cecilia. - In: ATTI DEL SEMINARIO MATEMATICO E FISICO DEL'UNIVERSITÀ DI MODENA E REGGIO EMILIA. - ISSN 1825-1269. - STAMPA. - LIII:(2005), pp. 173-206.
Giberti, Claudio; Vernia, Cecilia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/305137
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