# Centre for Nonlinear Dynamics Zagreb

## Bifurcation Theory

Focus of research:

A bifurcation of a dynamical system is a qualitative or topological change in its dynamics produced by varying parameters of a given family. Bifurcation theory provides a strategy for investigating the bifurcations that occur within a family.

Fractal analysis of bifurcation uses fractal invariants of an orbit to characterize a bifurcation. The basic fractal invariants are box dimension (also called box counting, Minkowski, Bouligand dimension or limit capacity), and Minkowski content. It was first discovered that the box dimension of a spiral trajectory tending to a singular point or limit cycle signals a moment of Hopf bifurcation, see D. Žubrinić, V. Županović,  Fractal analysis of spiral trajectories of some planar vector fields, Bulletin des Sciences Mathematiques, 129/6 (2005), 457-485. The Hilbert's 16th problem (still completely open) asks for upper uniform bound for number of limit cycles which could bifurcate from a planar polynomial system. This approach from the point of view of fractal geometry is shown to be efficient also for discrete real and complex dynamical systems. In the spirit of 16th Hilbert's problem, discrete systems are applied to a study of continuous systems, using Poincaré map, time-1 map and  holonomy map.

Key publications:

1. D. Žubrinić, V. Županović,  Fractal analysis of spiral trajectories of some planar vector fields, Bulletin des Sciences Mathematiques, 129/6 (2005), 457-485.
2. V. Županović, D. Žubrinić, Fractal dimensions in Dynamics, Encyclopedia of Mathematical Physics ; Oxford : Elsevier, Vol. 2 (2006), 394-402.
3. D. Žubrinić, V. Županović, Fractal analysis of spiral trajectories of some vector fields in R^3, Comptes rendus Mathématiques,342, 12 (2006), 959-963.
4. D. Žubrinić, V. Županović, Poincare map in fractal analysis of spiral trajectories of planar vector fields, Bulletin of the Belgian Mathematical Society - Simon Stevin.15 (2008) , 5; 947-960.
5. L. Korkut, D. Vlah, D. Žubrinić, V. Županović, Generalized Fresnel integrals and fractal properties of related spirals,
Applied Mathematics and Computation. 206 (2008) , 1; 236-244.
6. L. Korkut, D. Žubrinić, V. Županović, Box dimension and Minkowski content of the clothoid,  Fractals. 17 (2009) , 4; 485-492
7. J.P. Milišić, D. Žubrinić, V. Županović, Fractal analysis of hopf bifurcation for a class of completely integrable nonlinear Schroedinger Cauchy problems,  Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE). 60 (2010) ; 1-32.
8. P. Mardešić, M. Resman, V. Županović, Multiplicity of fixed points and  epsilon-neighborhoods of orbits, J. Differ. Eqn. 253 (2012), 2493-2514, arXiv:1108.4707v3.
9. L. Korkut, M. Resman, Oscillations of chirp-like functions,
Georgian Mathematical Journal. 19 (2012) , 4; 705-720.
10. G. Radunović, D. Žubrinić, V. Županović, Fractal analysis of Hopf bifurcation at infinity, International journal of bifurcation and chaos in applied sciences and engineering. 22 (2012) , 12; 1230043-1-1230043-15.
11. L. Horvat Dmitrović, Box dimension and bifurcations of one-dimensional discrete dynamical systems, Discrete Contin. Dyn. Syst 32 (2012), no. 4, 1287-1307, arXiv:1210.8202.
12. M. Resman, Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms, Discrete and continuous dynamical systems. 33 (2013) , 8; 3767-3790, arXiv:1207.2954.
13. M. Resman,  Invariance of the generalized Minkowski content with respect to the ambient space, Chaos, solitons and fractals. 57 (2013) ; 123-128, arXiv:1207.3279v1.
14. L. Horvat Dmitrović, Box dimension of Neimark-Sacker bifurcation, Journal of difference equations and applications. 20 (2014) , 7; 1033-1054,
15. M. Resman, Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations, Nonlinearity, accepted for publication (2014), arXiv:1307.0780v3.

Preprints:

1. L. Korkut, D. Vlah, V. Županović,  Fractal properties of Bessel functions, submitted, arXiv:1304.1762.
2. L. Korkut, D. Vlah, V. Županović, Geometrical and fractal properties of a class of systems with spiral trajectories in R3, submitted,  arXiv:1211.0918.
3. L. Korkut, D.  Vlah, D. Žubrinić, V. Županović, submitted, Wavy spirals and their fractal connection with chirps, submitted, arXiv:1210.6611.