# American Institute of Mathematical Sciences

2010, 7(1): 83-97. doi: 10.3934/mbe.2010.7.83

## Multistability, oscillations and bifurcations in feedback loops

 1 Department of Mathematics, University of Oklahoma, Norman, OK 73019-0315, United States 2 School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom

Received  May 2009 Revised  October 2009 Published  January 2010

Feedback loops are found to be important network structures in regulatory networks of biological signaling systems because they are responsible for maintaining normal cellular activity. Recently, the functions of feedback loops have received extensive attention. The existing results in the literature mainly focus on verifying that negative feedback loops are responsible for oscillations, positive feedback loops for multistability, and coupled feedback loops for the combined dynamics observed in their individual loops. In this work, we develop a general framework for studying systematically functions of feedback loops networks. We investigate the general dynamics of all networks with one to three nodes and one to two feedback loops. Interestingly, our results are consistent with Thomas' conjectures although we assume each node in the network undergoes a decay, which corresponds to a negative loop in Thomas' setting. Besides studying how network structures influence dynamics at the linear level, we explore the possibility of network structures having impact on the nonlinear dynamical behavior by using Lyapunov-Schmidt reduction and singularity theory.
Citation: Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83
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