Two-component feedback loops (TCFLs) are dynamical systems arising in mathematical biology that describe the time evolution of pairs of interacting molecules using complex network theory. These dynamical systems closely resemble a Hamiltonian system in %DIFDELCMD < \Bbb %%% R ^{2n}, but with the canonical equation for one of the variables in each conjugate pair rescaled by a number called the Turing instability parameter. The Turing instability parameter therefore measures the obstruction preventing a TCFL from being Hamiltonian where the Turing instability parameter equals one. To generalise TCFLs to symplectic manifolds in this paper we introduce and study the properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We describe why these objects have some interesting applications to symplectic geometry and discuss how their biological interpretation motivates new problems in Floer theory, mirror symmetry, and the study of %DIFDELCMD < \Bbb %%% D-K \"{a}hler manifolds . Since many questions in complex network theory can be translated into the topological setting, this paper therefore serves to bring a selection of ideas from biology to pure mathematics .
Networks of planar Hamiltonian systems closely resemble Hamiltonian system in %DIFDELCMD < \Bbb %%% \mathbb{R ^{2n}, but with the canonical equation for one of the variables in each conjugate pair rescaled by a number called the Turing instability parameter. To generalise these dynamical systems to symplectic manifolds in this paper we introduce and study the properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We describe why these objects have some interesting applications to symplectic geometry and discuss how their physical interpretation motivates new problems in Floer theory, mirror symmetry, and the study of %DIFDELCMD < \Bbb %%% \mathbb{D \"{a}hler manifolds .
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Two-component feedback loops (TCFLs) are dynamical systems arising in mathematical biology that describe the time evolution of pairs of interacting molecules using complex network theory. These dynamical
Networks of planar Hamiltonian
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\mathbb{R
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The Turing instability parameter therefore measures the obstruction preventing a TCFL from being Hamiltonian where the Turing instability parameter equals one. To generalise TCFLs
To generalise these dynamical systems
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biological
physical
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D-K
\mathbb{D
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. Since many questions in complex network theory can be translated into the topological setting, this paper therefore serves to bring a selection of ideas from biology to pure mathematics