Analysis of singularities and fixed points of complex dynamical systems (Acronym: ANAKODI)

Croatian Science Foundation (HRZZ) project, call IP-2025-02 no. IP-2025-02-1147

!!! A POSTDOC POSITION available on the project, at Department of Mathematics, Faculty of Science, University of Zagreb;

call to be opened end March / during April 2026; applications possible for 1 month after the call opens

duration: 12-18 months (finish date November 30^th 2027 the latest).

We are searching for a motivated candidate with a PhD in dynamical systems or ODE (earned before the date of the application for the position), willing to collaborate, to become a member of our group. It is not necessary that the candidate has previous experience exactly with topics of the project. Salary is the standard prescribed salary for a higher research assistant in Croatia. Means for travelling to conferences/workshops/collaborations are ensured by the project. If interested, please send an e-mail to maja.resman@math.hr as soon as possible. There is no teaching requirement.

Project members:

# The principal investigator: Maja Resman, PhD, Associate professor, University of Zagreb, Faculty of Science, Department of mathematics, Zagreb, Croatia, https://web.math.pmf.unizg.hr/~mresman/

# The team members:

· Vlatko Crnković, PhD, Higher research assistant, University of Zagreb, Faculty of Electrical Engineering and Computing, Zagreb, Croatia

https://www.fer.unizg.hr/vlatko.crnkovic

· Luka Kraljević, mag.math., PhD student and research assistant, University of Zagreb, Faculty of Science, Department of mathematics, Zagreb, Croatia

· Pavao Mardešić, PhD, MCF HD, Université Bourgogne Europe, Dijon, France

· Daniel Panazzolo, PhD, Professor, Université Haute Alsace, Mulhouse, France

http://www.lmia.uha.fr/Daniel_Panazzolo/Bienvenue.html

· Dino Peran, PhD, Assistant Professor, University of Split, Croatia, https://www.pmfst.unist.hr/team/dino-peran/

· Goran Radunović, PhD, Associate professor, University of Zagreb, Faculty of Science, Department of mathematics, Zagreb, Croatia

https://web.math.pmf.unizg.hr/~goranr/

· Maja Resman, PhD, Associate professor, University of Zagreb, Faculty of Science, Department of mathematics, Zagreb, Croatia (PI), https://web.math.pmf.unizg.hr/~mresman/

· Loic Teyssier, PhD, MCF, Université de Strasbourg, IRMA, Strasbourg, France

· Vesna Županović, PhD, Full Professor, University of Zagreb, Faculty of Electrical Engineering and Computing, Zagreb, Croatia, https://www.fer.unizg.hr/vesna.zupanovic

# The administering organization: University of Zagreb, Faculty of Science, Zagreb, Croatia,

https://www.pmf.unizg.hr/math/

# Total budget of the project: 159,547.45 EUR

# estimated duration of the project: 36 months (31/01/2026-30/01/2029)

The project summary:

The project is in the area of dynamical systems and deals, broadly, with local analysis of analytic dynamical systems (continuous/represented by vector fields or discrete/represented by iterates of a diffeomorphism) locally around singular resp. fixed points. The singularities are the simplest examples of invariant sets for the dynamics. Away from invariants sets the systems behave similarly if we change the initial point and can be rectified to simple parallel flows by an analytic change of variables. Singularities in general display interesting dynamical behavior of trajectories in their vicinity. The discrete system, which is sometimes easier to analyse, can be 'embedded' in the continuous one as its time map; normally one uses time-1 map or some return map/holonomy in the case of foliations with monodromies. Therefore, in general, it suffices to consider the discrete systems as 'representations' of the continuous ones which carry sufficient information on dynamics. In this project we deal with complex systems, complex functions and complex foliations in one or more complex variables. We are interested in classification problems for such systems, i.e. finding the invariants that tell us when the two systems can be transformed one to another by a change of variables. More precisely, the interesting problem in one-dimensional holomorphic discrete dynamics are currently irrational rotations in the linear part and classifications of parabolic germs with linear part equal to identity in dimension 2. Also, we return to the question of reading analytic invariants, but for generic one parameter unfoldings of vector fields in 1 dimension (whereas the parabolic point of multiplicity 2 unfolds in two simple points). Here, the main question that we pose is the following: can we classify the germ/the bifurcation considering just one local realization (orbit) of the unfolding? To that end, we analyse the orbits fractally, i.e. by integral transforms of their epsilon-neighbohoods.

The main objectives in the project (not exclusive):

O1 Analysis of singularities of integral transforms of orbits for various discrete dynamical systems

O2 Polycycle foliations in C^2

O3 Classifications of complex diffeomorphisms