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May  2008, 9(3&4, May): 701-730. doi: 10.3934/dcdsb.2008.9.701

A non-autonomous bifurcation theory for deterministic scalar differential equations

Carmen Núñez 1, and  Rafael Obaya 1,

1. 

Universidad de Valladolid, Departamento de Matemática Aplicada, ETSII, Paseo del Cauce s/n, 47011 Valladolid

Received  January 2007 Revised  June 2007 Published  February 2008

In the extension of the concepts of saddle-node, transcritical and pitchfork bifurcations to the non-autonomous case, one considers the change in the number and attraction properties of the minimal sets for the skew-product flow determined by the initial one-parametric equation. In this work conditions on the coefficients of the equation ensuring the existence of a global bifurcation phenomenon of each one of the types mentioned are established. Special attention is paid to show the importance of the non-trivial almost automorphic extensions and pinched sets in describing the dynamics at the bifurcation point.
Keywords: pitchfork bifurcation., almost periodic dynamics, almost automorphic dynamics, Non-autonomus dynamical systems, saddle-node bifurcation, transcritical bifurcation.
Mathematics Subject Classification: 37B55, 34C23, 37G9.
Citation: Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
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