Holomorphic dynamical systems (MTM2014-52209-C2-2-P)

University: Universitat de Barcelona
Project leaders: Núria Fagella
Researchers: Xavier Jarque, Antoni Garijo
Start date: 2015-01-01
End date: 2017-12-31
Funded by: Ministerio de Economia y Competitividad

This project focuses on low-dimensional dynamical systems, both real and complex, and its study from the topological, analytical and numerical points of view. It is divided into two subprojects:

  • DynSysTAN: ynamical Systems: Topology, Analysis and Numerics, and
  • Holodyn: Holomorphic Dynamical Systems
The first sub-project focuses on the real dynamic systems (discrete and continuous) while the second one focuses on dynamical systems arising from iterated holomorphic functions in the complex plane, including the rational and transcendental functions (those with an essential singularity at infinity), entire and meromorphic as well. Our group has made significant contributions in each of these lines and we intend to continue and improve the work on them. We also want to broaden horizons by opening modern new lines as the interaction between complex and real systems that occurs when complexifying certain analytical systems. The objectives of this project are to improve our understanding of these systems with an emphasis on the study of the following research lines, with their corresponding objectives, and with the specification of which subproject develops them.
  1. Topological and combinatorial dynamics in low dimension (DynSysTAN).
    1. Cycles and entropy for tree maps.
    2. Rotation theory and periods for graph maps.
    3. Linearization of periodic maps.
  2. Quasiperiodically forced systems (DynSysTAN).
    1. Numerical and analytical studies on SNA.
    2. "Forcing" theory of quasiperiodic two dimensional systems.
    3. Numerical and semi-analytical computation of invariant objects using wavelet basis.
  3. Applications to Astrodynamics and "Complex Networks" (DynSysTAN).
    1. Complex adaptive and non-adaptive networks.
    2. A map of heteroclinic connections between the t rajectories around the collinear libration points.
    3. Applications of the parameterization method to Astrodynamics.
  4. Qualitative theory of differential equations in the plane and the period function (DynSysTAN).
    1. Pass time through a saddle-node.
    2. Regular parameters and criticality in potential systems.
  5. Interaction between real and complex dynamical systems (DynSysTAN, Holodyn).
    1. Entropy in Hubbard tress and/or complex quasi-periodically forced systems.
  6. Holomorphic dynamics in the Riemann sphere (Holodyn).
    1. Singular perturbations.
    2. Numerical methods viewed as dynamical systems.
  7. Iteration of transcendental entire functions (Holodyn).
    1. Existence and topology of Fatou domains. Topology of the Julia set.
    2. Distribution and properties of invariant objects in the phase plane.
  8. Dynamics of meromorphic transcendental functions (Holodyn).
    1. Fatou components and Newton's method.
    2. Meromorphic functions with asymptotic values and parameter espaces.