Hopf and Neimark-Sacker bifurcations: applications to discrete-time hypercycles with functional shifts
| Autor: | Perona García, Júlia |
|---|---|
| Přispěvatelé: | Fontich Julia, Ernest, Guillamon Grabolosa, Antoni, Sardanyés Cayuela, Josep, Universitat Politècnica de Catalunya. Departament de Matemàtiques |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Difference equations
Hypercycle model Neimark-Sacker bifurcation 39 Difference and functional equations::39A Difference equations [Classificació AMS] Hopf bifurcation Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències [Àrees temàtiques de la UPC] Discrete dynamical systems Invariant curves Equacions en diferències |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Popis: | Hypercycles are cyclic catalytic sets of replicating macromolecules, where each one of the species catalyzes the replication of the next species of the set. This system has been widely studied within the framework of origins of life and, more recently, in cooperative systems within the subject of theoretical ecology. Hypercycles' dynamics has been mainly studied using continuous-time dynamical systems, mainly with autonomous ordinary differential equations. In this project, we study a discrete-time hypercycle model introduced by Hofbauer. First, we study the Hopf bifurcation in a two-dimensional family of differential equations depending on one parameter that describes the transition from a fixed point to a periodic orbit. We also study the Neimark-Sacker bifurcation in a two-dimensional family of discrete systems depending on one parameter that describes the same transition but with invariant curves instead of periodic orbits. We also introduce a theorem by Hofbauer and Iooss that proves the existence of attracting invariant cycles in a discretization of a differential equation. Finally, we study an application of this theorem to Hofbauer's discrete hypercycle system, where one species suffers a functional shift from cooperation to degradation of another species. Such a functional shift may appear realistic for ribozymes (catalytic RNAs), which, through mutation, could change from a catalytic to a degradative (RNA cleavage) function. In this model, by means of a singular change, it appears a Neimark-Sacker bifurcation and another rather degenerate bifurcation, which, to our knowledge, has not been studied yet. |
| Databáze: | OpenAIRE |
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